text:
\section{Introduction}
\label{introduction}

The Third Generation Partnership Project (3GPP) has recently introduced Narrow-Band Internet-of-Things (NB-IoT) and enhanced Machine-Type Communications (eMTC) features into the cellular standard protocols. These two narrow-band and complementary technologies expand the cellular networks to support low-power, wide-area (LPWA) cellular connectivity for a wide range of IoT use cases \cite{RRatasuk}.

In general, IoT devices need to operate under extreme power constraints. Consequently, they often communicate at very low signal-to-noise ratio (SNR), e.g., $-13$ dB or $0.03$ bits per transmission (translated to capacity) in NB-IoT protocols \cite{RRatasuk}. Also, they are often not equipped with advanced transceivers due to cost constraints. Therefore, the solution adopted in the standard is to use the legacy turbo codes or convolutional codes at moderate rates, e.g., the turbo code of rate $1/3$, together with many repetitions, e.g., up to $2048$ repetitions in NB-IoT. This implies effective code rates as low as $1.6 \times 10^{-4}$ are supported in such protocols. This repetition scheme has efficient implementations with computational complexity and latency effectively reduced to that of the outer code. However, it is expected that repeating a high-rate code to enable low-rate communication will result in rate loss and mediocre performance. As a result, studying  ultra-low-rate error-correcting codes for reliable communications in such low-capacity regimes becomes necessary \cite{MFereydounian}.

In \cite{MFereydounian}, the authors constructed an efficient repetition scheme with outer polar codes and showed that the proposed polar-repetition scheme outperforms the Turbo-repetition code, the proposed code design in the eMTC and NB-IoT (uplink) standards, over additive white Gaussian noise (AWGN) channel. In another related work, low-rate codes for binary symmetric channels were constructed by concatenating high-rate polar codes with repetitions \cite{DumerBSC}. Also, non-binary LDPC codes concatenated with multiplicative repetition codes were introduced in \cite{Kasai}. By multiplicatively repeating the $(2,3)$-regular non-binary LDPC mother code of rate $1/3$, they constructed rate-compatible codes of lower rates $1/6, 1/9, 1/12, \ldots$ which outperform the best low-rate binary LDPC codes at the cost of the increase in decoding complexity. Very recently, weakly-coded binary LDPC type code combined with polar code has been introduced in \cite{Dumerldpc}, \cite{Dumer} which is shown to outperform uncoded modulation over high noise memoryless channels. However, the complexity of the proposed scheme in \cite{Dumerldpc} and \cite{Dumer} is higher than that of a repetition-based scheme.

In \cite{Fariba}, we proposed {\em polar coded repetition}  to improve the asymptotic achievable rate of the repetition scheme, while ensuring that the overall encoding and decoding complexity is kept almost the same. A slightly modified polar codeword is transmitted in each repetition block by deviating from Ar{\i}kan's standard $2 \times 2$ kernel in a certain number of polarization recursions at each repetition block. It is shown that the polar coded repetition outperforms the  polar-repetition, in terms of the asymptotic achievable rate, for any given number of repetitions over the binary erasure channel (BEC).

In this paper, we propose an alternative mechanism for polar-repetition schemes, referred to as hybrid non-binary multiplicative repetition. In this scheme, the outer code is a hybrid binary and non-binary polar code constructed in two stages. The first stage of the outer encoder utilizes Ar{\i}kan's binary polarization kernel applied recursively, as in original polar codes \cite{Arikan}. The output bits of the first stage are grouped into $t$-tuples and are turned into symbols over the extension binary field $GF(2^t)$. Then Ar{\i}kan's kernel is again applied recursively. Hence, the output of the outer encoder consists of symbols over $GF(2^t)$. The inner code is a non-binary multiplicative repetition code.

The motivation behind constructing the outer code in two stages is to improve the decoding performance by capturing the correlation of the bits within each symbol in the decoding process.  The encoded symbols can be either turned into binary strings for transmission using a binary modulation scheme, e.g.,  binary  phase  shift  keying  (BPSK) or, alternatively, can be sent using a higher order modulation. The proposed structure allows for i) benefiting from the multiplicative repetition over an extension field as opposed to a simple repetition in a  binary polar-repetition scheme, and ii) keeping the complexity of the encoder/decoder almost the same as that of the polar-repetition scheme.
The simulation results show that with increasing list size in the CRC-aided SCL decoder, the proposed hybrid non-binary repeated polar code provides significant performance gain compared to a  polar-repetition scheme with comparable decoding complexity over AWGN and Rayleigh fading channels. Alternatively, the proposed scheme results in lower decoding complexity compared to the polar-repetition scheme with the same performance.

The rest of this paper is organized as follows. In Section II, we review some basics of polar codes as well as repetition codes. In Section III, we discuss the proposed scheme. The numerical results are discussed in Section IV. Finally, Section V concludes the paper.

\section{Background}

In this section,
a brief background on polar codes, their {\em successive cancellation} (SC) and {\em cyclic redundancy check-aided successive cancellation list} (CRC-aided SCL) decoders, as well as some background on repetition schemes, a simple and efficient scheme for low-capacity applications, are discussed.

\subsection{Polar Codes}

An $(n, k, \mathcal{F})$, with $n=2^m$, polar code based on the $2 \times 2$ polarization kernel $G_2=\footnotesize \begin{pmatrix}
1 & 0\\
1 & 1
\end{pmatrix}$ is a linear block code generated by $k$ rows of $G_n=G_2^{\otimes m}$, where $.^{\otimes m}$ is the $m$-th Kronecker power of a matrix \cite{Arikan}.
The set of frozen bits $\mathcal{F}$, with $|\mathcal{F}|=n-k$, is the set of the indices of sub-channels with the lowest reliabilities. Ar{\i}kan's polar codes are constructed by setting entries of the input vector $u_0^{n-1}$\footnote{$u_0^{n-1}$ is a row vector $(u_0, u_1, \ldots u_{n-1})$ and $u_0^i$ is its subvector $(u_0, u_1, \ldots u_i)$.} indexed by $\mathcal{F}$ to
some pre-defined values, e.g. zero, and the remaining $k$ bits are used to transmit the information. At the decoder side, the SC decoder, makes the decision on $u_i$, based on the previously decoded bits, $\hat{u}_0^{i-1}$, and channel output vector, $y_0^{n-1}$, according to the following log likelihood ratio (LLR) rule:

\begin{equation}
\hat{u}_i=
\begin{cases}
1, \  \  \   \ \  \  \  \  \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textup{if} \ \ \ i \in \mathcal{F}^C \ \& \\  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ln \frac{W_{n}^{(i)}(\hat{u}_0^{i-1},y_0^{n-1}|u_i=0)}{W_{n}^{(i)}(\hat{u}_0^{i-1}, y_0^{n-1}|u_i=1)} < 0\\
\text{the frozen value of $u_i$},  \ \  \   \  \textup{if} \ \ \ i \in \mathcal{F},  \label{uhat}
\end{cases}
\end{equation}
where $W_{n}^{(i)}$ is the $i$-th bit-channel \cite{Arikan}.

In order to improve the error correction performance of the SC decoder, the successive cancellation list (SCL) decoding algorithm was proposed by Tal and Vardy in \cite{Tal}. In SCL decoding, the $L$ most likely paths $u_0^{i-1}$ are tracked. When decoding $u_i$, for $i \in \mathcal{F}^C$, the decoder extends each path into two paths exploring both possibilities $u_i=0$ and $u_i=1$. If the number of obtained paths exceeds $L$, the decoder picks $L$ most likely paths as the surviving ones and prunes the rest based on a certain Path Metric (PM). Let $\hat{u}_i[l]$ denote the estimate of $u_i$ in the $l$-th path, for $l \in \{1, 2, \ldots, L\}$ and $S_{i}[l]$ denote its corresponding LLR. Then the corresponding PM is calculated as follows:
\begin{equation}
PM_l^{(i)}=
\begin{cases}
PM_l^{(i-1)}+|S_{i}[l]| \   \ \textup{if}   \ \hat{u}_i[l] \neq \frac{1}{2} (1-\textup{sgn} (S_{i}[l]))\\
PM_l^{(i-1)} \ \ \ \ \ \ \ \  \ \ \  \   \  \textup{otherwise},  \label{pathmetric}
\end{cases}
\end{equation}
where $PM_l^{(-1)}$ is set to $0$. Finally, the path with the smallest PM is selected as the estimated bits $\hat{u}_0^{n-1}$. As the list size $L$ grows large, the SCL decoder approaches the maximum-likelihood (ML) decoding performance. To further improve the performance of the SCL decoder, $p$-bits cyclic redundancy check (CRC) are appended to the information bits as an outer code.
The SCL decoder outputs the decoding path with the smallest PM among the paths which pass the CRC \cite{CRC}. Remarkably, Polar-CRC schemes have been adopted for control channels in the fifth generation of wireless networks (5G) due to demonstrating the state-of-the-art performance for a wide range of code parameters \cite{5Gspec}. Polar codes and polarization phenomenon have been also successfully applied to a wide range of problems including data compression~\cite{Arikan2,abbe2011polarization}, broadcast channels~\cite{mondelli2015achieving,goela2015polar}, multiple access channels~\cite{STY,MELK}, joint detection and decoding~\cite{Jalali}, physical layer security~\cite{MV}--\cite{wei2016polar}, fading channels ~\cite{Si}--\cite{Trifonovfading}, and  coded modulation~\cite {Seidl,MahdavifarBIM}.

\subsection{Repetition Schemes}\label{Repetition}
In low-capacity applications, a straightforward way of designing practical low-rate codes, that is also adopted in the standard, is through repetition schemes. Let $r$ denote the number of the repetitions and $N$ denote the total length of the code. For constructing a repetition scheme, first, a smaller outer code (e.g., a polar code) of length $n=N/r$ is designed and then each of its coded bits is repeated $r$ times.
The main advantage of repetition schemes is the relatively low computational complexity of their encoding and decoding algorithms, especially when $r$ is large. This is because the encoding/decoding complexity is effectively reduced to that of the outer code. Once the outer code is constructed, at the encoder, one just needs to repeat each of the coded bits $r$ times. At the decoder, the LLR of a coded bit repeated $r$ times is equal to the sum of the $r$ LLRs of the individual transmission. While this looks promising, one should note that the main drawback of repetition schemes is the loss in capacity. In general, we have $C(W^r) \leq rC(W)$, where $W^r$ is the channel resulting from repeating $r$ times transmission across $W$, and $C(\cdot)$ is the channel capacity. Note also that the ratio $C(W^r) / rC(W)$ vanishes as $r$ grows large. Consequently, for very large $r$, the repetition code might suffer from an unacceptable rate loss \cite{MFereydounian}.

\section{The Proposed Scheme}

As mentioned in Section II-B, the polar-repetition code is a practical scheme for low-capacity scenarios due to its low encoding/decoding complexity. However, from a channel coding prospective, repeating a high rate polar code to enable low-rate communication is sub-optimal \cite{MFereydounian}. To improve the performance of the polar-repetition scheme in low-capacity regime, we propose the {\em hybrid non-binary repeated polar code} scheme: the concatenation of a binary polar code with a non-binary random multiplicative repetition code.
In this scheme, the codewords are not generated by simple repetitions of the polar code but by random multiplicative repetitions of the non-binary symbols. Our simulation analysis of the proposed scheme under SC and CRC-aided SCL decoders shows that concatenating with random non-binary multiplicative repetition code improves the polarization rate of the code and reduces the number of the low-weight codewords (see Sec. IV).

A potential solution to benefit from the random  non-binary multiplicative code would be to concatenate it with {\em non-binary} polar codes. A similar solution is proposed in the context of LDPC coding, where gains are observed with multiplicative random non-binary LDPC codes compared to binary LDPC codes in the low-rate regime [7]. However, in the context of polar coding, the decoding complexity of non-binary polar codes is often very high for practical scenarios such as IoT, where the devices are required to be low complexity and low cost. Therefore, in this paper, we consider the {\em binary} polar codes as the outer code. In general, a straightforward way of concatenating a binary polar code with a non-binary one would result in a loss of the correlation information between the bits in a symbol and render the advantage of the non-binary outer code. However, our proposed scheme overcomes this problem by splitting the outer code into two stages to capture such correlation in the decoding process.

Owing to the recursive structure of the polar codes, one can consider the polarization transform kernel $G_n$ as the concatenation of $G_{n/t}$ and $G_t$, with $G_n=G_{n/t} \otimes G_t$, where $t=2^{m^{\prime}}, m^{\prime}=\{1, 2, 3, \ldots, \}$, \cite{Mahdavifar}. Figure \ref{fig:Concatenation} (a) shows the block diagram of the encoder of the polar code with this structure, where $x_0^{n-1}$ is the output of the transformation $G_t$. Figure \ref {fig:Concatenation} (c) shows the structure of the polar-repetition. In this scheme, the outer code is the polar code depicted in Figure \ref {fig:Concatenation} (a) and the inner code is the repetition code which repeats the output of the outer polar code, $z_0^{n-1}$, $r$ times and generates code $C_r$ as follows.
\begin{equation}
    \begin{aligned}
C_r &=\{c_0^{rn-1} | c_{(r-1)n+v}=z_v, \\
 \textup{for} & \ \ v=\{0, \ldots, n-1\}, \ \ c_0^{(r-1)n-1} \in C_{r-1}\}.
     \end{aligned}
\end{equation}
To improve the performance of the polar-repetition, the scheme depicted in Figure \ref{fig:Concatenation} (b) and (d) is proposed. The encoder and decoder of the proposed hybrid non-binary repeated polar codes are as follows.

\begin{figure*}[h]
\includegraphics[width=1\linewidth]{Fig1.eps}

\caption{(a) Layered factor graph representation of a polar code (b) Layered factor graph representation of the proposed hybrid non-binary polar code (c) Polar-repetition scheme (d) Hybrid non-binary repeated polar codes scheme.}
\label{fig:Concatenation}
\end{figure*}

\subsection{Encoding of the Proposed Hybrid Non-Binary Repeated Polar Codes}

For encoding the proposed scheme, in the first stage, the binary input bits $u_0^{n-1}$ are divided into subsets of bits of size $t$. Then, each of these $n/t$ $t$-tuples are encoded with binary polarization kernel $G_t$ over $GF(2)$ and the output is $x_{0}^{n-1}$. In the second stage, each of these outputs are grouped together as a $2^t$-bit symbol $a_i$, $a_i=(x_{it}, x_{it+1}, \ldots x_{(i+1)t-1}) \in GF(2^t)$, $i=\{0, 1, \ldots, n/t-1\}$. Then, the symbols $a_0^{n/t-1}$ are encoded with the binary polarization kernel $G_{n/t}$ over $GF(2^t)$ and generate code $C_1=\{z_0^{n/t-1} \in GF{(2^t)}\}$. Finally, coefficients $\rho_{n/t}^{rn/t-1}$ are chosen at random from $GF(2^t) \setminus \{0\}$ and are multiplied by $z_0^{n/t-1}$ to generate the code $C_r$, $r>1$, as follows.
\begin{equation}
    \begin{aligned}
C_r &=\{c_0^{rn/t-1} | c_{(r-1)n/t+v}=\rho_{(r-1)n/t+v}z_v, \\
 \textup{for} & \ \ v=\{0, \ldots, n/t-1\}, \ \ c_0^{(r-1)n/t-1} \in C_{r-1}\}.\label{Cr}
     \end{aligned}
\end{equation}

Note that the coefficients $\rho_{(j-1)n/t}^{jn/t-1}$, $j=\{2, 3, \ldots, r\}$ are the random multiplication coefficients for the $j$-th repetition. Algorithm 1 shows the process of encoding the proposed scheme. The inputs to this algorithm are binary input bits $u_0^{n-1}$, $t$, $n$ and $r$. The outputs is the code $C_r$. The $\gets$ symbol denotes appending an element to a list.

\begin{algorithm}
\small
\footnotesize
\DontPrintSemicolon
\SetAlgoLined
\SetNoFillComment
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}

\Input{$u_0^{n-1}, t, n, r$}
\Output{Code $C_r$ }
 Divide $u_0^{n-1}$ into sets of $t$ bits, $u_{it}^{(i+1)t-1}$, $i=\{0, 1, \ldots, n/t-1\}$. \\

\For {$ i = 0$ \KwTo $n/t-1$}{
     $x_{it}^{(i+1)t-1} =$ Encode each sets of bits, $u_{it}^{(i+1)t-1}$, with binary kernel $G_t$ over $GF(2)$. \\
     Group $x_{it}^{(i+1)t-1}$ together to make a $t$-bit symbol $a_i$.
}
  $C_r= \{z_0^{n/t-1}\} \gets$ Encode $a_0^{n/t-1}$ symbols with binary kernel $G_{n/t}$ over $GF(2^t)$.\\

  \For {$ j = 2$ \KwTo $r$}{
   $C_r \gets$ Choose $n/t$ coefficients $\rho_{(j-1)n/t}^{jn/t-1}$ uniformly at random from $GF(2^t) \setminus \{0\}$, multiply them with $z_0^{n/t-1}$.

  }
 \KwRet $C_r$
\caption{Encoding Algorithm of the Proposed Hybrid Non-binary Repeated Polar Codes}\label{encoder1}
\end{algorithm}

The outer encoding only requires standard binary polar encoding followed by binary to $2^t$-ary conversion. In the decoding process, and in order to capture the correlation of the bits in a symbol, the conversion from binary to $2^t$-ary is done  between Stage 1 and Stage 2 of decoding.

Note that the authors in \cite{Presman} constructed mixed kernels over alphabets of different sizes and improved the polarization properties of the kernel $G_n$. However, in this paper, by using the structure of the kernel $G_n$, we group the binary bits into symbols without modifying the polarization kernel $G_n$.

\textbf{Example 1:} Figure \ref{fig:example} (a) shows an example for $n=8$, $r=3$, and $t=4$. The polarization kernels of stage 1 and 2 are $G_4=B_4 G_2^{\otimes 2}$  and $G_2$, respectively, where $B_4$ is the bit-reversal permutation matrix.
By choosing the primitive $\alpha \in GF(2^4)$ with $\pi (\alpha)=\alpha^4+\alpha+1=0$, the output codewords of this example are $c_0^{5}=(1,  \alpha^{13}, \alpha^8, \alpha^{4}, \alpha^4, \alpha^{14})$.

\begin{figure*}[h]
\centering
\includegraphics[width=0.8\linewidth]{example16.eps}
\caption{Equivalent  encoders for the proposed scheme for $n=8$, $r=3$, $t=4$ based on (a)  Algorithm 1 and (b) Algorithm 2.}

\label{fig:example}
\end{figure*}

The encoded symbols $c_i$ are turned into binary strings for transmission over AWGN and Rayleigh fading channels using the BPSK modulation scheme, with the received signals for these two channels given by
\begin{equation*}
    y_i=(c_i)_M+w_i, \ \ \ i=\{1, 2, \ldots, N\}
\end{equation*}
and
\begin{equation*}
    y_i=h_i (c_i)_M+w_i, \ \ \ i=\{1, 2, \ldots, N\},
\end{equation*}
respectively, where $(c_i)_M$ is the modulated codeword, $h_i$ is the channel gain with Rayleigh distribution and $w_i$ is a zero mean
Gaussian noise, $w_i \sim \mathcal{N} (0, \sigma^2)$.
Here, we assume that $h_i$ and $w_i$ are independent, the receiver knows the channel state information (CSI)  and the transmitter knows only the distribution of the CSI.

In the block fading channel model, a transmission frame
of $N$ symbols is affected by $1 \leq B \leq N$ dependent fading realizations, resulting in a block of $\tau =N/B$ symbols being affected by the same fading realization. Different values $B$ represents different types of fading, e.g., $B=1$ is referred to fast fading and  $B=N$ is referred to slow fading.

\subsection{SCL Decoding of the Proposed Hybrid Non-Binary Repeated Polar Codes}
For decoding the proposed scheme, we use CRC-aided LLR-based SCL decoder. Algorithm \ref{ListDecoder} shows the details of the process. The SCL decoding of the proposed scheme involves mainly 4 parts, i.e. the initial LLRs, the LLRs of the multiplicative repetition, the LLRs of Stage $1$ and the LLRs of Stage $2$ of the outer polar code. The details of the LLRs calculations are as follow.

    \textbf{Initial LLRs:}
    The initial LLRs of the $i$-th symbol, $i \in \{0, 1, \ldots, rn/t-1\}$, with symbol value $s \in GF(2^t)$ for AWGN and block fading channels are defined as
    \begin{equation}
        {S}_{in,i}^{(s)}=\ln{\frac{W(y_i|(c_i)_M=0_M)}{W(y_i|(c_i)_M=s_M)}},\label{initialLLRsA}
    \end{equation}
    and
     \begin{equation}
        {S}_{in,i}^{(s)}=\ln{\frac{W(y_i|(c_i)_M=0_M, h_i)}{W(y_i|(c_i)_M=s_M, h_i)}},\label{initialLLRsF}
    \end{equation}
    respectively.

    Let us denote ${S}_{in,i}=\{S_{in,i}^{(s)}\}_{s \in GF(2^t)}$, the vector of the $i$-th initial LLRs for all possible symbols $s$ over $GF(2^t)$.

   \textbf{LLRs of the multiplicative repetition:}
    Since $GF(2^t)$ is a finite field, the non-zero elements can be expressed as powers of a primitive element $\alpha$ in the field, i.e., $1,\alpha,\dots, \alpha^{2^t-2}$. Therefore, multiplication by an arbitrary non-zero symbols $\rho_j = \alpha^{\tau}$, $j=\{n/t, \ldots, rn/t-1\}$, can be regarded as a cyclic shift of the field elements by $\tau$. As a result, the decoder permutes the LLR vector ${S}_{in,i}$ and outputs the vector $\pi_{\rho_j}({S}_{in,i})$.

    Finally, the LLR of an $r$-tuple consisting of $r$ independent transmissions of symbols is equal
    to sum of the LLRs of the individual channel outcomes after the permutations (see lines 2-8 of the Algorithm \ref{ListDecoder}).

    \textbf{LLRs update of Stage $2$:}
    In general, the LLR of the $i$-th symbol, $i \in \{0, 1, \ldots, n-1\}$ , with symbol value $s$ over kernel $G_n$, can be calculated according to the following formula, \cite{Arikan}:
    \begin{equation}
\begin{aligned}
 S_{i}^{(s)} & \overset{\Delta}{=} \ln \frac{\sum_{u_{i+1}^{n-1}  }R_{G_n}(\hat{u}_0^{i-1},0,u_{i+1}^{n-1})}{\sum_{u_{t+1}^{n-1}} R_{G_n}(\hat{u}_0^{i-1},s,u_{i+1}^{n-1})},\label{LLR}
 \end{aligned}
\end{equation}
where $R_{G_n}(u)=\exp(-\sum_{j=0}^{n-1}S_j^{(x_j)})$ and $x_j$ is the $j$-th index of the vector $x_0^{n-1}=u_0^{n-1}G_n$ and $s$ is an element from $GF(2^t)$. One can use the following equation for simplifying eq. (\ref{LLR}).
\begin{equation*}
    \ln (\sum_i e^{-f_i}) \approx - \min_i (f_i).
\end{equation*}

 Now, for our binary kernel $G_2$, consider two input LLR vectors ${S}_{+}$ and ${S}_{-}$ of size $2^t$. Then,  the  output LLR vectors $\hat{S}_{+} $ and $\hat{S}_{-} $ can be derived from (\ref{LLR}), as:
 \begin{equation}
     \begin{aligned}
      \hat{S}_{+}^{(s)} \approx & \min_{u_1 \in GF(2^t)}{(S_{+}^{(s+u_1)}+S_{-}^{(u_1)})}\\
      & - \min_{u_1 \in GF(2^t)}{(S_{+}^{(u_1)}+S_{-}^{(u_1)})},\\

      \hat{S}_{-}^{(s)} \approx & ~S_{+}^{(\hat{u}_0+s)} +S_{-}^{(s)}-S_{+}^{(\hat{u}_0)}-S_{-}^{(0)}.\label{fs2}
     \end{aligned}
 \end{equation}
 where $S_{+}^{(i)}$ and $S_{-}^{(j)}$ are the $i$-th and $j$-th index of the vectors ${S}_{+}$ and ${S}_{-}$, respectively and $\hat{u}_0$ is the previously decoded symbol, while $u_1$ is the yet to be decoded symbol.

\textbf{LLRs update of Stage $1$:}
For updating the LLRs of Stage $1$, consider the input non-binary LLR vector $\hat{S}$. The output binary LLRs $\hat{S}_{b_i}$, $i=\{1, 2, \ldots, t\}$, derived from eq. (\ref{LLR}), are as follows.

\begin{equation}
\begin{aligned}
 \hat{S}_{b_i} &  \approx   \min_{u_{i+1}^{t-1}} \hat{S}^{([v^{(1)}]_{2^t})} -\min_{u_{i+1}^{t-1}} \hat{S}^{([v^{(0)}]_{2^t})},\label{LLRstage1}
 \end{aligned}
\end{equation}
where $[v^{(\beta)}]_{2^t}$ is the representation of the binary vector $v^{(\beta)}=(\hat{u}_0^{i-1},\beta,u_{i+1}^{t-1})G_t$ as element of $GF(2^t)$ for $\beta=\{0, 1\}$ and $\hat{S}^{([v_\beta]_{2^t})}$ is the $[v^{(\beta)}]_{2^t}$-th index of the vector $\hat{S}$.

In Eq. (\ref{LLRstage1}) we show how to calculate the output binary LLRs $\hat{S}_{b_i}$ given the input non-binary LLR $\hat{S}$ in one step. However, updating the LLRs of Stage 1 can be done recursively as well. To this end, Stage 1 of the encoder in Figure \ref{fig:Concatenation} (b) needs to be modified. Instead of encoding $t$ bits with binary polarization kernel $G_t$ over $GF(2)$, $G_2$ is applied recursively on the inputs symbols over $GF(2^{2^{k^{\prime}-1}})$ in $k^{\prime}$ levels ($1 \leq k^{\prime} \leq m^{\prime}$). Then the obtained symbols are grouped into $2^{k^{\prime}}$-bit symbols $a_0^{(n/2^{k^{\prime}})-1}$. Finally, the output of the last level, $t$-bit symbol $a_i$, $i=\{0, 1, \ldots (n/t)-1\}$, is fed into Stage $2$. Part of Stage 1 for this modified encoder is demonstrated in Figure \ref{fig:Recurively}. The process of the encoding for this structure is presented in Algorithm \ref{encoder2}.

\begin{algorithm}
\small
\footnotesize
\DontPrintSemicolon
\SetAlgoLined
\SetNoFillComment
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}

\Input{$u_0^{n-1}, m^{\prime}, t, n, r$}
\Output{Code $C_r$ }

Define List $\text{A}=\{\}$\\
$\text{A} \gets u_0^{n-1}$\tcp*{Init.}
\For {$ k^{\prime} = 1$ \KwTo $m^{\prime}$}{
\For {$ i = 0$ \KwTo $(n/2^{k^{\prime}})-1$}{
     $x_{2i}^{2(i+1)-1} =$ Encode each sets $(A[2i], A[2(i+1)-1])$ with binary kernel $G_2$ over $GF(2^{2^{k^{\prime}-1}})$. \\
     Group $x_{2i}^{2(i+1)-1}$ together to make a $2^{k^{\prime}}$-bit symbol $a_i$.
}
$A = \{\}$ \tcp*{empty List $A$}
$A \gets a_0^{(n/2^{k^{\prime}})-1}$
}
lines 6-9 of Algorithm \ref{encoder1}.\\
 \KwRet $C_r$
\caption{Encoding of the Proposed Hybrid Non-binary Repeated Polar Codes  for Stage $1$}\label{encoder2}
\end{algorithm}

\textbf{Example 2:} Figure \ref{fig:example} (b) demonstrates a encoder structure for the code discussed in Example 1 based on Algorithm 2. The polarization kernels of Stage 1 (Layer 1 and Layer 2) and Stage 2 is $G_2$.
The output codewords in this example are $c_0^{5}=(1,  \alpha^{7}, \alpha^8, \alpha^{13}, \alpha^4, \alpha^{8})$ over $GF(16)$.

\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{Fig4.eps}
\caption{Part of Stage 1 of the encoder based on Algorithm 2.}
\label{fig:Recurively}
\end{figure}

In order to update the LLRs of the layers of Stage 1, based on Algorithm $2$, the following recursive formulas can be used for calculating $\hat{S}_{+}$ and $\hat{S}_{-}$ from $\hat{S}$.
\begin{equation}
\begin{aligned}
 \hat{S}_{+}^{(s)} &  \approx \min_{u_1 \in GF(2^{t^{\prime}})} \hat{S}^{([v^{(s)}]_{2^{t^{\prime}}})}-\min_{u_1 \in GF(2^{t^{\prime}})} \hat{S}^{([v^{(0)}]_{2^{t^{\prime}}})}\\
 \hat{S}_{-}^{(s)} &  \approx  \hat{S}^{([\hat{v}^{(s)}]_{2^{t^{\prime}}})}-\hat{S}^{([\hat{v}^{(0)}]_{2^{t^{\prime}}})}, \label{LLRstage1v2}
 \end{aligned}
\end{equation}
where $0 \leq  j \leq m^{\prime}-1$, $t^{\prime}=2^{j}$, $[v^{(\beta)}]_{2^{t^{\prime}}}$ is the representation of the binary vectors $v^{(\beta)}=(\beta,u_1)G_2$ as elements of $GF(2^{t^{\prime}})$, and $[\hat{v}^{(\beta)}]_{2^{t^{\prime}}}$ is the representation of the binary vectors $\hat{v}^{(\beta)}=(\hat{u}_0,\beta)G_2$ as elements of $GF(2^{t^{\prime}})$ for $\beta=\{0, s\}$.

The CRC-aided SCL decoder after updating the LLRs and calculating the PM for different paths based on the values of the $\hat{S}_{b_i}$  chooses the path with the smallest PM which passes the CRC.

\begin{algorithm}
\small
\footnotesize
\DontPrintSemicolon
\SetAlgoLined
\SetNoFillComment
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}

\Input{List size $L$, $r$, $n$, $t$, $\rho_{n/t}^{rn/t-1}$ and $y_0^{rn/t-1}$}
\Output{The estimated bits $\hat{u}_0^{n-1}$ }

${S}_{in,i} =$ Calculate the initial LLRs with eq. (\ref{initialLLRsA}) or eq. (\ref{initialLLRsF}) for all $i=\{0, 1, \ldots, rn/t-1\}$ by using $y_0^{rn/t-1}$.\\

 ${S}_{\textup{inner},i} = {S}_{in,i}, i=\{0, \ldots, n/t-1\}$\tcp*{Init.: Pick the first $n/t$ elements of ${S}_{in,i}$.}

 \For {$ i = n/t$ \KwTo $rn/t-1$}{
 $\pi_{\rho_i}({S}_{in,i})$ \tcp*{Permute the vector ${S}_{in,i}$ based on the random coefficient $\rho_i$.}
 }
 \For{$ j = 2$ \KwTo $r$}{
 ${S}_{\textup{inner},i} = {S}_{\textup{inner},i}+\pi_{\rho_i}({S}_{in,i})[(j-1)n/t:jn/t-1]$.\tcp*{Output LLRs of the inner code}
 }

\For {$ k^{\prime} = 0$ \KwTo $n-1$ \KwBy $t$}{

     Using the LLRs ${S}_{\textup{inner},i}$, for $i=\{0, \ldots, n/t-1\}$, update the LLRs for Stage $2$ with eq. (\ref{fs2}).\\
     \For {$ j = 1$ \KwTo $t$}{

    Using the updated LLRs from Stage $2$, update the LLRs for Stage $1$ with eq. (\ref{LLRstage1}) and obtain $\hat{S}_{b_j}$.\\
     Calculate the PM for $\hat{S}_{b_j}$ with eq. (\ref{pathmetric}) for each of the $L$ paths.\\
     Update bits for Stage $1$.\\
    }
    Update the symbols for Stage $2$.\\
}
 CRC-aided SCL decoder with list size $L$ chooses the path with the smallest PM which passes the CRC and outputs $\hat{u}_0^{n-1}$.\\
\KwRet $\hat{u}_0^{n-1}$

\caption{List Decoding Algorithm for the Proposed Hybrid Non-binary Repeated Polar Codes}
\label{ListDecoder}
\end{algorithm}

\section{Analysis and Numerical Results}
In this section, we first provide complexity analysis of the proposed scheme and compare it with those of the polar-repetition scheme and the scheme in \cite{Dumerldpc}. Then, we analyze the numerical result of the proposed scheme given certain underlying binary field extensions in the construction, i.e., over GF($4$) and GF($16$), and compare it with the polar-repetition scheme and the scheme in \cite{Dumerldpc} over AWGN and Rayleigh fading channels.
Finally, we analyze the performance of the hybrid non-binary repeated polar code.

\subsection{Complexity Analysis} \label{Complexity}

In this subsection, we count the exact number of operations needed for calculating the LLRs of the proposed scheme, polar-repetition, and the scheme in \cite{Dumerldpc} to show the performance-complexity trade-off in subsection \ref{NA}. Note that to demonstrate the computational complexity, we used the total number of the summation and comparison operations for calculating the LLRs of the outer and inner codes for the SC decoder. The complexity advantages of the entire system using CRC-aided SCL decoder will be shown in Sec. IV-B.

      \textbf{Polar-repetition code:} To update the LLRs of the  polar-repetition, one needs to compute
      \begin{itemize}
          \item $(n-1)r$ operations for decoding the inner repetition code of size $r$.
          \item $  (\frac{n}{2} \times 4+ \frac{n}{2} \times 1) \log n$ operations for decoding the outer polar code of size $n$.
      \end{itemize}
      Consequently, the total complexity is $nr+2.5 n \log n$ operations.

      \textbf{Hybrid non-binary repeated polar code:}  Since the two methods proposed in Algorithms \ref{encoder1} and \ref{encoder2} have the same computational complexities, we only count the number of the operations of the first method as follows.
    \begin{enumerate}
        \item  \textbf{Inner multiplicative repetition code:}  According to (\ref{initialLLRsA}) or (\ref{initialLLRsF}), the size of the initial LLR vector of the proposed scheme, $S_{in,i}$, is $2^t$ and $S_{in,i}^{(0)}=0$. Consequently $(2^t-1) rn/t$ operations are needed for computing  lines 6-8 of  Algorithm \ref{ListDecoder} for decoding the inner multiplicative repetition code.

        \item  \textbf{Stage2:} For updating the LLRs of Stage 2, one need to compute $\hat{S}_{+}^{(s)}$ and $\hat{S}_{-}^{(s)}$ given in (\ref{fs2}).

       To obtain $\hat{S}_{+}^{(s)}$, one should compute
        \begin{itemize}
        \item  $S_+^{(s+u_1)}+S_-^{(u_1)}$, $\forall s, u_1 \in GF(2^t)$, with $(2^t-1) \times (2^t-1)$ operations. \footnote{Note that $S_+^{(u_1)}+S_-^{(u_1)}$ is a special case of $S_+^{(s+u_1)}+S_-^{(u_1)}$ for $s=0$ and $\hat{S}_+^{(0)}=0$, $\hat{S}_-^{(0)}=0$.}
        \item  $\min_{u_1 \in GF(2^t)}(S_+^{(s+u_1)}+S_-^{(u_1)})$, $\forall s \in GF(2^t)$, with $2^t \times (2^t-1)$.
        \item  $\hat{S}_+^{(s)}$ with $2^t-1$ operations for $s \in GF(2^t)\setminus\{0\}$.

        \end{itemize}
        Therefore, the total number of the operations for computing $\hat{S}_{+}^{(s)}$ is $(2^{t}-1)\times 2^{(t+1)}$.

       To obtain $\hat{S}_{-}^{(s)}$, one should compute
        \begin{itemize}
        \item $S_+^{(\hat{u}_0+s)}+S_-^{(s)}$, $\forall s \in GF(2^t)$, with $2^t-2$ operations.
        \item  $\hat{S}_{-}^{(s)}$ with $2^t-1$ operations for $s \in GF(2^t)\setminus \{0\}$.
         \end{itemize}
        Therefore, in total $2^{t+1}-3$ operations are needed for computing $\hat{S}_{-}^{(s)}$.

    Finally, for the Ar\i kan's kernel of size $n/t$,
    $(\log \frac{n}{t}) [\frac{n}{2t} ((2^{t}-1)\times 2^{(t+1)})+\frac{n}{2t} (2^{t+1}-3)]$ operations are needed for updating the LLRs of  Stage $2$.

        \item \textbf{Stage 1:} According to (\ref{LLRstage1}), for each $b_i, i \in \{1, 2, \ldots, t\}$, one needs to compute
        \begin{itemize}
            \item $\min_{u_{i+1}^{t-1}} \hat{S}^{([v^{(\beta)}]_{2^t})}$, $\beta=\{0,1\}$ with $2^{t-i}-1$ operations.
            \item $\hat{S}_{b_i}$ with $1$ operation.
        \end{itemize}
        As a result, in total, $\frac{n}{t} \sum_{i=1}^t (2(2^{t-i}-1)+1)$ operations are needed to update the LLRs of the ${n}/{t}$ kernel $G_t$ of size $t$.
    \end{enumerate}
\textbf{Scheme proposed in \cite{Dumerldpc}:} This scheme is the concatenation of the $(n_l,k_l)=(8192,128)$ LDPC with $(n_p,k_p)=(128,80)$ polar code. In this scheme, one needs $8192(8\log(128)+32)=720896$ operations for decoding the LDPC code and $110469$ operations for decoding the polar code under SCL with list size $L=32$. As a result the total number of operations for this scheme is $831365$.

In a nutshell, we conclude that the complexity of the total decoding process of the polar-repetition and the proposed schemes are $O(nr+ n \log n)$ and $O(2^t rn/t+2^{2t} (n/t) \log (n/t) + 2^t n/t)$, respectively. Here, if $t$ and $r$ be  constants, the complexity of both schemes will be $O(n \log n)$. Moreover, the complexity of the scheme proposed in \cite{Dumerldpc} is $O(n_l (\log n_p +L)+Ln_p \log n_p)$. However, for numerical comparison purposes, we use the aforementioned exact number of operations for the different schemes. The complexity comparison of the three discussed schemes is shown in Table \ref{complexitycomparison} for $N=8192$ and three different values $r=16, 32, 64$.

\begin{table*}
\begin{center}
\scalebox{0.9}{
\begin{tabular}{|P{4cm}||P{3.5cm}|P{4.5cm}|P{4.5cm}|P{1cm}|}
 \hline
  \multirow{2}{*}{Parameters} & \multicolumn{4}{|P{11cm}|}{{Repetition Scheme}} \\
  \cline{2-5}
  & \multicolumn{1}{|P{2.5cm}|}{{Inner rep. $n(r-1)$ }} & \multicolumn{2}{|P{7cm}|}{{Outer polar code \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ $2.5 \times n \log n$}} & \multicolumn{1}{|P{1cm}|}{{Total }}\\
\hline
 $n=512$, $r=16$ & $7680$ & \multicolumn{2}{|P{7cm}|}{$11520$} & $19200$  \\
\hline
$n=256$, $r=32$  & $7936$ & \multicolumn{2}{|P{7cm}|}{$5120$} &   $13056$     \\
\hline
$n=128$, $r=64$ & $8064$ & \multicolumn{2}{|P{7cm}|}{$2240$} &  $10304$      \\
\hline
\hline
  \multirow{2}{*}{Parameters} & \multicolumn{4}{|P{12cm}|}{{Proposed Scheme over GF(4)}} \\
  \cline{2-5}
  & \multicolumn{1}{|P{3cm}|}{{Inner rep.
  $\frac{n(r-1)}{t}(2^t-1)$}} & \multicolumn{1}{|P{3.5cm}|}{{Stage 2
  $(2^{2t}-3/2)\frac{n}{t}\log \frac{n}{t}$}} & \multicolumn{1}{|P{3cm}|}{{Stage 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $2n$}} & \multicolumn{1}{|P{1cm}|}{{Total }}\\
\hline
 $n=512$, $r=16$, $t=2$ & $11520$ & \multicolumn{1}{|P{3.5cm}|}
 $29696$} & \multicolumn{1}{|P{3cm}|}{$1024$} & $42240
 \\
\hline
$n=256$, $r=32$, $t=2$  & $11904$ & \multicolumn{1}{|P{3.5cm}|}
$12992$} & \multicolumn{1}{|P{3cm}|}{$512$} &
$25408$\\
\hline
$n=128$, $r=64$, $t=2$ & $12096$ & \multicolumn{1}{|P{3.5cm}|}
$5568$} & \multicolumn{1}{|P{3cm}|}{$256$}&
$17920$\\
\hline
\hline
  \multirow{2}{*}{Parameters} & \multicolumn{4}{|P{12cm}|}{{Proposed Scheme over GF(16)}} \\
  \cline{2-5}
  & \multicolumn{1}{|P{3cm}|}{{Inner rep.
  $\frac{n(r-1)}{t}(2^t-1)$}} & \multicolumn{1}{|P{3.5cm}|}{{Stage 2
  $(2^{2t}-3/2)\frac{n}{t}\log \frac{n}{t}$}} & \multicolumn{1}{|P{4cm}|}{{Stage 1 $\frac{n}{t} \sum_{i=1}^{t} (2(2^{t-i}-1)+1)$}} & \multicolumn{1}{|P{1cm}|}{{Total }}\\
\hline
 $n=512$, $r=16$, $t=4$ & $28800$ & \multicolumn{1}{|P{3.5cm}|}
 $228032$}
 & \multicolumn{1}{|P{3.5cm}|}{$3328$} &
 $260160$\\
\hline
$n=256$, $r=32$, $t=4$  & $29760$ & \multicolumn{1}{|P{3.5cm}|}
$97728$} & \multicolumn{1}{|P{3.5cm}|}{$1664$} &
$129152 $    \\
\hline
$n=128$, $r=64$, $t=4$ & $30240$ & \multicolumn{1}{|P{3.5cm}|}
$40720$} & \multicolumn{1}{|P{3.5cm}|}{$832$}&
$71792 $\\
\hline

\end{tabular}%
}
\end{center}
\caption{ Complexity comparison of different schemes}
\label{complexitycomparison}
\end{table*}

\subsection{Numerical Analysis}\label{NA}

In this subsection, we provide numerical results for the proposed scheme given $GF(4)$ and $GF(16)$ as the underlying binary field extension in the construction\footnote{For the simulation, Algorithm 1 is used. Although both Algorithm 1 and 2 show the same performance.}. The communication is assumed over AWGN and Rayleigh fading channels with BPSK modulation, where encoded symbols are turned into binary strings and then modulated. Also, the decoding is performed under the SC and CRC-aided SCL decoding algorithms. Then, we compare the performance of the proposed scheme with those of the polar-repetition scheme as well as the scheme proposed in \cite{Dumerldpc}. Note that the SCL decoder is implemented using the randomized order statistic algorithm for the selection of the $L$ most likely paths in each stage, which has the complexity $O(L)$, where $L$ is the list size. Moreover, the construction for the proposed hybrid non-binary repeated polar code and polar-repetition schemes is based on the Monte-Carlo simulation at an optimized design SNR, denoted by $\gamma$, to be specified separately for each case.

It is worth noting that in each instance of the simulation, a different set of the random coefficients $\rho_{(j-1)n/t}^{jn/t-1}$, $j=\{2, 3, \ldots, r\}$  is used. At the end, the average is taken over all the instances. An alternative approach is to choose the set of the coefficients in a certain way and then use them in all the instances of the simulation. The choice of such coefficients could be optimized, however, solving such a combinatorial optimization problem is complex and beyond the scope of this paper.

Figures \ref{fig:512k80}, \ref{fig:256k80}, and \ref{fig:128k80} compare the performance of the proposed scheme for $N=8192$, $k=80$, and
over $GF(4)$ and $GF(16)$ with that of the straightforward polar-repetition scheme over AWGN channel. To illustrate the trade-off between the rate of the outer polar code and the number of the repetitions, we consider three different values $r=16, 32, 64$. The optimized design SNR $\gamma$ for each simulated case is specified in the legend of the corresponding plot.

It can be seen that the proposed scheme over $GF(16)$ outperforms the one over $GF(4)$ and the repetition scheme, under SC and CRC-aided SCL decoder with the same list size and $6$-bit CRC. For $r=16$ under CRC-aided SCL, the performance of the proposed scheme over $GF(16)$ with $L=8$ is almost the same as that of the proposed scheme over $GF(4)$  with $L=32$ and the polar-repetition scheme with $L=128$. For $r=32$ under CRC-aided SCL, the proposed scheme over $GF(16)$  with $L=8$ outperforms the proposed scheme over $GF(4)$  with $L=32$ and that one outperforms the repetition scheme  with $L=128$. Moreover, the performance of the proposed scheme over $GF(16)$ with $L=4$ is almost the same as that of the repetition scheme  with $L=128$. As we increase the number of the repetitions to $64$, due to the loss in the asymptotic achievable rate, the performance of polar-repetition scheme degrades. As a result, the performance of the proposed scheme over $GF(4)$ with $L=8$ outperforms that of the repetition scheme  with $L=128$.

For comparison, the performance of the scheme proposed in \cite{Dumerldpc} for low-SNR regime is also considered.
It can be seen that for $r=16$, the proposed schemes over $GF(4)$ under CRC-aided SCL with $L=32$ and $GF(16)$ with $L=8$ outperform the scheme in \cite{Dumerldpc}. Moreover, the performance of the scheme in \cite{Dumerldpc} is the same as that of the proposed scheme over $GF(4)$  with $L=8$ and repetition scheme with $L=32$. For $r=32$, the proposed scheme over $GF(16)$  with $L=8$ outperforms the scheme in \cite{Dumerldpc}. On the other hand, the performance of the proposed scheme over $GF(4)$  with $L=32$ is almost the same as that of \cite{Dumerldpc}. However, for $r=64$, the scheme in \cite{Dumerldpc} outperforms the proposed scheme over $GF(16)$ with $L=32$.

Figures \ref{fig:512k40}, \ref{fig:256k40}, and \ref{fig:128k40} compare the performance of the proposed scheme over $GF(4)$ and $GF(16)$ with that of the straightforward polar-repetition scheme over AWGN channel for $N=8192$, $k=40$. Similarly, the proposed scheme over $GF(16)$ outperforms the one over $GF(4)$ and repetition scheme under both SC and CRC-aided SCL decoder.
The scheme in \cite{Dumerldpc} can not be realized for these parameters.

\begin{figure*}[tp!]
 \centering
\begin{subfigure}[t]{0.45\textwidth}
 \centering
\includegraphics[width=1.1\linewidth]{Fig512r16k80_v4.eps}
\caption{$N=8192$, $n=512$, $k=80$, $r=16$, $t=2, 4$.}
\label{fig:512k80}
\end{subfigure}%
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=1.1\linewidth]{512k40r16_v4}
\caption{$N=8192$, $n=512$, $k=40$, $r=16$, $t=2, 4$.}
\label{fig:512k40}
\end{subfigure}

\medskip
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=1.1\linewidth]{n256r32k80_v4.eps}
\caption{$N=8192$, $n=256$, $k=80$, $r=32$, $t=2, 4$.}
\label{fig:256k80}
\end{subfigure}%
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=1.1\linewidth]{Fig256k40r32_v4.eps}
\caption{$N=8192$, $n=256$, $k=40$, $r=32$, $t=2, 4$.}
\label{fig:256k40}
\end{subfigure}

\medskip
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=1.1\linewidth]{Fig128r64k80_v4.eps}
\caption{$N=8192$, $n=128$, $k=80$, $r=64$, $t=2, 4$.}
\label{fig:128k80}
 \end{subfigure}%

\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=1.1\linewidth]{Fig128k40r64_v4.eps}
\caption{$N=8192$, $n=128$, $k=40$, $r=64$, $t=2, 4$.}
\label{fig:128k40}
\end{subfigure}
\caption{Performance comparison of the proposed scheme with the polar-repetition scheme and the scheme in \cite{Dumerldpc} over AWGN channel for different values of $n$, $r$ and $k$. }
\end{figure*}

\begin{figure*}[tp!]
     \centering
    \begin{subfigure}[t]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{ComplexityPerformance512k80.eps}
         \caption{Performance}
         \label{Performance512}
     \end{subfigure}%
    \begin{subfigure}[t]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{Complexity512k80ebno15.eps}
         \caption{Complexity}
                  \label{Complexity512}
        \end{subfigure}
        \caption{CRC-aided SCL decoding over AWGN channel at $Eb/No = 1.5 $ dB for $N=8192$, $n=512$, $r=16$, $k=80$.}

        \medskip
         \begin{subfigure}[t]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{PerformanceComplexity256k80.eps}
         \caption{Performance}
                  \label{Performance256}
\end{subfigure}%
    \begin{subfigure}[t]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{Complexity256k80.eps}
         \caption{Complexity}
                \label{Complexity256}
        \end{subfigure}
         \caption{CRC-aided SCL decoding over AWGN channel at $Eb/No = 1.5$ dB for $N=8192$, $n=256$, $r=32$, $k=80$.}

        \medskip
         \begin{subfigure}[t]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{PerformanceComplexity_128k80.eps}
         \caption{Performance}
         \label{Performance128}
\end{subfigure}%
    \begin{subfigure}[t]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{Complexity128k80.eps}
         \caption{Complexity}
                  \label{Complexity128}
        \end{subfigure}
         \caption{CRC-aided SCL decoding over AWGN channel at $Eb/No = 1.5$ dB for $N=8192$, $n=128$, $r=64$, $k=80$.}

\end{figure*}
Figures \ref{Performance512}, \ref{Performance256} and \ref{Performance128} present the simulation results of the hybrid non-binary repeated polar code and the polar repetition scheme under CRC-aided SCL decoder as a function of  $L$ over AWGN channel at $Eb/N0 = 1.5$ dB for $N=8192$, $k=80$ and three different number of repetitions, $r=16, 32, 64$, respectively. It can be seen that the hybrid non-binary repeated polar code over $GF(16)$ requires significantly lower list size $L$ to achieve the same performance as the polar-repetition scheme over $GF(4)$. Moreover, this gap grows with $L$. For comparison, the performance of the scheme proposed in \cite{Dumerldpc} with $L=32$ is also provided.

Figure \ref{Complexity512} presents the performance of the CRC-aided SCL decoding of the schemes in Figure \ref{Performance512} in terms of the decoding complexity. Observe that the proposed scheme over $GF(4)$ can provide better performance than polar-repetition scheme with the same decoding complexity for FER $\leq 6 \times 10^{-3}$ ($L> 4$ for the proposed scheme over $GF(4)$ and $L>8$ for polar-repetition scheme). This is due to the higher slope of the corresponding curve in Figure \ref{Performance512}, which eventually enables one to compensate for the relatively high complexity of the LLR computation presented in Table \ref{complexitycomparison}. The proposed scheme over $GF(16)$ has greater processing complexity than the one over $GF(4)$, so that its curve intersects the polar-repetition scheme at FER $=2 \times 10^{-3}$ ($L\geq 4$ for the proposed scheme over $GF(16)$ and $L\geq 64$ for polar-repetition scheme). Moreover, the proposed schemes over $GF(4)$ and $GF(16)$ have the same FER $= 8 \times 10^{-4}$ for $L\geq 64$  and $L > 8$, respectively. Similar behaviour is observed for $r=32$ repetitions in Figure \ref{Complexity256}. However, as we increase $r$ from $16$ to $32$, the decoding complexity decreases, while the performance of all the schemes degrades. As a result, the intersections of the curves happen at lower FER and smaller number of the operations. For $r=64$ repetitions in Figure \ref{Complexity128}, a significant degradation in the performance occurs due to the reduced achievable rate of the outer polar code due to having a shorter length. For any $L$, the hybrid non-binary repeated polar code over $GF(16)$ provides the best performance for the same decoding complexity.

It can be observed that for $r=16$ repetitions (see Figure \ref{Complexity512}), the hybrid non-binary repeated polar code and polar-repetition scheme outperform the scheme in \cite{Dumerldpc} with the same decoding complexity. For $r=32$  repetitions (see Figure \ref{Complexity256}), the scheme in \cite{Dumerldpc} slightly outperforms the polar-repetition, but not the proposed scheme for the same decoding complexity. Only for $64$ repetitions (see Figure \ref{Complexity128}), the scheme in \cite{Dumerldpc} outperforms the others with the same decoding complexity.

Figure \ref{fig:fading} compares the performance of the proposed scheme for $N=8192$, $k=80$, $r=16$ over $GF(4)$ and $GF(16)$ with that of the straightforward polar-repetition scheme over Rayleigh fading channel. For each repetition block, we consider one distinct fading coefficient. As a result, in each frame of size $8192$, there are $16$ distinct fading coefficients and each block of length $512$ symbols in each repetition will be affected by the same fading coefficient.
\begin{figure}[h]
\centering
\includegraphics[width=1.1\linewidth]{Plot_fading2.eps}
\caption{Performance comparison of the proposed scheme with the polar-repetition over Rayleigh fading channel for $N=8192$, $n=512$, $k=80$, $r=16$, $t=2, 4$.}

\label{fig:fading}
\end{figure}
Similar to AWGN channel, it can be seen that the proposed scheme over $GF(16)$ outperforms the one over $GF(4)$ and the repetition scheme, under SC and CRC-aided SCL decoder with the same list size and $6$-bit CRC.

\subsection{Performance Analysis}

In this subsection, we analyze the performance of the proposed scheme to gain insight into its better performance under the CRC-aided SCL decoder.
Since the SCL decoder with large list size achieves performance very close to that of the  ML decoder, we study the performance of the proposed scheme under ML decoding to have better analytical understanding.

The block error probability under ML decoding can be estimated via the truncated union bound as follows, \cite{Sason}.
\begin{equation}
P_e^{\textup{ML}} \leq \sum_{i=d}^{n} A_i Q(\sqrt{2 i R E_b/N_o)},
\end{equation}
where $A_i$ is the number of the codewords of weight $i$ and $d$ is the minimum distance of the code.
At high SNR, upper bound on $P_e^{\textup{ML}}$ depends primarily on $d$ and $A_d$. Hence, to obtain a good performance under ML decoding, one needs to eliminate low-weight non-zero (LWNZ) codewords from the code.

To enumerate the LWNZ codewords, we transmit the all-zero codeword in the extremely high SNR regime under SCL decoder with very large list size, \cite{Li}. In this case, it is expected that the list most likely contains only the codewords with the least Hamming weights.

Table \ref{lowweightcodewords} compares the number of low-weight codewords of the proposed scheme with the polar-repetition ones over AWGN channel for $N=8192$, $n=512$, $r=16$, $k=40$.
As it is expected the number of LWNZ codewords for the proposed scheme over $GF(4)$ is less than that of the polar-repetition scheme. For $GF(16)$, although the minimum distance is slightly reduced, the number of low-weight codewords is still much smaller compared to the polar-repetition scheme.

\begin{table*}
\begin{center}
\scalebox{0.7}{
\begin{tabular}{|P{3cm}|P{3cm}|P{2.2cm}|P{1.5cm}|P{2.2cm}|P{1.5cm}|P{2.2cm}|P{1.5cm}|P{2.2cm}|P{1.5cm}|}
 \hline
 \multirow{1}{*}{Parameters} & Schemes &$A_{1000}- A_{1024}$ &$A_{64\times 16}$ & $A_{1329}-A_{2047}$& $A_{128 \times 16}$ & $A_{2049}-A_{3071}$ & $A_{192 \times 16}$ & $A_{3073}-A_{4095}$&$A_{256 \times 16}$\\
\hline
   \multirow{7}{*}{$n=512$, $r=16$ } & Prop. Sch., $t=2$ & $0$ & $0$ & $<10$ & $0$ & $<30$ & $3$ & $<230$ & $230$\\
   \cline{2-10}
   \multirow{7}{*}{$k=40$} & Prop. Sch., $t=4$ & $2$ & $0$ &$32766$& $0$ &$0$& $0$ & $0$ & $0$\\
\cline{2-10}
    & Rep. Sch. &$0$ & $105$ &$0$& $1365$ &$0$& $5005$ & $0$ & $22819$\\
\cline{2-10}
&  & $A_{4097}-A_{5119}$& $A_{320\times 16}$ & $A_{5121}-A_{6143}$& $A_{384 \times 16}$ & $A_{6145}-A_{7167}$ & $A_{448 \times 16}$ & $>A_{7169}$ &\\
\cline{2-10}
 & Prop. Sch., $t=2$ &  $<230$ & $0$& $<20$ & $0$ & $0$ & $0$ & $0$& \\
 \cline{2-10}
 & Prop. Sch., $t=4$ &  $0$ & $0$& $0$ & $0$ & $0$ & $0$ & $0$ &\\
\cline{2-10}
 & Rep. Sch. & $0$ & $3003$ &$0$& $455$ &$0$& $15$ & $0$ &\\
\hline
\end{tabular}%
}
\end{center}
\caption{Number of Low-Weight Codewords}
\label{lowweightcodewords}
\end{table*}

\section{Conclusion}
In this paper, we proposed a new concatenation scheme to improve upon the performance of the polar-repetition scheme in the low-SNR regime. The proposed scheme is the concatenation of a hybrid non-binary polar code with a multiplicative repetition code. Extensive simulation and numerical results show that the proposed scheme provides a better trade-off between the decoding complexity and performance compared with the polar-repetition scheme, under CRC-aided SCL decoder over AWGN and Rayleigh fading channels.

There are several directions for the future research. Finding the best permutation for the inner non-binary multiplicative repetition code is an interesting problem. Reducing the decoding complexity of our proposed constructions by pruning the decoding trellis and simplifying it \cite{Alamdaryazdi}, and reducing the latency to become sub-linear \cite{MondelliLatency} are other interesting directions. Also, analyzing the polarization properties of the proposed scheme with the optimal permutation in terms of error exponent, \cite{AT}, and scaling exponent, \cite{HassaniSE}, \cite{MITI2020}, and comparing them with polarization based on kernels constructed in \cite{Lin}--\cite{FaribaTVT2020}, are other interesting open problems.

\section*{Acknowledgment}
The authors would like to thank Navid Gharavi and Prof. Ilya Dumer for providing the simulation results of their proposed scheme in \cite{Dumerldpc}.

redpajama_set_name: RedPajamaArXiv


Today, Jobling Purser develop and produce a wide range of British Standard products which are used all around the world.
As specialists in producing highway maintenance solutions we apply our knowledge of bitumen formulating to a wide range of industry applications. Our high quality products make roads safer and play a crucial part in a diverse range of manufacturing sectors.
We are a family owned and run business with a trading history which reaches back to the 1830's.
Jobling Purser have been involved in a wide variety of different sectors in their long and successful history. The first trading records show activity in North African phosphate and manganese which were sold as fertiliser and colouring agents for glass, respectively.
In the 1860's, when the second generation of the family took over, they introduced oil and grease production into the business portfolio. From here the business expanded and in 1920's, Shell issued a licence to Jobling Purser to manufacture bitumen emulsions.
As the years passed, the phosphates side of the business declined and so the company moved forward in becoming a major manufacturer of bitumen products for Road and Highways industries as well as a bitumen formulator for various manufacturing sectors.
In 1971, our Chairman moved to Newcastle to take on the management of Jobling Purser and set up agreements with Shell to distribute bitumen to the North East of England and Scotland. He also expanded the business by growing a complementary highway maintenance contracting operation.
In 1987, the business was sold to Shell with the manufacturing side being bought back by the Jobling Purser family in 1992. In 1993 and 1996 we welcomed the arrival of two members of the next generation of, Jobling Purser's. With their enthusiasm and the weight of expectation from the family on their shoulders, they have helped to introduce new products and technologies which have seen the business grow year on year.
In 2000, Jobling Purser ventured into asphalt and coated stone production purchasing an existing plant in Blaydon on Tyne. More recently in 2012 we have after many years of loyal service, retired our Blaydon equipment and invested in a brand new state of the art Benninghoven Asphalt plant which is now based on the same site as our manufacturing base on the banks of the river Tyne in Newcastle upon Tyne.
The key to our success over the years, and what we continue to strive for today, is providing quality products with a service and technical excellence which is unrivalled within the industry.
©2000 – 2018 James A Jobling & Co Ltd t/a Jobling Purser | All Rights Reserved.


text:
Peabodies 'Politically Motivated'
The Cambodian government reacts to Who Killed Chea Vichea?'s Peabody Award. From The Cambodia Daily:
Phay Siphan, spokesman at the [Cambodian government] Council of Ministers, who saw the documentary last year, said the Peabody Awards were likely a "politically motivated" institution" ... "It's a campaign against the government....It's more like propaganda," said Mr. Siphan, adding that anyone can submit a complaint to the Ministry of Interior about the killing of Chea Vichea if they want to.
Chea Vichea Documentary Receives Prestigious Award (April 6, 2012)
In the Phnom Penh Post, the director of the Peabody Board, Horace Newcomb, and the director of Who Killed Chea Vichea?, Bradley Cox, respond to Siphan's accusation:
"This is simply not the case," said Newcomb. "A review of all the award recipients over the years shows a concern for social issues, but also for entertainment, education, public service and so on. Such a review would show no political motivation other than our interest in excellent media production."
"The Cambodian government has been trying to block our movie from the beginning, so it's no surprise that they would disparage it again now, but their comments, as usual, are completely without merit," Cox said.
Peabody Puts Banned Film Back in Spotlight (April 12, 2012)
Speaking to Voice of America, Newcomb added "The Peabody Award is not politically motivated ... It looks for excellence in media production, excellence of many different sorts, excellence on its own terms, and that's what we found in this documentary." (VoA, April 18, 2012)
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Olson's video Miscellaneous Images Galore uses a screen saver function to show the many images he encounters daily in our image-saturated culture. Lycan's body of work Junk Drawerconsiders the everyday items that get thrown in a drawer rather than discarded. Photographic enlargements of these items, from paperclips to unused balloons, are presented on the gallery floor as a sculptural installation.
Kelly Lycan received her BFA from the Nova Scotia College of Art and Design, and her MFA from the University of California, Santa Barbara and Los Angeles. She has exhibited in solo and group shows in Canada and the United States, most recently at Gallery TPW in Toronto. She is also a member of Instant Coffee, a service oriented artist collective tat builds social structures, where ideas, material and actions are explored. Instant Coffee has exhibited in Canada, South America, Europe and the United States.
Daniel Olson (b. 1955) completed degrees in mathematics and architecture before obtaining a BFA in 1986 from the Nova Scotia College of Art and Design (Halifax) and an MFA in 1995 from York University (Toronto). Olson's work- which includes sculpture, multiples, installation, photography, performance, audio, video and artist's books – has been exhibited widely since 1986 and is documented in several catalogues. He is unhappy with the world, but continues to live there for personal reasons. Olson has been seen recently in Montreal, where he is rumoured to be working on his famous disappearing act.


It has been a very long time since I have written a blog post. So long, in fact, that I am a little embarrassed to start back up again. But since I'm fairly confident that my only current readership is immediate family (half of whom also act as my editors and are obligated to read this stuff), I feel okay about it.
I won't even try to cover the events since my last post. (Moved back to America, started a business, went to a 4-star, acquired new horses, had some successes, cried over spilt milk a number of times…) Oh, look there, didn't take as long as I thought it would. But I gotta tell you guys, there are some things that happened this past year that were pretty hilarious. Perhaps by recounting them here, we can all learn together and head into next year's competition season with the confidence that these mistakes will never, ever, happen again. Seriously.
This past April I took Whitey, aka Share Option, to Rolex. Even writing it down, 8 months after it already happened, sounds crazy to me. I was driving through the gates of the Kentucky Horse Park three days before the first horse inspection and I could not believe it was happening. I had spent months prepping and conditioning my horse in every way while simultaneously preparing myself for the potential disappointment of not making it there (we all know how quickly plans change in this sport). Perhaps it would have been better had I spent those months visualizing being there and imagining the scenarios and situations. Because, let me tell you, when I finally arrived, I was like – Where am I and how did I get here?
About half an hour before I was supposed to get on, someone mentioned to me that we were supposed to sew our competition number onto the saddle pad. What?! We quickly found a needle and thread and frantically sewed what we thought was my number onto my pad. I got on Whitey and made the 10 minute hack down to the warm up arena. I only ever allow about 20 minutes warm up because Whitey is a very relaxed horse and he gets lazy if I ride him for too long. Even still, I was the first person down there so I went straight to the ring reserved for the rider about to head down to the main arena.
After warming up for maybe 10 minutes, the ring steward arrives and tells me I am not allowed in this ring! I try to explain to her that I am the first rider of the day so I am, in fact, allowed in the ring. She looks at me in confusion and tells me that I am not scheduled to ride until this afternoon. Now I'm confused. I glance down at my number, which has been beautifully sewn onto my saddle pad. IT IS THE WRONG NUMBER. In my haste to get it on there, I had grabbed another competitor's number from our shared tack room and sewn it onto my pad. Yep, that's right. I had arrived at the dressage warm up at Rolex, an event in which I was only competing one horse, with the wrong number on my saddle pad. And only managed to figure it out 10 minutes before I needed to be in the ring. I am positive this is the first time something like this has ever happened at this event. So, I calmly walk over to the fence where about 100 people are already standing and watching me warm up and said to my crew "ummm, I have the wrong number on".
Now, if you've ever been to Kentucky, you know that the barns are about 10 minutes away from the ring. My wonderful groom, Savannah, managed to get there and back in record time. Meanwhile, I am back in the warm-up ring and I couldn't resist going over to my very focused friend Hannah Burnett in the ring next to me and tell her what I did. All the tension and nerves we were feeling just dissipated because we could not stop giggling hysterically. I spent the last 5 minutes before I went into the ring cutting the wrong number off and fastening my real number on (with a safety pin after all and not a needle and thread… the horror!) David was trying to tell me I still needed to canter, practicing collection and moving forward; the stewards were yelling at me to enter the ring and all I could do was get the silly number on. I literally picked up the canter from right where I was and headed down the chute into the ring for the biggest moment of my career to date.
Now I got into that ring, with the right number, and I did the test. To be honest, Whitey was great and I'm not sure a different warm up would have changed anything. I got through the dressage, I tackled the cross country (with one mistake that was all mine and not my horse) and I had a great show jumping round with one rail.
Yes, it could have gone better. And yes, there were at least 10 more times that week where I did something really silly that only a first-timer would do. But, it was the most fun I have ever had and if there is anything I want to do again in my life, it is to go back and get another shot. And I am 100% sure that, when I get there, I will definitely, without-a-doubt, absolutely-not-no-way, wear the wrong number to dressage. Chances are still good I'll grab the wrong number bib for cross country, use someone else's saddle for show jumping, or somehow manage to mount the wrong horse… but I am positive I will never sew the wrong number to my saddle pad again. And I think that if I manage to learn from my mistakes and perhaps laugh at myself along the way, I'm still headed in the right direction.


Gratefulness.org, a world group providing on-line and community-based academic packages and practices which encourage and information a dedication to grateful residing, not too long ago featured ChangingAging in its collection "Grateful Changemakers." Learn an excerpt beneath or the full story here.
With a few years of mixed expertise in long-term care reform and advocacy, Dr. Invoice Thomas and Kavan Peterson based ChangingAging to usher in a brand new tradition of human getting old. They've constructed a worldwide community at ChangingAging.org, which presents content material that explores the terrain of getting old, and thru their nonfiction theater tour which travels the nation and blends medical science, storytelling, and stay music to shatter our tradition's damaging myths about getting old. Their latest start-up "Minka Houses and Communities" helps individuals stay on their very own phrases with modern homes that meet the wants of everybody. Co-founder and editor of ChangingAging.org, Kavan Peterson tells us extra about how they're working to create a tradition of getting old constructed round love, kindness, and gratefulness.
There was an fascinating motion for nearly three a long time, constructed round altering the tradition inside our long-term care system. ChangingAging was particularly a motion that Dr. Invoice Thomas and I launched within the mid-2000's with the concept that we might leverage new instruments, like social media expertise, to introduce a broader vary of individuals to this tradition shift.
For essentially the most half, individuals keep away from excited about getting old and keep away from experiencing locations like nursing properties except they're pressured to do it, so lots of people don't learn about this motion except they work in senior providers instantly or they've a beloved one who will get previous and experiences the present long-term care system.
We acknowledged that it's nice to share info and be related digitally, however if you wish to change individuals's minds, it's a must to change their hearts.
Dr. Invoice Thomas and I have been on the lookout for a technique to have a broader impression, to achieve extra individuals, and to alter the best way they considered getting old, each on a private and societal degree. The ChangingAging platform initially was a digital platform that we constructed into a global following. We used that platform and following to mix the digital expertise with a stay in-person expertise — the ChangingAging Tour — in order that we might impression higher social change. We acknowledged that it's nice to share info and be related digitally, however if you wish to change individuals's minds, it's a must to change their hearts. Probably the most highly effective method to do this is thru in-person engagement, leveraging the instruments of theater and the stay expertise. It's much more impactful than a Fb put up.
The Tour gave us this unprecedented alternative to journey round North America. We partnered with long-term care organizations like AARP, so we had a mandate to convene neighborhood conversations in any respect 130 of our tour stops the place we have been might both carry collectively and introduce age-friendly community-building ideas or the place there have been age-friendly communities already underway and we might assist their efforts, join with them, and supply a chance for the neighborhood to have a good time. We need to take heed to individuals and ask them in regards to the points they care about in addition to their hopes and desires for their very own getting old.
In virtually each neighborhood that we visited and each dialog we had, essentially the most pressing subject individuals talked about was housing and safety. It's essential for us to acknowledge that longevity is a big success, and so quite than at all times framing it as this drawback that previous individuals trigger, we have to acknowledge that we will be taught from this expertise, to create higher communities and higher properties. This is a chance now — quite than do nothing and let thousands and thousands of individuals endure by having insufficient housing and insufficient communities — to start out this age-friendly revolution and design our properties and communities in order that they're extra supportive of what older individuals want.
Out of this expertise of touring round and listening to what individuals have been involved about, we launched an organization known as Minka that has designed homes that may be constructed very quick, that may be constructed wherever, and that may be assembled in individuals's backyards or as entire communities the place each home is seen, universally designed, and digitally related to satisfy the wants of individuals of all ages.
One in every of issues we developed as a form of mannequin for our design is that this acronym known as MAGIC, in order that no matter we're doing, it's in assist of Multi-Means, multi-Generational, Inclusive Neighborhood.
A part of creating ChangingAging was the popularity that the issue is just not previous age. The issue is just not what occurs whenever you develop previous, even the destructive issues. The issue is just not the insurance policies and the laws and the destructive issues inside getting old providers and the long-term care system. There are points there that create institutional settings that folks don't thrive in, however these should not the true drawback.
The core drawback that must be tackled is rampant ageism inside our tradition. All of these destructive issues are signs of a tradition that's too consumer-based and too centered on portraying youth as the top of perfection. Prior to now 50 years in our tradition — that is my opinion and an opinion shared by others — our hyper-consumerism created this cultural impact that checked out youth as excellent. That's simply ridiculous, as a result of youth is completely flawed.
Even the fiercest advocates for individuals who stay with dementia and Alzheimer's illness will be accountable for creating this narrative of tragedy and loss that demeans and dehumanizes, inflicting individuals residing with dementia to endure.
ChangingAging and this concept of disrupting getting old has actually been about other ways to get individuals to see past their very own ageism, a socially imposed ageism. It's straightforward to level to the extremely seen examples of age discrimination. We all know it occurs. We all know that ladies are dramatically extra impacted by it. They really feel extra strain to keep up this phantasm of youthfulness with a view to be seen in society. That's an actual strain. However ageism goes method deeper than that. It actually permeates every part we do and every part we take into consideration getting old.
ChangingAging and the ChangingAging tour has had this mission to disrupt ageism. Particularly, a giant a part of the tour that's now shaping the housing and the neighborhood improvement work has been the concept that we might even disrupt dementia. We took the concept of disrupting getting old to the subsequent degree, saying, "Okay, if we're going to actually deal with individuals's worst fears about getting old, let's discuss dropping your reminiscence." For many individuals, that's the final factor that they concern — dropping their identification and their reminiscence.
Due to that deep-rooted concern, we culturally deal with individuals who expertise reminiscence loss very badly. Even the fiercest advocates for individuals who stay with dementia and Alzheimer's illness will be accountable for creating this narrative of tragedy and loss that demeans and dehumanizes, inflicting individuals residing with dementia to endure. ChangingAging has had this mission to be the main voice, saying we will disrupt this tragedy narrative and inform a special story about mindfulness and presence as beneficial attributes that we will truly study from individuals who expertise cognitive change or reminiscence loss.
How do you see ChangingAging as being related to grateful residing?
Once you stay with dementia you do expertise cognitive losses, however you acquire different heightened senses to compensate. It's effectively documented. You acquire a heightened means to speak and acknowledge individuals emotionally, a capability that comes from mainly being solely within the current second, grounded in ways in which individuals with hyper-cognition are incapable of. A whole lot of the difficulties in communication and our interactions with individuals residing with dementia is as a result of we now have such a tough time being current. Folks residing with reminiscence loss are completely current and tuned into our — that's, individuals with hyper-cognition — lack of consideration.
We discovered that by spending time with individuals residing with dementia, inviting them to share their tales, and weaving these tales into our performances, we now have been in a position to disrupt this tragedy narrative, which has had a ripple impact of dramatically altering the best way individuals take into consideration getting old typically. In case you can demystify and scale back their worst fears, you open area for a richer understanding of getting old and fewer concern and denial about what's going to occur after they age.
What do you see because the lasting impression of ChangingAging's work?
We actually imagine in each private and neighborhood transformation. We go on tour and create these stay occasions in order that we will be with individuals and provides them the chance for private transformation. Then we construct the community and leverage expertise in order that now we're truly constructing properties and communities. We do imagine that this subject round a rising getting old inhabitants is a optimistic alternative to construct higher communities. We predict it's a chance that might be misplaced if we don't act. If we're apathetic in regards to the scenario, our communities will worsen. We have now a novel alternative to rework our communities for the higher if we embrace the concept of bringing elders into the middle of our neighborhood.
In our opinion, people have developed in a method for millennia that has put elders on the middle of our communities, they usually have at all times been the glue that has held individuals collectively. We've had this bizarre experiment for just a few a long time through which we eliminated elders, institutionalizing them and segregating our communities, and I believe there's a number of proof that it's not turning out very effectively.
I don't need to be hyperbolic, however for those who take a look at consultants like Robert Putnam, writer of Bowling Alone, who has checked out sociological adjustments in our tradition, a definite decline has been measured within the social cloth that holds us collectively, a decline within the social capital that may assist individuals thrive and be extra related. We imagine that creating communities the place older adults are included and acknowledged not for what they devour or do — which is how we measure the value of youthful individuals — however as a substitute are measured by their knowledge and their storytelling, we'll improve social capital which is definitely extra beneficial than monetary capital relating to thriving as we age.
What evokes you personally about this work?
In my very own life, I'm a part of a neighborhood in Seattle known as Momentia. Momentia is a neighborhood of people that stay with dementia and reminiscence loss and who've come collectively, reclaiming the phrase and all the expertise. I've been impressed by this concept that we will create our personal story about our expertise, no matter it's. Momentia particularly impressed this Disrupt Dementia motion and the neighborhood design work that we're doing to create dementia-friendly and Minka communities.
Momentia is a neighborhood of people that stay with dementia and reminiscence loss and who've come collectively, reclaiming the phrase and all the expertise.
I'm impressed additionally by Dr. Invoice Thomas' household. The prototype Minka was constructed for Haley Jane Thomas. She's one in all Invoice's daughters who was born and has lived with a profound neurological dysfunction known as Ohtahara Syndrome that arrested her improvement in infancy. She is nonverbal, blind, and restricted bodily. I've recognized her since she was 11.
Haley is in her early 20s, and she or he now has a Minka home of her personal designed to satisfy her wants with all the finest design and expertise accessible. Her total life, she's been an inspiration to Invoice, his spouse Jude, and everybody in our world that has executed this work. All of them have impressed us to create higher communities and providers. Haley is a lead inspiration.
How does ChangingAging plan to develop or broaden?
We have now a number of Minka communities within the works now. The following section for us is certainly going to be having a presence in these communities. I don't know precisely what that's going to seem like but, however that's going to be the subsequent section of those new age- and dementia-friendly co-living communities.
The MAGIC idea was developed in partnership with a neighborhood in Evansville, Indiana, which included the College of Southern Indiana and a broad vary of neighborhood members. We introduced the ChangingAging tour there, and we had a number of neighborhood occasions. The primary MAGIC Minka village can be a co-student/elder housing neighborhood on the College of Southern Indiana. The primary of its type on the earth. On Oct. 24, 2018, we unveil the primary MAGIC home, after which we're constructing the neighborhood.
How does gratefulness encourage you to make change on the earth?
I believe in our discipline of labor, the gratefulness that we expertise working with elders is the first inspiration that drives our work. There's gratefulness for the truth that we stay in an period the place so many individuals can efficiently stay this lengthy in life. Elders have a lot to share.
My mother simply retired in Might after 35 years of working a nursing residence. Lots of people come into the sector of getting old and long-term care due to experiences, particularly unhealthy experiences, that they had with members of the family or relations within the long-term care system. I got here into it having grown up within the discipline — elders are the those that I frolicked with rising up and those that I beloved. I do know that I, in addition to Invoice and others who do that work, are pushed by our gratitude for having been in a position to spend time with elders in our neighborhood. That's completely the factor we're most grateful for.
Next articleWhat to Do in Your Final Trimester?


As a sport, tennis is on the rise in Canada. With young stars Milos Raonic, Eugenie Bouchard and Vasek Pospisil, it's no wonder Canadians nation-wide are inspired to pick up a racquet.
In a 2014 national research study, Tennis Canada announced that the sport is growing by leaps and bounds. In the 12 months preceding the survey, over 6.5 million Canadians played tennis at least once. 1.7 million of those are considered 'frequent players' who hit the courts at least twice a month. In children, more than 600,000 kids aged 6-11 played tennis. With this increase in popularity, players need to make sure that they are approaching the sport with the right equipment.
This means having not just an appropriate racquet, but also the right court-specific footwear and potentially an orthotic or insole.
Let's start with court shoes. There are different court shoes depending on the surface you play on.
Grass court (Wimbledon) is the fastest type of court surface due to hard-packed soil and slippery grass which is less firm. As a result, you want a shoe with a good amount of traction. Grass court shoes have little nubs or mini cleats to give you the traction you need. The grass also reduces bounce which means players generally rely more on their serve than with other surfaces.
Hard Court (Australian & US Opens) is the most common type of surface. It allows for a high bounce and a medium to fast-paced game. Due to the firm nature of the surface, you need a shoe with a durable outsole. The outsole patterns are generally in the chevron style.
Clay Court (French Open) is made up of crushed brick, stone, shale or basalt. This slows the ball down and allows for a high bounce. Clay court shoes have a tight tread pattern so even if they pick up a lot of the clay the shoes will still provide traction.
An orthotic for court shoes is equally specific. Not regarding the type of surface you play on, but because the biomechanics of tennis and court sports are different from those of other sports such as soccer, hockey and golf.
Agility and quick feet are an asset to all tennis players. The sport requires short, quick bursts of motion in forwards, sideways and occasionally backwards directions. Players are required to rely on single leg strength, as they often have to stop, balance and shift their body weight and momentum from one leg to another.
The split step, important in groundstrokes, service return and volley preparation, lengthens the quadriceps, provides a stable base prior to motion and results in a quick transfer of load from one leg to the other. This creates a notable shearing force on the foot and shoe. Studies at the University of Western Australia are currently looking at the correlation between effective knee flexion-extension during the service action and concurrent lower loading of the shoulder and elbow.
Custom orthotics for tennis are a light-weight, durable design. While the components will change depending on each individual's needs, there are a few things that remain standard throughout. Forefoot materials include a high-rebound rubber which helps with forwarding propulsion, cuts shearing forces and provides durable cushioning.
The topcover is a micro-perforated, high-density foam that is breathable and lightweight. Overall, a court orthotic is designed to improve ground contact and stability during your game.


The loud drill gave her husband a headache. Now that you have finished your work, you are free to go home. I couldn't save him. He will sit for hours reading books. They're going to shoot him. He was thrown from the car. What classes are you taking? We wanted to talk about them. Jurevis took off his shoes and socks and rolled up his pant legs. Jayesh would make a very good spy.
I was happy for him. I am not sure. I've been really lonely these past two weeks. You think too much! Vince looks strong. Emil told Evan that John was in trouble. Without "ethical culture," there is no salvation for humanity. You don't really want me to kiss you, do you? Can you pass me the potatoes? There is little chance of success.
Rogue wants to see us again. Christofer takes a shower every morning before breakfast. I just kissed Kim. Which meat tastes the best? She began to like him right away. Are you aware of the time, Love? People here love you. Mormon missionaries work in pairs. My father's sister's husband is my uncle. The dove symbolizes peace.
Shirley opened a can of soup and emptied it into a saucepan. You must be getting tired of listening to everyone's personal problems. That is just not possible. I came to talk to you about Teresa. Aaron was in his office all morning. The eagle flies through the sky. I can't wait to eat it. Everyone is against me. He was in time for the last bus. I don't see that there's a choice.
Dan was hit on the forehead with a hammer. I'm not in the mood to help you. Ernest and Claire picked some wildflowers by the river. The restaurant will reopen under new management. I can't understand how your father managed to see through that man. Where's your girlfriend? Aren't we missing something? I want you to get me out of here. Drew heard that. I can't repeat what Murph said.
When will he leave for Beijing? "Mr Chairman, what needs moving next, again?" "Scoreboard, giant ball and the tug-of-war rope." Bonnie owns a yellow sports car. Please don't touch this. Jayesh hasn't yet learned how to write his own name. Everyone knows that two plus two equals four. You're not upset, are you? I still don't know your name. The magazine is aimed at teenagers. They move from place to place.
What's it all mean? I didn't wind the clock last night. Leon may be back before 2:30. When Monica rented a car at the airport rental agency, he agreed to bring it back in seven days. Her husband intends to bring out a new monthly magazine. Can you arrange it? We need to spread more awareness. They have to live on his small income. We all want to go home. The referee, according to principle of advantage, raised his hand and motioned for the advance to continue.
That's hard to imagine. The girl was afraid of her own shadow. Let the children play. I want to see this movie. I caught him by surprise. Gil said that he feels much better today. We looked for the puppy, but it was in vain. Peggy just did it. The rapid spin of the Earth along with the liquid, hot metallic core causes a magnetic field to surround the Earth. "Look what I just bought." "What is it?"
Tarmi left the group and was never seen again. Rebecca was the one who actually wrote the letter. I'm offering Claudio a job. May I approach the bench? Lowell translated the contract into French. That's so 2015. The intern is lazy. Enlisting in the special forces is a strange fate for a boy who has never had a gun in his hands. What are they trying to tell me? This guy is quick in seducing women so you better beware.
What are they going to find? Lance laughed before he realized Milner wasn't joking. I never said that I wanted that. Would you like Pravin to help you? Children are the flowers of our lives. Nothing decorates a house better than children carefully set in the corners. I want your help on this. It's the longest journey of my life! Was it you that left the door open last night? My contact lenses make my eyes dry. Should I use drops? Is the poorest country in the European Union poorer than the poorest state in the United States?


text:
Will Facebook's New Offers Product Appeal to Merchants?
May 7, 2012 May 7, 2012 by Steven Jacobs
Facebook rolled out its rehashed deals play Offers to businesses on Friday after three months in private tests with a handful of partners. The self-service platform allows businesses to create and distribute deals free of charge on a merchant's brand pages as well as in users' news feeds.
The social network's massive scale has already posed some issues for small businesses — a hotel in Australia claimed over the weekend that an offer which it ran on Facebook reached over 3 million people in less than five days. Facebook has seen little success in the hyperlocal space to date, but the company appears to be making a big push in the SMB market in the run-up to its highly anticipated IPO later this month.
Jake Cohen, founding team member at Privy said it's interesting to see Facebook reapproach the offers space after the company's unsuccessful bid last year, but that the play was likely too self-serve to really appeal to merchants: "After talking to some of our customers who saw and engaged with the offers functionality, the response between them all was the same: easy to use, but not a ton of value there," Cohen told Street Fight. "With no transaction online, it is unclear what it will do aside from simply be posted on the site. Business owners don't want different activities/strategies on Facebook, their website, Foursquare and other platforms — they want one controlled panel so they can communicate with their staff clearly and understand how the combined efforts are working. You need a different service than just Facebook Offers for that."
Scoutmob co-founder Michael Tavani, agreed: "We feel that local merchants just aren't ready to create their own deals and get their staff up to speed. We have to account manage and hand-hold these local merchants a lot in order to put out offers that are compelling and run smoothly when consumers walk-in. Self-serve in today's day in age with local merchants will not work. LivingSocial Instant & Foursquare's Offers have shown that to some extent.
Tavani said that as big as Facebook's reach is, the company isn't the best platform for all types of businesses. "I'm not convinced that consumers want to hear (unfiltered) from their HVAC guy or local pizza place or real estate agent on Facebook," said Tavani. "There seems to be a trend away from a cluttered feed and most of these local businesses don't have the compelling content (or the bandwidth) to keep the conversation interesting to consumers. Facebook will not be all things to all people, and I think local is so hard to pull off really well that just a huge reach alone won't solve the problem. They'll get beat by more focused local players.
Meanwhile, Nimble Commerce CEO Prashant Nedungadi was a little more positive about the service's prospects, calling it "a smart solution for small businesses looking to go beyond a simple 'like' relationship with customers." He suggested that the move fits the next phase of the offers/couponing industry in connecting supply with demand.
Steven Jacobs is an associate editor at Street Fight
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Tags: Facebook features Foursquare Jake Cohen LivingSocial Michael Tavani Nimble Commerce Prashant Nedungadi Privy scoutmob
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IBC is blessed to be multi‑racial with many various backgrounds and talents. Our diversity adds depth and opportunity. Throughout the year we have church get-togethers, some for the entire family, such as our annual Labor Day Picnic. Some are for all the adults, such as our annual Adult Christmas Party. Of course, teens and children have their activities as well. A Marriage Retreat with a guest speaker is offered early each year at a resort only an hour or two away. The camaraderie and fellowship are important parts of a Biblical church family.
The IBC Choir and orchestra provide special music at the Sunday Morning Service. Choir Director, Mr. Brian Wessel, along with our IBC musicians, seek to glorify God in the selection and presentation of traditional gospel music.
Ladies Fellowships, bridal showers, baby showers, Sunday School class, meals for new mothers, special outings and activities help the IBC women to encourage one another.
The men of Independent Baptist Church also have opportunities for fellowship. Men meet on Saturday mornings at 7:00 a.m. to pray together. 3 or 4 times a year the men have a Prayer Breakfast. In the fall they enjoy a muskie fishing retreat in the north woods of Michigan.
Adult Singles often meet informally at an area restaurant on Sunday evenings following the church service. Recent activities include attending professional sports games in Chicago, plays, concerts, and museums. They also attend Singles Retreats in the area and are key participants on mission trips.
IBC Seniors, age 55 and up, go on various trips and activities throughout the year such as visiting Shipshewana or Fair Oaks Farm in Indiana. They enjoy dinner together after seeing a musical or play and occasionally meet for a senior luncheon at church.


This winter, the Museum of Contemporary Art Chicago presents two performances by Oslo-based artist Ingri Fiskdal that blend fashion, movement, sound, and light: STATE (Feb 7-10) and Diorama (Feb 11-13).
In STATE, a small society manifests through ritual dances, music, and costumes, with an imaginative design co-created with sound artist Lasse Marhaug and avant-garde fashion designer Henrik Vibskov. Diorama takes place outdoors and is timed to follow the Chicago winter sunset and the city's flickering skyline from the perspective of Millennium Park. Here, Fiksdal's performers, dressed in sequined costumes by Fredrik Floen, incorporate and absorb the reflective power of Anish Kapoor's Cloud Gate.
Co-presented with the School of the Art Institute of Chicago as part of Chicago Theater Week 2019, Ingri Fiksdal and Jonas Corell Peterson: STATE takes place at the MCA from Thursday to Saturday, February 7-9, at 7:30 pm with an additional performance on Sunday, February 10 at 2 pm, and Ingri Fiksdal: Diorama is free to attend and takes place at Millennium Park from Monday to Wednesday, February 11-13, at 4 pm.
Ingri Fiksdal's creative process places equal emphasis on music, sound, light design, scenography, costume, and dance/choreography. Her awards include the Natt&Dag Award for Night Tripper, the Norwegian Critics' Association Dance Award for HOODS, and a Hedda Award nomination for Cosmic Body, which made its North American premiere at the MCA Chicago in 2016. She tours throughout Norway and has been presented at brut Wien, Vienna; Kampnagel, Hamburg; Harare International Festival of the Arts, Zimbabwe; the Homo Novus festival, Lithuania; In Between Time, Great Britain; the ANTI festival, Finland; and the Armory Show, New York. She was a research fellow at the Oslo National Academy of the Arts.


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8 August 2014 at 12:04pm
Is this the world's largest poppy?
Is this the world's largest poppy? Credit: ITV News Anglia
A giant poppy made from over 60,000 living poppies has been unveiled today at Skylark Garden Centre, near March, Cambridgeshire.
It is believed to be the biggest poppy made from poppies in the world.
It's made in a field full of poppies Credit: Jonathon Stevens
Children from Thomas Eaton School, Wimblington, Cambridgeshire, helped plant the poppies two months ago.
The giant poppy made from 60,000 Flanders poppies is 40m in diameter and covers an area of 1300 square meters, the size of an Olympic swimming pool.
Skylark Garden Centre are waiting to hear from Guinness World Records to see if their claim has been accepted.
Children helped plant the poppies Credit: Jonathon Stevens
"We are delighted and humbled at the effort that has been taken by Skylark Garden Centre into creating this live Poppy field in the shape of a Poppy. Since the end of the Great War the Poppy has become the emblem of Remembrance and by involving the local school children in this project it helps them to remember those that fell and hopefully the legacy is that our children and their children will carry on remembering those who sacrificed their lives for another 100 years. It is fitting that the Poppies have flowered in the week that marks the centenary of Britain going to war in 1914, maybe the Poppies knew the best time."
– Bryan Jones
The poppies in full bloom. Credit: Jonathon Stevens
Last updated Fri 8 Aug 2014
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How high-end housing developments use sustainability as a selling point
Sustainable housing development
Times reports taylor wimpey
Developments emphasize communities
Incorporating green building
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Our net revenue was up 109.2% year-over-year and 27.8% quarter-over-quarter to 838.2 million RMB, which exceeded the high end.
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For some brands, sustainability is a no-brainer. Swedish fast fashion giant H&M has pledged to use 100 per cent recycled or sustainable. the industry must not focus on recyclability as a selling.
How high-end housing developments use sustainability as a selling point New projects in California and Florida sell a better life now, and a better future tomorrow By Patrick Sisson May 23, 2019.
For instance, the premises we are operating from we will no longer use. The market value. in one building, and selling off others to ensure we have the right resources to deliver services to our.
Take a look at the current restaurant options at One Fulton Square in downtown Flushing – a mixed-use development home to luxe condos. wave of Chinese residents moving into the area's high-end real.
Eventually this seminar paper will enlighten the role of the planning systems in achievement of sustainable housing development, with the help of some major issues in new housing development, like transportation, higher density, social inclusion, mixed land use development and use of brown field sites.
As sustainable as they are affordable, these three LEED Platinum housing units in Philadelphia were all priced at under $300,000. When the developers of Los Angeles' Auburn 7 housing complex broke ground in 2007, they weren't sure whether people would pay a premium for eco-friendly amenities. Today.
Ground rent scandal' shakes Taylor Wimpey How the ground rents worked. Between 2007 and 2011, taylor wimpey created and sold a number of long leaseholds to homebuyers who paid a lump sum for a home and agreed to pay an annual ground rent fee. Sunday times reports taylor wimpey ground rent scandal, as executives meet leaseholders By admin on Nov 28, 2016 09:22 am The Taylor Wimpey ground.
It's a fine phone, capable of far more than I need, but like high-end consumers everywhere. so many old iPhones for recycling, or sell so many new iPhones to sustainably minded consumers like me.
8 Executive summary How to measure the social sustainability of new housing development How to measure the social sustainability of new housing development Executive summary 9 the framework has been designed to build on what is known about creating and supporting thriving communities. it has then been tested on four developments, to assess how they
Our picks for America's best green housing developments emphasize communities working toward sustainability by incorporating green building, energy efficiency and reuse of previously developed land. Several are enrolled in the U.S. Green Building Council's LEED (Leadership in Energy and.
Price of a place on the housing ladder exceeds 200,000 St. Catharines to push for former GM property tax sale tax sale properties niagara Falls, Ontario: The subject property would be sold on auction since the property owner is behind more than 3 years of Municipal property taxes.. St. Catharines Next Post Tax sale properties Wainwright No. 61.In high-price areas such as London it is the only chance for many wanting to escape private rentals and have a place. the property ladder. He bought a 40% share of a £190,000 flat in Wisteria.