text:
\section{Introduction}
\IEEEPARstart{P}{ower}
efficiency is important for higher-order modulation formats in communication systems. For an additive white Gaussian noise (AWGN) channel with an average power constraint, signal shaping is able to reduce the well-known asymptotic $1.53$ dB gap \cite{forney84} between the channel capacity and the achievable information rate (AIR) with a uniform signal (such as the most widely used quadrature amplitude modulation (QAM)), by adjusting the distribution of the transmitted signal alphabet to the capacity-achieving distribution, which is the Gaussian distribution. There are two flavors of shaping, geometric shaping by rearranging the positions of equally likely constellation points \cite{fengwensun93,forney84}, and probabilistic shaping by changing the distribution of a regular constellation \cite{calderbank90, kschischang93}. Much work has been devoted to the design of Gaussian-like constellations with feasible complexity in two \cite{gilbert52, campopiano62, foschini74, bin18,kadir20} and higher dimensions \cite{forney89a} for the AWGN channel.

The concept of Voronoi constellations (VCs) was first proposed in 1983 by Conway and Sloane, as a finite set of points of a lattice enclosed by a translated scaled-up version of its Voronoi region \cite{conway83}. Very fast algorithms for mapping integers to constellation points and vice versa were presented in \cite{conway83,conway82decoding}. In \cite{alijlt20,aliecoc}, Mirani \emph{et al.} designed multidimensional scaled VCs with up to $10^{28}$ constellation points utilizing this concept for the AWGN channel and nonlinear fiber channel, and showed high shaping gains and coding gains, and significant bit error rate (BER) and symbol error rate (SER) gains over QAM in uncoded system as well.

Forney generalized this concept in 1989 to VCs based on an arbitrary lattice partition $\Lambda/\Lambda_\mathrm{s}$ \cite{forney89b}, where $\Lambda$ is referred to as the coding lattice and $\Lambda_\mathrm{s}$ the shaping lattice. They can be different, as long as the shaping lattice is a sublattice of the coding lattice. Forney presented encoding and decoding algorithms for certain choices of the shaping lattice, such as so-called mod-2 or mod-4 binary lattices.

Feng \emph{et al.} presented a more general method to enumerate the points in a VC based on an arbitrary lattice partition \cite{feng13}, which is reviewed by Zamir in \cite[Ch.~9]{zamir14book}, \cite{zamir14}. This enumeration admits encoding and decoding algorithms as fast as Conway and Sloane's. An equally simple but less general method called ``rectangular encoding'' was proposed by Kurkoski in \cite{kurkoski18} for VCs whose shaping lattice and coding lattice both have triangular generator matrices. This method is applicable to a variety of coding lattices and shaping lattices.

Ferdinand \emph{et al.} proposed a two-step VC construction method called ``systematic Voronoi shaping'' in \cite{ferdinandTWC} based on the concept of ``systematic shaping'' proposed by Sommer \emph{et al.} in \cite{sommer09}, combining a high-dimensional coding lattice defined by a lower-triangular parity check matrix and a lower-dimensional shaping lattice, to achieve high coding gains and high shaping gains.  The SER performance was evaluated when low-density lattice codes \cite{sommer08} are used as the coding lattice and some common multidimensional lattices with low-complexity quantization algorithms are used for the shaping lattice. For the shaping step, algorithms to map the integers to points in VCs with a cubic coding lattice and vice versa were explicitly described in \cite{ferdinand14}.

In our conference paper \cite{ourISIT}, we studied VCs with a cubic coding lattice, for which we compared Feng's, Ferdinand's, and Kurkoski's encoding and decoding algorithms. To minimize the BER, we applied pseudo-Gray labeling to these algorithms, and evaluated the performance of the designed VCs in terms of the Gray penalty \cite{simon1973,smith1975} and BER performance for some common multidimensional shaping lattices both in uncoded and coded systems. In our proposed scheme, coding is completely separated from shaping and performed using error-control coding, outside the Voronoi shaping.

We study VCs with a cubic coding lattice for the following three reasons. First, the decoding algorithm is much simpler than for VCs with rescaled coding and shaping lattices \cite{conway83, alijlt20}, since the search for the closest lattice point, which dominates the decoding complexity for high-dimensional lattices, is simple dimension-wise integer rounding for a cubic coding lattice. Second, although the proposed VCs have no coding gain, the shaping gain is still achievable, and the lack of coding gain can be compensated by error-correction coding that is usually performed anyway. After error-correction coding, the proposed VCs can have better BER performance than the conventional scaled VCs because of their inherently good pseudo-Gray labeling feature. Third, for mapping integers to bits, the proposed VCs allow for improved granularity in spectral efficiencies.

 In this paper, we extend our work \cite{ourISIT} on the VCs with a cubic coding lattice. We propose a new method based on the concept of importance sampling to estimate the mutual information (MI) for very large constellations, and exemplify it for VCs. This has been considered as a challenging issue for such large constellations since the exact MI calculation requires enumerating all constellation points. The MI of VCs is evaluated here for the first time to our knowledge. The AIR performance of VCs in combination with a low-density parity-check (LDPC) code is also presented. For mapping integers to bits, we improve the granularity in spectral efficiencies by rotating the shaping lattice by $\ang{45}$ and scaling it by $\sqrt{2}$. We also investigate the trade-off between shaping gain and decoding complexity in higher dimensions.

\emph{Notation:} Bold lowercase symbols denote row vectors and bold uppercase symbols denote matrices or random vectors. The elements of a vector $\boldsymbol{u}$ are denoted by $u_i$, the rows of a matrix $\boldsymbol{P}$ are denoted by $\boldsymbol{p}_i$, and the element at row $i$, column $j$ of a matrix $\boldsymbol{P}$ are denoted by $P_{ij}$. The sets of integer, real, complex, and natural numbers are denoted by $\mathbb{Z}$, $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{N}$, respectively. Rounding a vector to its nearest integer vector is denoted by $\lfloor \cdot \rceil$, in which ties are broken arbitrarily. The largest integer not greater than a given real number is denoted by $\lfloor \cdot \rfloor$.

\section{Preliminaries}

Given a set of $n$ linearly independent \emph{basis vectors,} a \emph{lattice} is the set of all linear combinations of these vectors with integer coefficients. If the basis vectors are arranged row-wise into a matrix $\boldsymbol{G}$, then the lattice is
\begin{align} \label{eq:latticedef}
\Lambda \triangleq \{ \boldsymbol{u} \boldsymbol{G} :\; \boldsymbol{u} \in \mathbb{Z}^n \}
.\end{align}
Without loss of generality, we assume that the generator matrix has dimension $n \times n$.%
\footnote{The $n$ basis vectors must have dimension at least $n$ in order to be linearly independent. If $n$ basis vectors are given in more than $n$ dimensions, then an $n$-dimensional lattice, which is equivalent to the original lattice in Euclidean geometry, can be defined by rotating (e.g., QR-decomposing) $\boldsymbol{G}$.} From the definition, any lattice includes the all-zero vector $\boldsymbol{0}$. The generator matrix of a given lattice is not unique. Two generator matrices $\boldsymbol{G}$ and $\boldsymbol{G}'$ generate the same lattice if and only if $\boldsymbol{G}' = \boldsymbol{U} \boldsymbol{G}$, where $\boldsymbol{U}$ is an integer matrix with determinant $\pm 1$ \cite[p.~10]{conway99book}.

The \emph{closest lattice point quantizer} $\mathcal{Q}_{\Lambda}(\cdot)$ maps an arbitrary vector $\boldsymbol{x}\in\mathbb{R}^n$ to its closest lattice point in $\Lambda$
\begin{align}
    \mathcal{Q}_{\Lambda}(\boldsymbol{x})=\argmin_{\boldsymbol{\lambda} \in \Lambda}\|\boldsymbol{x}-\boldsymbol{\lambda}\|^2.
\end{align}

If an affine transformation function $T(\cdot)$ is applied to the lattice $\Lambda$, e.g., a scaling, rotation, or/and a shift, then it follows that
\begin{align}
    \mathcal{Q}_{T(\Lambda)}(\boldsymbol{x} )=T(\mathcal{Q}_{\Lambda}(T^{-1}(\boldsymbol{x}))).\label{eq:T}
\end{align}

The \emph{fundamental Voronoi region} of a lattice $\Lambda$ is the set of vectors in Euclidean space having the all-zero vector as its closest lattice point, i.e.,
\begin{align}
\Omega(\Lambda) \triangleq \{\boldsymbol{x} \in \mathbb{R}^n :\; \mathcal{Q}_{\Lambda}(\boldsymbol{x})=\boldsymbol{0} \}
.\end{align}
Given an $n$-dimensional coding lattice $\Lambda$, an $n$-dimensional shaping lattice $\Lambda_\mathrm{s}$, and an \emph{offset vector} $\boldsymbol{a} \in \mathbb{R}^n$, where $\Lambda_\mathrm{s}$ is a sublattice of $\Lambda$, i.e., $\Lambda_\mathrm{s}\subset \Lambda$, a VC in its general form defined by Forney \cite{forney89b} is
\begin{align}
\Gamma \triangleq (\Lambda-\boldsymbol{a}) \cap \Omega(\Lambda_\mathrm{s})\label{eq:VC}
.\end{align}
We assume that no points in $\Lambda-\boldsymbol{a}$ fall on the boundary of $\Omega(\Lambda_\mathrm{s})$.%
\footnote{An arbitrarily small perturbation of the offset $\boldsymbol{a}$ in a random direction prevents points from falling on the boundary with probability one.} The number of points in the VC is
\begin{align}
M \triangleq |\Gamma| = \frac{|\!\det\boldsymbol{G}_\mathrm{s}|}{|\!\det\boldsymbol{G}|}
,\end{align}
where $\boldsymbol{G}_\mathrm{s}$ is a generator matrix of $\Lambda_\mathrm{s}$. This relation can be verified by recognizing $|\!\det\boldsymbol{G}|$ and $|\!\det\boldsymbol{G}_\mathrm{s}|$ as the volumes of $\Omega(\Lambda)$ and $\Omega(\Lambda_\mathrm{s})$, respectively \cite[p.~4]{conway99book}. The average symbol energy is
\begin{align}
E_{\text{s}}=\frac{1}{M}\sum_{\boldsymbol{x}\in\Gamma}\|\boldsymbol{x}\|^{2}.\label{eq:Es}
\end{align}

To compare the performance of different constellations, the following relevant figures of merit were defined and widely used in the literature.

\emph{1)} The \emph{spectral efficiency} \cite{forney89a, kschischang93, mkeaofc12} of a constellation is defined as
\begin{align}
    \beta=2\log_{2}(M)/n ~\text{[bits/symbol/dimension-pair]}.
\end{align}

\emph{2)} The \emph{asymptotic power efficiency} (APE) \cite[Eq.~(5.8)]{Benedetto99book},\cite{agrell09} is defined as
\begin{align}
\gamma=\frac{d_{\text{min}}^{2}\log_{2}(M)}{4E_{\text{s}}},
\end{align}
where $d_{\text{min}}$ is the minimum Euclidean distance of the constellation. Usually, the pulse amplitude modulation (PAM) is chosen as the benchmark \cite{forney89a,kschischang93}, which has an APE of
\begin{align}
\gamma_{\text{PAM}}=\frac{3\beta}{2(2^{\beta}-1)}.
\end{align}
This also applies to the geometric extension of a PAM, i.e., an $n$-dimensional cubic constellation constructed by the Cartesian product of $n$ equal one-dimensional PAM constellations.

\emph{3)} The \emph{APE gain} of a constellation over a cubic constellation at the same spectral efficiencies is quantified as
\begin{align}
    g=10\log_{10}\frac{\gamma}{\gamma_{\text{PAM}}}~\text{[dB]}.
\end{align}
It can be divided into a \emph{coding gain} $g_{\text{c}}$ obtained by packing the constellation points in a cubic shape more densely, and a \emph{shaping gain} $g_{\text{s}}$ obtained by making the boundary of the cubic-packing points more spherical \cite{forney89a,mkeaofc12}. For the fundamental Voronoi region of the lattice $\Lambda_\mathrm{s}$, the (asymptotic) shaping gain is defined as $g_{\text{s}}(\Lambda_\mathrm{s})=1/(12G(\Omega(\Lambda_\mathrm{s})))$ \cite{forney89a}, where $G(\Omega(\Lambda_\mathrm{s}))$ is the normalized second moment of this region \cite[Eq.~(9)]{conway82Gvalue}.

\section{Mapping integers to VCs}
 Feng \emph{et al.} proposed encoding and decoding algorithms, i.e., mapping integers to constellation points and vice versa, for arbitrary VCs in \cite{feng13}, which relies on the Smith normal form \cite[Ch.~15]{brownbook93}. In \cite{ferdinand14}, Ferdinand \emph{et al.} proposed encoding and decoding algorithms specifically for VCs with a cubic coding lattice. Kurkoski proposed a ``rectangular encoding'' method in \cite{kurkoski18}, which is applicable to VCs with a shaping lattice described by a triangular generator matrix and a cubic coding lattice. In this section, the three algorithms are reviewed and compared specifically when they are applied to VCs with a cubic coding lattice.

Table \ref{tab:algorithms} summarizes the three algorithms for the special case of the lattice partition $\mathbb{Z}^n/\Lambda_\mathrm{s}$, which is the focus of this paper. The more general versions of Feng's algorithms for arbitrary $\Lambda$ are described in \cite{feng13}. In Ferdinand's algorithms described in \cite{ferdinand14}, the basis vectors are column vectors and the decoding goes from low to high dimensions. Here for the convenience of comparison, we describe them using row vectors as basis vectors, and rewrite the decoding process of \cite[Eqs.~(19)--(21)]{ferdinand14}, with a reverse order of decoding, which however does not change the performance of their algorithms.

The closest lattice point quantizer $\mathcal{Q}_{\Lambda_\mathrm{s}}(\cdot)$ in these algorithms is well-studied for many common lattices \cite[Ch.~20]{conway99book}, \cite{conway82decoding, conway84}, and other lattices can be handled by general algorithms \cite{agrell02,Ghasemmehdi11}.

\begin{table*}[tbp]
  \newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
  \centering
  \begin{tabular}{| m{3cm} | m{4.75cm}| m{4.1cm} | m{4.1cm}|}
    \hline
    Algorithms & Feng's algorithms\cite{feng13}& Ferdinand's algorithms \cite{ferdinand14}& Kurkoski's algorithms\cite{kurkoski18}\\
    \hline
    \hline
    \emph{Preprocessing:} Given the generator matrix $\boldsymbol{G}_\mathrm{s}$ of the shaping lattice $\Lambda_\mathrm{s}$ and the generator matrix of the cubic coding lattice is $\boldsymbol{G}=\boldsymbol{I}_{n}$.
    &Find two integer matrices $\boldsymbol{S}$ and $\boldsymbol{T}$ with determinant $\pm 1$ such that $\boldsymbol{J}=\boldsymbol{S}\boldsymbol{G}_\mathrm{s}\boldsymbol{T}$ is the Smith normal form of $\boldsymbol{G}_\mathrm{s}$. Then let $u_i \in \{0,\ldots, J_{ii}-1\}$ for $i=1,\ldots,n$.
    &Find an integer matrix $\boldsymbol{S}$ with determinant $\pm 1$ such that $\boldsymbol{L}=\boldsymbol{S}\boldsymbol{G}_\mathrm{s}$ is a lower-triangular matrix with positive diagonal elements. Let $u_i \in \{0,\ldots, L_{ii}-1\}$ for $i=1,\ldots,n$.
    &Find an integer matrix $\boldsymbol{S}$ with determinant $\pm 1$ such that $\boldsymbol{L}=\boldsymbol{S}\boldsymbol{G}_\mathrm{s}$ is a lower-triangular matrix with positive diagonal elements. Let $u_i \in \{0,\ldots, L_{ii}-1\}$ for $i=1,\ldots,n$. \\
    \hline
    \emph{Encoding:} Input $\boldsymbol{u}$. Output $\boldsymbol{x}$.
    &\tabincell{l}{1) Let $\boldsymbol{c} \leftarrow \boldsymbol{u} \boldsymbol{T}^{-1} - \boldsymbol{a}$\\2) Let $\boldsymbol{z} \leftarrow \mathcal{Q}_{\Lambda_\mathrm{s}}(\boldsymbol{c})$\\3) Let $\boldsymbol{x} \leftarrow \boldsymbol{c}-\boldsymbol{z}$}
    & \tabincell{l}{1) Let $\boldsymbol{d} \leftarrow \left ( \frac{u_{1}}{L_{11}},\frac{u_{2}}{L_{22}},\ldots,\frac{u_{n}}{L_{nn}} \right )$\\2) Let $\boldsymbol{c}=\boldsymbol{d}\boldsymbol{L}-\boldsymbol{a}$\\3) Let $\boldsymbol{z} \leftarrow \mathcal{Q}_{\Lambda_\mathrm{s}}(\boldsymbol{c})$\\4) Let $\boldsymbol{x} \leftarrow \boldsymbol{c}-\boldsymbol{z}$}
    & \tabincell{l}{1) Let $\boldsymbol{c} \leftarrow \boldsymbol{u} - \boldsymbol{a}$\\2) Let $\boldsymbol{z} \leftarrow \mathcal{Q}_{\Lambda_\mathrm{s}}(\boldsymbol{c})$\\3) Let $\boldsymbol{x} \leftarrow \boldsymbol{c}-\boldsymbol{z}$}\\
    \hline
    \emph{Decoding:} Input $\boldsymbol{y}$. Output $\boldsymbol{u}$.
    &\tabincell{l}{1) Let $\boldsymbol{c} \leftarrow \lfloor \boldsymbol{y}+\boldsymbol{a} \rceil$\\2) Let $\boldsymbol{u} \leftarrow \boldsymbol{c} \boldsymbol{T}$\\3) Let $u_i \leftarrow u_i \mod J_{ii} $, $\forall i=1,\ldots,n$}
    & \tabincell{l}{1) Let $\boldsymbol{u} \leftarrow \lfloor \boldsymbol{y}+\boldsymbol{a} \rceil$\\2) For $i=n,n-1,\ldots,1,$ do\\~~~~~$\left\{\begin{matrix}
v_i \leftarrow \lfloor u_i/L_{ii} \rfloor\\ \boldsymbol{u} \leftarrow \boldsymbol{u}-v_i \boldsymbol{l}_i \end{matrix}\right.$}
    &\tabincell{l}{1) Let $\boldsymbol{u} \leftarrow \lfloor \boldsymbol{y}+\boldsymbol{a} \rceil$\\2) for $i=n,n-1,\ldots,1,$ do\\~~~~~$\left\{\begin{matrix}
v_i \leftarrow \lfloor u_i/L_{ii} \rfloor\\ \boldsymbol{u} \leftarrow \boldsymbol{u}-v_i \boldsymbol{l}_i \end{matrix}\right.$}\\
    \hline
  \end{tabular}
  \caption{A Comparison of different encoding and decoding algorithms for VCs with a cubic coding lattice.}
  \label{tab:algorithms}
\end{table*}

Every matrix over a principal ideal domain has a Smith normal form, and the integers form a principal ideal domain \cite{brownbook93,feng13}. In Feng's algorithm, given an integer generator matrix $\boldsymbol{G}_\mathrm{s}$, the Smith normal form is first computed in the preprocessing stage as $\boldsymbol{J}=\boldsymbol{S}\boldsymbol{G}_\mathrm{s}\boldsymbol{T}$, where $\boldsymbol{J}$ is a diagonal matrix with positive diagonal elements and $J_{ii}$ divides $J_{ii+1}$ for $i=1,\ldots,n-1.$ Then the integer vectors $\boldsymbol{u}$ are enumerated according to the diagonal elements of $\boldsymbol{J}$. The $M=\!\det \boldsymbol{J}$ possible values of $\boldsymbol{u}\boldsymbol{T}^{-1}$ can be uniquely mapped to the $M$ constellation points $\boldsymbol{x}\in\Gamma$ by
\begin{align} \label{eq:enumerationFeng}
\boldsymbol{x} = \boldsymbol{u} \boldsymbol{T}^{-1} + \boldsymbol{v} \boldsymbol{S}\boldsymbol{G}_\mathrm{s} - \boldsymbol{a},
\end{align}
where $\boldsymbol{v}\in\mathbb{Z}^n$. The Smith normal form $\boldsymbol{J}$ is unique given a $\boldsymbol{G}_\mathrm{s}$, whereas $\boldsymbol{S}$ and $\boldsymbol{T}$ are not unique; thus Feng's algorithms can generate different mapping rules for the same $\boldsymbol{G}_\mathrm{s}$.

In Ferdinand's algorithms, a lower-triangular generator matrix $\boldsymbol{L}$ for $\Lambda_\mathrm{s}$ is first computed (the algorithms are applicable only if $\boldsymbol{L}$ exists), then the integer vectors $\boldsymbol{u}$ are enumerated according to the diagonal elements of $\boldsymbol{L}$. In the encoding process, $\boldsymbol{d}\boldsymbol{L}$ are defined to be integer vectors in the fundamental parallelotope of $\Lambda_\mathrm{s}$. The constellation points $\boldsymbol{x} \in \Gamma$ can be labeled by unique values of $\boldsymbol{d}\boldsymbol{L}$ and $\boldsymbol{v}\in \mathbb{Z}^n$ as
\begin{align} \label{eq:enumerationFerdinand}
\boldsymbol{x} = \boldsymbol{d} \boldsymbol{L} + \boldsymbol{v} \boldsymbol{L} - \boldsymbol{a}.
\end{align}
 However, $\boldsymbol{d}\boldsymbol{L}$ are not guaranteed to be integer vectors for arbitrary shaping lattices, thus making Ferdinand's algorithms less general than Kurkoski's algorithms. The condition for Ferdinand's algorithms to be applicable is
\begin{align}
    \frac{L_{ij}}{L_{ii}}\in \mathbb{Z}, \forall i={1,\ldots,n}, ~1\leq j<i.
\end{align}
The decoding can be done sequentially, beginning from $u_n$, thanks to the triangular structure of $\boldsymbol{L}$.

Kurkoski's algorithms are also applicable only if the lower-triangular matrix $\boldsymbol{L}$ exists. Then every point $\boldsymbol{x}\in\Gamma$ can be uniquely enumerated by vectors $\boldsymbol{u}$ and $\boldsymbol{v}$ such that
\begin{align} \label{eq:enumeration}
\boldsymbol{x} = \boldsymbol{u} + \boldsymbol{v} \boldsymbol{G}_\mathrm{s} - \boldsymbol{a},
\end{align}
where $u_i \in \{0,\ldots, L_{ii}-1\}$ for $i=1,\ldots,n$ and $\boldsymbol{v} \in \mathbb{Z}^n$. There are $M=\!\det \boldsymbol{L} = \prod_i L_{ii}$ possible values of $\boldsymbol{u}$, and each of them occurs exactly once among all points $\boldsymbol{x}\in\Gamma$. The decoding process is the same as Ferdinand's.

Given the same $\boldsymbol{G}_\mathrm{s}$, Feng's and Ferdinand's algorithms can have different mapping rules, and Kurkoski's algorithms provide a third mapping rule, see the numerical example in this section. When Feng's $\boldsymbol{u}\boldsymbol{T}^{-1}$ and Ferdinand's $\boldsymbol{d}\boldsymbol{L}$ are equal, their algorithms are equivalent. For most commonly used shaping lattices, e.g., $D_{4}$, $E_{8}$, $\Lambda_{16}$, and $\Lambda_{24}$ \cite[Ch.~4]{conway99book}, Ferdinand's algorithms are applicable. Feng's algorithms can have the same mapping rule as Ferdinand's algorithms, when the generator matrices are written in nice lower-triangular matrices as in \cite[Ch.~4]{conway99book}.

Apparently Feng's algorithms are more general than Kurkoski's, which are in turn slightly more general than Ferdinand's. However, for the case where all three algorithms are applicable, e.g., for commonly used multidimensional shaping lattices, Kurkoski's algorithms generally provide better labelings of constellation points. When a Gray code is applied to label the integer coordinates of $\boldsymbol{u}$, i.e., converting $u_{i}$ to a binary reflected Gray code for $i=1,\ldots,n$, each pair of nearest $\boldsymbol{u}$ in terms of Euclidean distance differs by exactly one bit. Kurkoski's algorithms allow for directly mapping $\boldsymbol{u}$ to constellation points, thus making most of the constellation points have Gray neighbors, which we call pseudo-Gray labeling. In Feng's and Ferdinand's encoding algorithms, $\boldsymbol{u}$ is multiplied by a matrix, which changes the neighbor relationships between constellation points. Kurkoski's algorithms allow for this pseudo-Gray labeling for VCs, and can thus achieve a smaller Gray penalty and better BER performance \cite{ourISIT} than Feng's and Ferdinand's algorithms. Throughout this paper, we adopt Kurkoski's encoding and decoding and this pseudo-Gray labeling scheme.

\begin{figure*}[tbp]
\centering
\subfloat[Feng's algorithms.]{\includegraphics[width=6.6in]{./figures/VC/FengVC.pdf}%
\label{Fig:FengVC}}
\hfil
\subfloat[Ferdinand's algorithms.]{\includegraphics[width=3.1in]{./figures/VC/FerdinandVC.pdf}%
\label{Fig:FerdinandVC}}
\hspace{0.4in}
\subfloat[Kurkoski's algorithms.]{\includegraphics[width=3.1in]{./figures/VC/ourVC.pdf}%
\label{Fig:ourVC}}
\caption{Example: different integer mapping rules for a two-dimensional VC. The blue filled points are encoded into points in the shifted Voronoi region $\boldsymbol{a}+\Omega(\Lambda_\mathrm{s})$ (the light blue region) in encoding.}
\label{Fig:VC}
\end{figure*}

\emph{Example:} We consider a two-dimensional VC for which the generator matrices of $\Lambda_\mathrm{s}$ and $\Lambda$ are
\begin{align}
\boldsymbol{G}_\mathrm{s}=\begin{pmatrix}
6 & 0\\
4 & 4
\end{pmatrix},\;
\boldsymbol{G}=\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix},
\end{align}
and the offset vector $\boldsymbol{a}=(-1/2, 0)$.

In Feng's algorithms, the Smith normal form $\boldsymbol{J}=\boldsymbol{S} \boldsymbol{G}_\mathrm{s} \boldsymbol{T}$ is
\begin{align}
    \boldsymbol{J}=\begin{pmatrix}
    2&0\\
    0&12
    \end{pmatrix},\;
    \boldsymbol{S}=\begin{pmatrix}
    1&1\\
    -2&-3
    \end{pmatrix},\;
    \boldsymbol{T}=\begin{pmatrix}
    1&2\\
    -2&-5
    \end{pmatrix}.
\end{align}
Then the integer vectors $\boldsymbol{u}=(u_1,u_2)$ are defined as ${u_1 \in \{0,1\}}$ and $u_2 \in \{0,\ldots, 11\}$. The vectors $\boldsymbol{u}\boldsymbol{T}^{-1}$ are calculated as
\begin{align}
\boldsymbol{u}\boldsymbol{T}^{-1}=
    \begin{pmatrix}
    u_{1}& u_{2}
    \end{pmatrix}
    \begin{pmatrix}
    5& 2\\
    -2& -1
    \end{pmatrix}
=\begin{pmatrix}
    5u_{1}-2u_{2}&2u_{1}-u_{2}
    \end{pmatrix}.
\end{align}

In Ferdinand's algorithms, $\boldsymbol{L}=\boldsymbol{G}_\mathrm{s}$ and $\boldsymbol{u}$ are enumerated as $u_1 \in \{0,\ldots, 5\}$ and $u_2 \in \{0,\ldots, 3\}$. The vectors $\boldsymbol{d}\boldsymbol{L}$ are computed by
\begin{align}
\boldsymbol{d}\boldsymbol{L}=
    \begin{pmatrix}
    u_{1}/6& u_{2}/4
    \end{pmatrix}
    \begin{pmatrix}
    6& 0\\
    4& 4
    \end{pmatrix}
=\begin{pmatrix}
    u_{1}+u_{2}&u_{2}
    \end{pmatrix}.
\end{align}
In Kurkoski's algorithms, $\boldsymbol{L}=\boldsymbol{G}_\mathrm{s}$ and $\boldsymbol{u}$ are enumerated as $u_1 \in \{0,\ldots, 5\}$ and $u_2 \in \{0,\ldots, 3\}$.

Fig.~\ref{Fig:VC} illustrates the three different mapping rules for this example, where the vectors $\boldsymbol{u}\boldsymbol{T}^{-1}$ in Feng's algorithms, $\boldsymbol{d}\boldsymbol{L}$ in Ferdinand's algorithms, and $\boldsymbol{u}$ in Kurkoski's algorithms are highlighted. The enumeration of points in a rectangular shape of Kurkoski's algorithms help achieve a lower Gray penalty and a lower BER \cite{ourISIT}.

\section{Design of VCs}
A VC based on the lattice partition $\Lambda/\Lambda_\mathrm{s}$ is defined by \eqref{eq:VC}. In this section, we discuss the choice of parameters of VCs based on the lattice partition $\mathbb{Z}^n/\Lambda_\mathrm{s}$.

 To minimize the average symbol energy defined by \eqref{eq:Es}, the offset vector $\boldsymbol{a}$ is optimized using an iterative algorithm given in \cite{conway83}. This algorithm in may converge to a suboptimal vector for $8$ and higher dimensions, and cannot be exactly calculated when $M$ is very large. However, as $M$ increases, the performance difference between VCs generated using the optimal $\boldsymbol{a}$ and a random $\boldsymbol{a} \in \Omega(\Lambda)$ decreases, and can be neglected for large VCs, see \cite[Fig.~3]{alijlt20}. In this paper, for a small or moderate-size VC ($M\leq 2^{17}\approx 1.3\times10^5$), $\boldsymbol{a}$ was optimized using the method in \cite{conway83}, whereas for very large VCs where we can only approximate the average symbol energy by Monte Carlo simulations, a random $\boldsymbol{a}$ uniformly distributed in $\Omega(\Lambda)$ was selected.

The shaping lattice $\Lambda_\mathrm{s}$ should have a high shaping gain and low-complexity quantization algorithm. The most commonly-used multidimensional shaping lattices: the $4$-dimensional checkerboard lattice $D_{4}$, $8$-dimensional lattice $E_{8}$, $16$-dimensional Barnes--Wall lattice $\Lambda_{16}$, the $24$-dimensional Leech lattice $\Lambda_{24}$ \cite[Ch.~4]{conway99book}, and a suboptimal $32$-dimensional lattice $L_{32}$ are considered in this paper. The $32$-dimensional lattice $L_{32}$ is constructed by applying Construction B \cite[Ch.~5]{conway99book} to the $(32,6,16)$ first order Reed--Muller code. This lattice has a sublattice $2D_{32}$ of index $64$, which is beneficial, since the closest lattice point quantizer dominates the complexity for the encoding and decoding, especially for high-dimensional lattices. Compared with $\Lambda_{24}$ which has a sublattice $4D_{24}$ of index $8192$ \cite{conway84}, the complexity is reduced by a factor of $128$, while the shaping gain is only $0.091$ dB less. Thus, $L_{32}$ can be considered as a good trade-off between the shaping gain and decoding complexity. Table \ref{tab:gammas} shows the normalized second moment $G(\Omega(\Lambda_\mathrm{s}))$ and shaping gains $g_{s}(\Lambda_\mathrm{s})$ for these considered shaping lattices.

\begin{table}

  \renewcommand{\arraystretch}{1.3}
  \caption{Asymptotic shaping gains and normalized second moments of Voronoi regions of some lattices.}
  \label{tab:gammas}
  \centering
  \begin{tabular}{c c c c c c}
    \hline
    $\Lambda_\mathrm{s}$ &$D_{4}$& $E_{8}$&$\Lambda_{16}$& $\Lambda_{24}$ &$L_{32}$\\
    \hline \hline
    $G(\Omega(\Lambda_\mathrm{s}))$ & $0.0766$ & $0.0717$ & $0.0683$ & $0.0658$ & $0.0671$\\
    \hline
    $g_{\text{s}}(\Lambda_\mathrm{s})$ [dB] & $0.366$ & $0.653$ & $0.864$ & $1.026$& $0.935$\\
    \hline
  \end{tabular}
\end{table}
A VC based on the lattice partition $\Lambda/\Lambda_\mathrm{s}$ with ${M=|\!\det\boldsymbol{G}_\mathrm{s}|/|\!\det \boldsymbol{G}|}$ constellation points can be scaled by any integer, i.e., $\Lambda/m\Lambda_\mathrm{s},\;m\in \mathbb{Z}$. The new VC after scaling has a spectral efficiency of
\begin{align}
    \beta&=\frac{2\log_{2}(|\!\det m^n\boldsymbol{G}_\mathrm{s}|/|\!\det\boldsymbol{G}|)}{n} \notag\\
    &=\frac{2\log_{2}(M)}{n}+2\log_{2}(m).\label{eq:beta}
\end{align}

A rotation of $\ang{45}$ together with a scaling of $\sqrt{2}$ of the shaping lattice $\Lambda_\mathrm{s}$ can provide $1$ more bit/symbol/dimension-pair. This can be done for an even $n$ by multiplying the generator matrix $\boldsymbol{G}_\mathrm{s}$ from the right side with an $n$-by-$n$ matrix \cite{forney88}
\begin{align}
\boldsymbol{R}=\begin{pmatrix}
 1& 1 & 0 & 0 &0  & \cdots & 0 &0 \\
 -1&  1&  0&  0&  0&  \cdots&  0&0 \\
 0& 0 & 1 & 1 &  0&  \cdots&  0&0 \\
 0& 0& -1& 1&  0& \cdots & 0 &0 \\
 0& 0&  0&  0&  \ddots& ~&  \vdots& \vdots\\
 \vdots& \vdots&  \vdots&  \vdots& ~& \ddots &  0& 0\\
 0& 0&  0&  0&  \cdots&  0& 1 & 1\\
 0& 0&  0&  0&  \cdots&  0& -1& 1
\end{pmatrix},
\end{align}
which operates on every dimension-pair of $\Lambda_\mathrm{s}$, with a determinant of $2^{n/2}$. The generator matrix of the new shaping lattice $m\Lambda_\mathrm{s} \boldsymbol{R}$ (which remains a sublattice of $\Lambda$) is $m\boldsymbol{G}_\mathrm{s}\boldsymbol{R}$, and the closest point quantizer becomes
\begin{align}
    \mathcal{Q}_{m\Lambda_\mathrm{s}\boldsymbol{R}}(\boldsymbol{x})=m\mathcal{Q}_{\Lambda_\mathrm{s}}(\frac{1}{m}\boldsymbol{x}\boldsymbol{R}^{-1})\boldsymbol{R}
\end{align}
by \eqref{eq:T}. The new VC based on the lattice partition $\Lambda/m\Lambda_\mathrm{s}\boldsymbol{R}$ has a spectral efficiency of
\begin{align}
    \beta&=\frac{2\log_{2}(|\!\det m^n\boldsymbol{G}_\mathrm{s}\boldsymbol{R}|/|\!\det\boldsymbol{G}|)}{n} \notag\\
    &=\frac{2\log_{2}(M)}{n}+2\log_{2}(m)+1.\label{eq:beta_rotation}
\end{align}

In order to make bit mapping possible, both $M$ and $m$ must be powers of $2$. With $m$ an arbitrary power of $2$, $\Lambda/m \Lambda_s$ gives a resolution of $2$ bits/symbol/dimension-pair according to \eqref{eq:beta}. However, $\Lambda/m \Lambda_s \boldsymbol{R}$ offers the intermediate spectral efficiencies as in \eqref{eq:beta_rotation}, which decreases the overall resolution to $1$ bit/symbol/dimension-pair. Also, the combination of different shaping and coding lattices can lead to a more granular set of possible spectral efficiencies, which allows us to have more flexibility in choosing data rates as needed. In Fig.~\ref{Fig:shapinggain}, the shaping gain $g_{\text{s}}$ as a function of $\beta$ is presented for VCs with a cubic coding lattice and the considered shaping lattices in Table \ref{tab:gammas}. Larger markers show the spectral efficiencies for bit mapping; smaller markers show the finer-grained granularity of spectral efficiencies without considering bit mapping, where $m$ does not have to be a power of $2$.

\begin{figure}[tbp]
\centering
\begin{tikzpicture}
	\begin{axis}[
		xmin=0,
		xmax=13.5,
		ymin=0.2, ymax=1.2,
		xlabel={$\beta$ [bits/symbol/dimension-pair]},
		ylabel={APE gain $g$ [dB]},
		ylabel style={at={(axis description cs:-0.01,0.5)}, anchor=north},
		cycle list name=myCycleList,
	    legend pos=north west,
		legend cell align=left,
		legend style={fill=white,fill opacity=0.8, draw opacity=1,text opacity=1},
		ylabel style={yshift=-.05cm},
		xlabel style={xshift=-.05cm},
		height =0.4\textwidth,
		width=0.5\textwidth,
	]

	\addplot+[lines-1] table[
		x=SE,
		y=gammas,
	] {./figures/SE/gammasD4.txt};\addlegendentry{$\mathbb{Z}^{4}/D_{4}$}
 	\addplot+[lines-2] table[
 		x=SE,
 		y=gammas,
 	] {./figures/SE/gammasE8.txt};\addlegendentry{$\mathbb{Z}^{8}/E_{8}$}
 	\addplot+[lines-3] table[
		x=SE,
 		y=gammas,
 	] {./figures/SE/gammasL16.txt};\addlegendentry{$\mathbb{Z}^{16}/\Lambda_{16}$}
 	 \addplot+[lines-4] table[
		x=SE,
 		y=gammas,
 	] {./figures/SE/gammasL24.txt};\addlegendentry{$\mathbb{Z}^{24}/\Lambda_{24}$}
 	\addplot+[lines-5] table[
		x=SE,
 		y=gammas,
 	] {./figures/SE/gammasL32.txt};\addlegendentry{$\mathbb{Z}^{32}/L_{32}$}

 	 \addplot+[dashed,color=black,mark=none] coordinates {(-2,0.3659) (19,0.3659)};\addlegendentry{$g_{\text{s}}(\Lambda_\mathrm{s})$}

 	\addplot+[only marks,lines-1,mark=o,mark size=1] table[
		x=SE,
		y=gammas,
	] {./figures/SE/gammas_scaled_D4.txt};
	\addplot+[only marks, lines-2,mark=star,mark size=1] table[
		x=SE,
		y=gammas,
	] {./figures/SE/gammas_scaled_E8.txt};
	\addplot+[only marks, lines-3,mark=square,mark size=1] table[
		x=SE,
		y=gammas,
	] {./figures/SE/gammas_scaled_L16.txt};
	\addplot+[only marks, lines-4,mark=diamond,mark size=1] table[
		x=SE,
		y=gammas,
	] {./figures/SE/gammas_scaled_L24.txt};
	\addplot+[only marks, lines-5,mark=triangle,mark size=1] table[
		x=SE,
		y=gammas,
	] {./figures/SE/gammas_scaled_L32.txt};
 	 \addplot+[dashed,color=black,mark=none] coordinates {(-2,0.6530) (19,0.6530)};
 	 \addplot+[dashed,color=black,mark=none] coordinates {(-2,0.8640) (19,0.8640)};
 	 \addplot+[dashed,color=black,mark=none] coordinates {(-2,1.0259) (19,1.0259)};
 	 \addplot+[dashed,color=black,mark=none] coordinates {(-2,0.9345) (19,0.9345)};
	\end{axis}
\end{tikzpicture}
\caption{The APE gain $g$ as a function of $\beta$ for VCs with a cubic coding lattice. The smaller markers on top of the lines represent the cases in which the scaling factor is not a power of $2$. The black dashed lines are the asymptotic shaping gains $g_{\text{s}}(\Lambda_\mathrm{s})$ for these shaping lattices stated in Table \ref{tab:gammas}.}
\label{Fig:shapinggain}
\end{figure}

\section{Achievable Information Rates}
AIRs indicate the maximum amount of information that can be transmitted reliably over the underlying channel. It assumes a perfect channel code with an infinite blocklength, thus can be used as an upper bound of the performance of coded modulation \cite{alvarado18}. For very large VCs, the calculation of MI is challenging, since it requires the coordinates of all constellation points. In this section, we propose an MI estimation method and a log-likelihood (LLR) estimation method for very large constellations and apply these methods to VCs with a cubic coding lattice. The MI of our designed VCs is investigated, and their AIRs are compared with conventional scaled VCs when an LDPC code is applied.
\subsection{MI estimation method for very large constellations}\label{subsec:MIest_method}
For a memoryless discrete channel, the MI between the equally probable transmitted symbols $\boldsymbol{X}$ and the received noisy symbols $\boldsymbol{Y}$ can be written as
\begin{align}\label{eq:MI1}
    I(\boldsymbol{X};\boldsymbol{Y})\triangleq \frac{1}{M}\sum_{i=1}^{M} \int_{\mathbb{C}^n}  f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x}_{i})\log \frac{f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x}_{i})}{f_{\boldsymbol{Y}}(\boldsymbol{y})}d\boldsymbol{y},
\end{align}
where $\boldsymbol{x}_{i}$ for $i=1,\ldots,M$ form the constellation $\mathcal{X}$.
By applying Monte Carlo integration, the MI can be approximated as
\begin{align}
        I(\boldsymbol{X};\boldsymbol{Y})&\approx \frac{1}{N_{\text{s}}} \sum_{i=1}^{N_{\text{s}}}\int_{\mathbb{C}^n}  f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x}_{i})\log \frac{f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x}_{i})}{f_{\boldsymbol{Y}}(\boldsymbol{y})}d\boldsymbol{y}\\
        &\approx\frac{1}{N_{\text{s}}} \sum_{i=1}^{N_{\text{s}}} \log \frac{f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}_{i}|\boldsymbol{x}_{i})}{f_{\boldsymbol{Y}}(\boldsymbol{y}_{i})},\label{eq:MImontecarlo}
\end{align}
where $\boldsymbol{x}_{i}$ for $i=1,2,\ldots,N_{\text{s}}$ are $N_{\text{s}}$ symbols drawn uniformly from $\mathcal{X}$, and given a certain $\boldsymbol{x}_{i}$, $\boldsymbol{y}_{i}$ is drawn from the conditional distribution of the channel $f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}_{i}|\boldsymbol{x}_{i})$.

To numerically estimate \eqref{eq:MImontecarlo}, the key point is to calculate the marginal distribution of $\boldsymbol{y}$
\begin{align}
     f_{\boldsymbol{Y}}(\boldsymbol{y}) &= \frac{1}{M} \sum_{\boldsymbol{x}\in \mathcal{X}} f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})\label{eq:fy}.
\end{align}
The exact calculation of $f_{\boldsymbol{Y}}(\boldsymbol{y})$ using \eqref{eq:fy} requires storing all $M$ constellation points, which is infeasible when $M$ is very large. Furthermore, approximating $f_{\boldsymbol{Y}}(\boldsymbol{y})$ by standard Monte-Carlo techniques is very inaccurate, because for most realistic channel laws $f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})$, only a tiny fraction of all constellation points $\boldsymbol{x}$ contribute significantly to the sum in \eqref{eq:fy}.

To approximate \eqref{eq:fy}, we propose a method based on \emph{importance sampling}, which is a weighted sampling method that oversamples from the important region we are interested in, thus making Monte Carlo feasible \cite[Ch.~9]{mcbook}. This concept has been used in the machine learning field to estimated the MI between the original data and the learned representation \cite{ML1}, but as far as we know never to estimate the MI of a communication channel. The classic importance sampling approach is to sample from a new importance distribution that is proportional to the product of the distribution of the
integral variable and the integrand \cite[Eq.~(9.3)]{mcbook}, to resemble the true expectation. For example, to simulate \eqref{eq:fy} in our case, the importance distribution of $\boldsymbol{x} \in \mathcal{X}$ should be proportional to $(1/M) f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})$. For the MI estimation problem, we refine this concept by using a step-wise sampling rule, in which we only sample symbols from an important set $\mathcal{I}(\boldsymbol{y})$ which contains points that have an important contribution to $f_{\boldsymbol{Y}}(\boldsymbol{y})$, and no samples are sampled from the complementary set $\mathcal{X}-\mathcal{I}(\boldsymbol{y})$ since their contribution to $f_{\boldsymbol{Y}}(\boldsymbol{y})$ is negligible compared with $\mathcal{I}(\boldsymbol{y})$. The important set $\mathcal{I}(\boldsymbol{y})$ is divided into $D$ disjoint subsets $\mathcal{I}_{d}(\boldsymbol{y})$ for $d=1,\ldots,D$. The contribution to $f_{\boldsymbol{Y}}(\boldsymbol{y})$ from each subset $\mathcal{I}_{d}(\boldsymbol{y})$ can be either calculated by enumerating every symbol in $\mathcal{I}_{d}(\boldsymbol{y})$, or estimated by performing a uniform Monte Carlo sampling in $\mathcal{I}_{d}(\boldsymbol{y})$. Thus, the sum in \eqref{eq:fy} can be approximated as
\begin{align}
     f_{\boldsymbol{Y}}(\boldsymbol{y}) &\approx \frac{1}{M}\sum_{\boldsymbol{x} \in \mathcal{I}(\boldsymbol{y}) } f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})\label{eq:estfy_C}\\
     &\approx \frac{1}{M}\sum_{d=1}^D \frac{|\mathcal{I}_d(\boldsymbol{y})|}{K_d} \sum_{j=1}^{K_d} f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x}_{d,j}) \label{eq:estfy_Cr},
\end{align}
where $\boldsymbol{x}_{d,j}$ for $j=1,\ldots,K_{d}$ are all points from $\mathcal{I}_{d}(\boldsymbol{y})$ if ${K_{d}=|\mathcal{I}_d(\boldsymbol{y})|}$, or $K_{d}$ uniform random samples from $\mathcal{I}_{d}(\boldsymbol{y})$ if ${K_d < |\mathcal{I}_d(\boldsymbol{y})|}$. For accurate results, the sampling sets $\mathcal{I}(\boldsymbol{y})$ and $\mathcal{I}_{d}(\boldsymbol{y})$, but not necessarily their sizes $K_{d}$, should be chosen as functions of $\boldsymbol{y}$.

We propose \eqref{eq:estfy_Cr} as a very general way to estimate $f_{\boldsymbol{Y}}(\boldsymbol{y})$ and thereby the MI and related quantities. The proposed MI estimation method is a special case of importance sampling. Unlike the importance distribution in classical importance sampling, which should be more ``continuous'', our refined step-wise sampling rule makes random generation of $\boldsymbol{x}$ easier. As the number of the elements in $\mathcal{I}(\boldsymbol{y})$ increases, the estimated distribution $f_{\boldsymbol{Y}}(\boldsymbol{y})$ in \eqref{eq:estfy_C} should converge to the exact value. As a special case, setting $\mathcal{I}(\boldsymbol{y}) = \mathcal{X}$ and $D=1$ in \eqref{eq:estfy_Cr} yields the exact expression \eqref{eq:fy} if $K_1 = M$ and a standard Monte Carlo estimate thereof if $K_1 < M$. The same idea can be applied to other structured constellations, or other similar problems. The readers can derive their own estimation rules from \eqref{eq:estfy_Cr} for a specific channel and constellation.

\subsection{MI estimation for the designed VCs}\label{subsec:MIest_forVC}
We consider an $n$-dimensional real AWGN channel, which has the conditional distribution
\begin{align}
    f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})=\frac{1}{(2\pi \sigma^2/n)^{n/2}}\exp{(-\frac{\left\| \boldsymbol{y}-\boldsymbol{x}\right\|^2}{2\sigma^2/n})},
\end{align}
where $\sigma^2$ is the total noise power. We define the signal-to-noise ratio (SNR) as $E_{\text{s}}/\sigma^2$.

For estimating the MI of our VCs with a cubic coding lattice, first, given a received noisy symbol $\boldsymbol{y}$, we define a Euclidean ball $\mathcal{B}(\boldsymbol{y},R)$ containing all $n$-dimensional points in the translated cubic coding lattice having a Euclidean distance within $R$ from the nearest integer vector of $\boldsymbol{y}+\boldsymbol{a}$, i.e.,
\begin{align}
    \mathcal{B}(\boldsymbol{y},R)\triangleq\{\boldsymbol{x}:\|\boldsymbol{x}+\boldsymbol{a}-\left \lfloor \boldsymbol{y}+\boldsymbol{a}  \right \rceil\| \leq R,~\boldsymbol{x}+\boldsymbol{a} \in \mathbb{Z}^n\},
\end{align}
where the squared radius $R^2 \in \mathbb{N}$. The Euclidean ball $\mathcal{B}(\boldsymbol{y},R)$ consists of $R^2+1$ Euclidean ``shells'', each of which contains all $n$-dimensional points in the translated cubic coding lattice having a Euclidean distance of $r$ from $\left \lfloor \boldsymbol{y}+\boldsymbol{a}  \right \rceil$, i.e.,
\begin{align}
\mathcal{S}(\boldsymbol{y},r)\triangleq \{\boldsymbol{x}:  \|\boldsymbol{x}+\boldsymbol{a}-\left \lfloor \boldsymbol{y}+\boldsymbol{a} \rceil \right \| =r, ~\boldsymbol{x} +\boldsymbol{a} \in \mathbb{Z}^n\},
\end{align}
for $r^2=0,1,\ldots,R^2$. The number of points in each shell $\mathcal{S}(\boldsymbol{y},r)$ of a Euclidean ball $\mathcal{B}(\boldsymbol{y},R)$ are listed in Table \ref{tab:numpointsinR}.

\begin{table}[tbp]

  \renewcommand{\arraystretch}{1.3}
  \caption{The number of points in each shell $\mathcal{S}(\boldsymbol{y},r)$ of the Euclidean ball $\mathcal{B}(\boldsymbol{y},R)$ ($n\geq 4 $ in this table).}
  \label{tab:numpointsinR}
  \centering
  \begin{tabular}{c c c c c c c}
    \hline
    $r^2$ & $0$ & $1$ & $2$ & $3$ & $4$ & $\ldots$ \\
    \hline \hline
    $|\mathcal{S}(\boldsymbol{y},r)|$ & $1$ & $2n$ & $2^2{n\choose2}$ & $2^3{n\choose3}$& $ 2^4{n\choose4} + 2{n\choose1}$& $\ldots$\\
    \hline
  \end{tabular}
\end{table}
The important set is defined as all points in $\mathcal{B}(\boldsymbol{y},R)$ that belong to $\Gamma$ at the same time, i.e.,
\begin{align}
    \mathcal{I}(\boldsymbol{y},R)=\mathcal{B}(\boldsymbol{y},R) \cap \Gamma \label{eq:I},
\end{align}
which consists of $D=R^2+1$ disjoint subsets $\mathcal{I}_{d}(\boldsymbol{y})$ for $d=1,\ldots,D$. Each subset $\mathcal{I}_{d}(\boldsymbol{y})$ contains all points in $\mathcal{S}(\boldsymbol{y},\sqrt{d-1})$ that also belong to $\Gamma$, i.e., ${\mathcal{I}_{d}(\boldsymbol{y})=\mathcal{S}(\boldsymbol{y},\sqrt{d-1}) \cap \Gamma}$.

How should $K_{d}$ and $D$ be chosen for accurate estimation? Heuristically, we have found that $K_d = \min\{|\mathcal{I}_d(\boldsymbol{y})|, 10^4\}$ works well for all $d=1,\ldots,D$, which means that the contributions to \eqref{eq:estfy_Cr} from small enough subsets are computed exactly, whereas large subsets are sampled using $10^4$ random points. To determine $D$, we evaluate \eqref{eq:estfy_Cr} for increasing values of $D$ until the relative increase is less than $0.5\%$. More precisely, denoting the estimated $f_{\boldsymbol{Y}}(\boldsymbol{y})$ using \eqref{eq:estfy_Cr} for a certain $D$ by $f_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})$, we choose the smallest integer $D$ for which
\begin{align}
\max\left(  \frac{f_{\boldsymbol{Y}}^{(D+1)}(\boldsymbol{y})-f_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})}{f_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})} \right) < 0.5\%,\label{eq:criteria}
\end{align}
where the maximum is taken over multiple random vectors $\boldsymbol{y}$ and multiple Monte-Carlo realizations of \eqref{eq:estfy_Cr}. If a range of SNRs is being considered, we usually apply the criterion \eqref{eq:criteria} to the lowest SNR, which intuitively needs the largest number of subsets, and then use the obtained value of $D$ throughout the SNR range. Alternatively, \eqref{eq:criteria} can be evaluated separately for each SNR to save complexity. With these parameters, \eqref{eq:estfy_Cr} is able to estimate $f_{\boldsymbol{Y}}(\boldsymbol{y})$ accurately for VCs with up to $M=16777216$ points for all SNRs. For larger constellations where we have no benchmarks to compare with, we conjecture that this method is still valid. After having reliable estimated values of $f_{\boldsymbol{Y}}(\boldsymbol{y})$, the MI can be estimated accurately by \eqref{eq:MImontecarlo}.

\emph{Example 1:} We consider a moderate-size VC based on the lattice partition $\mathbb{Z}^4/16D_{4}$ with $M=131072$ constellation points. For better visualization, we show $pf_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})$ as a function of $D$ upon receiving a noisy symbol $\boldsymbol{y}$ in Fig. \ref{Fig:fyD4}, where $p=(2\pi \sigma^2/n)^{n/2}$ is a constant for a given $\sigma^2$. The benchmark values $pf_{\boldsymbol{Y}}(\boldsymbol{y})$ are calculated using \eqref{eq:fy}. As $D$ increases, the estimated values all converge to the exact values for different SNRs in Fig. \ref{Fig:fyD4}.

\begin{figure}[tbp]
\centering
\begin{tikzpicture}

	\begin{semilogyaxis}[
		xmin=1,
		xmax=21,
		xticklabels={$1/1$,$5/89$,$9/321$,$13/761$,$17/1281$,$21/4785$},
		xtick={1,5,9,13,17,21},
		ymin=5e-8, ymax=1e-3,
		xlabel={$D$ and $|\mathcal{B}(\boldsymbol{y},\sqrt{D-1})|$ },
		ylabel={$pf_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})$},
		ylabel style={at={(axis description cs:-0.01,0.5)}, anchor=north},
		cycle list name=myCycleList,
	    legend pos=north west,
		legend cell align=left,
		legend style={fill=white},
		ylabel style={yshift=.2cm},
		xlabel style={xshift=-.05cm},
		height =0.4\textwidth,
		width=0.48\textwidth,
	]
	\addplot+[lines-1] table[
		x=radius,
		y=snr14,
	] {./figures/fy/estfy_D4.txt};
	\addplot+[lines-2] table[
		x=radius,
		y=snr16,
	] {./figures/fy/estfy_D4.txt};
	\addplot+[lines-3] table[
		x=radius,
		y=snr18,
	] {./figures/fy/estfy_D4.txt};
	\addplot+[lines-4] table[
		x=radius,
		y=snr20,
	] {./figures/fy/estfy_D4.txt};
	\addplot+[lines-5] table[
		x=radius,
		y=snr22,
	] {./figures/fy/estfy_D4.txt};
	\addplot+[lines-6] table[
		x=radius,
		y=snr24,
	] {./figures/fy/estfy_D4.txt};
	\addplot+[lines-7] table[
		x=radius,
		y=snr26,
	] {./figures/fy/estfy_D4.txt};
	\addplot+[lines-8] table[
		x=radius,
		y=snr28,
	] {./figures/fy/estfy_D4.txt};
  	 \addplot+[dashed,color=black,mark=none]  coordinates {(0,0.00036156) (20,0.00036156)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,0.00014394) (20,0.00014394)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,3.9156e-05) (20,3.9156e-05)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,2.0295e-05) (20,2.0295e-05)};
  	 \addplot+[dashed,color=black,mark=none]  coordinates {(0,9.2801e-06) (20,9.2801e-06)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,3.0845e-06) (20,3.0845e-06)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,5.871e-07) (20,5.871e-07)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,7.0641e-08) (20,7.0641e-08)};
  	  \node[shape=rectangle] (a) at (18.5, 5e-4) {\footnotesize{SNR $=14\;$dB}};
  	  \node[shape=rectangle] (a) at (18.5, 2e-4) {\footnotesize{SNR $=16\;$dB}};
  	  \node[shape=rectangle] (a) at (18.5, 5.5e-5) {\footnotesize{SNR $=18\;$dB}};
  	  \node[shape=rectangle] (a) at (18.5, 2.9e-5) {\footnotesize{SNR $=20\;$dB}};
  	  \node[shape=rectangle] (a) at (18.5, 1.4e-5) {\footnotesize{SNR $=22\;$dB}};
  	  \node[shape=rectangle] (a) at (18.5, 4.2e-6) {\footnotesize{SNR $=24\;$dB}};
  	  \node[shape=rectangle] (a) at (18.5, 8e-7) {\footnotesize{SNR $=26\;$dB}};
  	  \node[shape=rectangle] (a) at (18.5, 1e-7) {\footnotesize{SNR $=28\;$dB}};
 	\end{semilogyaxis}
\end{tikzpicture}
\caption{The estimated value $pf_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})$ as a function of $D$ for different SNRs (solid curves with markers). The black dashed lines are the corresponding benchmark values $pf_{\boldsymbol{Y}}(\boldsymbol{y})$. The number after the `/' is the corresponding $|\mathcal{B}(\boldsymbol{y},\sqrt{D-1})|$. }
\label{Fig:fyD4}
\end{figure}

\emph{Example 2:} We consider a larger-size VC of the lattice partition $\mathbb{Z}^8/8E_{8}$ with $M=16777216$ constellation points. Table \ref{tab:r2E8} lists the minimum required $D$ found by \eqref{eq:criteria} and the corresponding number of symbols in $\mathcal{B}(\boldsymbol{y},\sqrt{D-1}))$ at different SNRs, which shows that $|\mathcal{B}(\boldsymbol{y},\sqrt{D-1})|$ grows fast as the SNR decreases. Fig. \ref{Fig:fyE8} shows the convergence of $pf_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})$ to $pf_{\boldsymbol{Y}}(\boldsymbol{y})$ at medium SNR range. We observe from Fig. \ref{Fig:fyE8} that the chosen values of $D$ in Table~\ref{tab:r2E8} are sufficient to approach $f_{\boldsymbol{Y}}(\boldsymbol{y})$ correctly for these SNR values.

\begin{table}

  \setlength{\tabcolsep}{4.5pt}
  \renewcommand{\arraystretch}{1.3}
  \caption{The minimum required $D$ and number of points in $\mathcal{B}(\boldsymbol{y},\sqrt{D-1})$ for the convergence of $f_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})$ to $f_{\boldsymbol{Y}}(\boldsymbol{y})$ for the VC of the lattice partition $\mathbb{Z}^8/8E_{8}$ at different SNRs.}
  \label{tab:r2E8}
  \centering
  \begin{tabular}{c c c c c c c}
    \hline
    SNR & $16$ & $14$ & $12$ & $10$ & $8$ &$6$\\
    \hline \hline
    $D$  & $5$ & $6$ & $8$ & $11$ &$16$& $23$\\
    \hline
    $|\mathcal{B}(8,\sqrt{D-1})|$ &$1713$&  $3729$& $12369$& $47921$& $231185$&$1025649$ \\
    \hline
  \end{tabular}
\end{table}

\begin{figure}[tbp]
\centering
\begin{tikzpicture}

	\begin{semilogyaxis}[
		xmin=1,
		xmax=31,
		ymin=1e-10, ymax=3e-4,
		xlabel={$D$},
		ylabel={$pf_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})$},
		ylabel style={at={(axis description cs:-0.02,0.5)}, anchor=north},
		cycle list name=myCycleList,
	    legend pos=north west,
		legend cell align=left,
		legend style={fill=white},
		ylabel style={yshift=.2cm},
		xlabel style={xshift=-.05cm},
		height =0.4\textwidth,
		width=0.5\textwidth,
	]
	\addplot+[lines-1,mark=none] table[
		x=radius,
		y=snr6,
	] {./figures/fy/estfy_E8.txt};
	\addplot+[lines-2,mark=none] table[
		x=radius,
		y=snr8,
	] {./figures/fy/estfy_E8.txt};
	\addplot+[lines-3,mark=none,mark=none] table[
		x=radius,
		y=snr10,
	] {./figures/fy/estfy_E8.txt};
	\addplot+[lines-4,mark=none] table[
		x=radius,
		y=snr12,
	] {./figures/fy/estfy_E8.txt};
	\addplot+[lines-5,mark=none] table[
		x=radius,
		y=snr14,
	] {./figures/fy/estfy_E8.txt};
	\addplot+[lines-6,mark=none] table[
		x=radius,
		y=snr16,
	] {./figures/fy/estfy_E8.txt};
	\addplot+[lines-7,mark=none] table[
		x=radius,
		y=snr18,
	] {./figures/fy/estfy_E8.txt};
	\addplot+[lines-8,mark=none] table[
		x=radius,
		y=snr20,
	] {./figures/fy/estfy_E8.txt};

	\addplot+[only marks,color=lines-1] table[
		x=radius,
		y=snr6,
	] {./figures/fy/estfy_sampled_E8.txt};
	\addplot+[only marks,color=lines-2] table[
		x=radius,
		y=snr8,
	] {./figures/fy/estfy_sampled_E8.txt};
	\addplot+[only marks,color=lines-3] table[
		x=radius,
		y=snr10,
	] {./figures/fy/estfy_sampled_E8.txt};
	\addplot+[only marks,color=lines-4] table[
		x=radius,
		y=snr12,
	] {./figures/fy/estfy_sampled_E8.txt};
	\addplot+[only marks,color=lines-5] table[
		x=radius,
		y=snr14,
	] {./figures/fy/estfy_sampled_E8.txt};
	\addplot+[only marks,color=lines-6] table[
		x=radius,
		y=snr16,
	] {./figures/fy/estfy_sampled_E8.txt};
	\addplot+[only marks,color=lines-7] table[
		x=radius,
		y=snr18,
	] {./figures/fy/estfy_sampled_E8.txt};
	\addplot+[only marks,color=lines-8] table[
		x=radius,
		y=snr20,
	] {./figures/fy/estfy_sampled_E8.txt};

  	 \addplot+[dashed,color=black,mark=none]  coordinates {(0,7.1861e-05) (30,7.1861e-05)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,1.1446e-05) (30,1.1446e-05)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,2.2074e-06) (30,2.2074e-06)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,6.2954e-07) (30,6.2954e-07)};
  	 \addplot+[dashed,color=black,mark=none]  coordinates {(0,1.105e-07) (30,1.105e-07)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,1.2456e-08) (30,1.2456e-08)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,1.9325e-09) (30,1.9325e-09)};
  	 \addplot+[dashed,color=black,mark=none] coordinates {(0,1.986e-10) (30,1.986e-10)};
  	  \node[shape=rectangle] (a) at (27.5, 1.2e-4) {\footnotesize{SNR $=6\;$dB}};
  	  \node[shape=rectangle] (a) at (27.5, 2e-5) {\footnotesize{SNR $=8\;$dB}};
  	  \node[shape=rectangle] (a) at (27.5, 4e-6) {\footnotesize{SNR $=10\;$dB}};
  	  \node[shape=rectangle] (a) at (27.5, 1.1e-6) {\footnotesize{SNR $=12\;$dB}};
  	  \node[shape=rectangle] (a) at (27.5, 2e-7) {\footnotesize{SNR $=14\;$dB}};
  	  \node[shape=rectangle] (a) at (27.5, 2e-8) {\footnotesize{SNR $=16\;$dB}};
  	  \node[shape=rectangle] (a) at (27.5, 3.5e-9) {\footnotesize{SNR $=18\;$dB}};
  	  \node[shape=rectangle] (a) at (27.5, 3.5e-10) {\footnotesize{SNR $=20\;$dB}};
 	\end{semilogyaxis}
\end{tikzpicture}
\caption{The estimated value $pf_{\boldsymbol{Y}}^{(D)}(\boldsymbol{y})$ as a function of $D$ for different SNRs. Solid lines without markers are estimated with $K_{d}=|\mathcal{I}_d(\boldsymbol{y})|$ for all subsets. The markers are estimated with $K_{d}=10^4$ uniform samples from $\mathcal{I}_d(\boldsymbol{y})$ for subsets with $d>8$ and $K_{r}=|\mathcal{I}_d(\boldsymbol{y})|$ for subsets with $1 \leq d\leq 8$. The black dashed lines are the corresponding benchmark values $pf_{\boldsymbol{Y}}(\boldsymbol{y})$.}
\label{Fig:fyE8}
\end{figure}

\subsection{LLR approximation for very large constellations}
We propose an LLR approximation method for very large constellations and exemplify it for our designed VCs and the scaled VCs.

For transmission of a constellation $\mathcal{X}$ through a given channel with conditional distribution $f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})$, upon receiving a noisy symbol $\boldsymbol{y}$, the LLR of a certain bit $b_{i}$, for $i=1,2,\ldots,\log_{2}(|\mathcal{X}|)$, is defined as
\begin{align}
    \text{LLR}(b_{i}|\boldsymbol{y})&\triangleq\log{\frac{\text{Pr}(b_{i}=0|\boldsymbol{y})}{\text{Pr}(b_{i}=1|\boldsymbol{y})}}\notag\\
    &=\log \frac{\sum_{\boldsymbol{x}\in \mathcal{X}^{(i,0)}}f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})}{\sum_{\boldsymbol{x}\in \mathcal{X}^{(i,1)}}f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})}\label{eq:exactLLR},
\end{align}
where $\mathcal{X}^{(i,0)}$ and $\mathcal{X}^{(i,1)}$ are the sets of constellation points with $0$ and $1$ at position $i$, respectively. The exact LLR calculated using the whole constellation is accurate but too complex to compute, which can be approximated by only considering the most likely constellation point with $b_{i}=0$ (or $b_{i}=1$), i.e.,
\begin{align}
    \text{LLR}(b_{i}|\boldsymbol{y})\approx \log \frac{\max_{\boldsymbol{x}\in \mathcal{X}^{(i,0)}}f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})}{\max_{\boldsymbol{x}\in \mathcal{X}^{(i,1)}}f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})}.\label{eq:LLR}
\end{align}
However, for very large constellations, searching for the most likely constellation point with $b_{i}=0$ and $b_{i}=1$ from the whole constellation is still infeasible.

We propose a similar technique as our MI estimation method in \ref{subsec:MIest_method} to further approximate the LLR. Instead of searching points from $\mathcal{X}$, we only search the closest point from an important set $\mathcal{I}(\boldsymbol{y})$. Specifically, the LLR in \eqref{eq:LLR} is further approximated as
\begin{align}
    \text{LLR}(b_{i}|\boldsymbol{y})\approx \log \frac{\max_{\boldsymbol{x}\in \mathcal{I}^{(i,0)}(\boldsymbol{y})}f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})}{\max_{\boldsymbol{x}\in \mathcal{I}^{(i,1)}(\boldsymbol{y})}f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})},\label{eq:approxLLR}
\end{align}
where $\mathcal{I}^{(i,0)}(\boldsymbol{y}) = \mathcal{X}^{(i,0)} \cap \mathcal{I}(\boldsymbol{y})$ and $\mathcal{I}^{(i,1)}(\boldsymbol{y}) = \mathcal{X}^{(i,1)} \cap \mathcal{I}(\boldsymbol{y})$. If there is no constellation point in $\mathcal{I}^{(i,0)}(\boldsymbol{y})$ (or $\mathcal{I}^{(i,1)}(\boldsymbol{y})$), we assume the most likely constellation point has a small probability, i.e., setting $\max_{\boldsymbol{x}\in \mathcal{I}^{(i,0)}(\boldsymbol{y})}f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})$ (or $\max_{\boldsymbol{x}\in \mathcal{I}^{(i,1)}(\boldsymbol{y})}f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})$) to a small
default value.

For the AWGN channel, the widely-used approximate LLR derived from \eqref{eq:LLR} is
\begin{align}
    &\text{LLR}(b_{i}|\boldsymbol{y})\approx \notag\\ &-\frac{1}{2\sigma^2/n}\left (\min_{\boldsymbol{x}\in\mathcal{X}^{(i,0)}}(\|\boldsymbol{y}-\boldsymbol{x}\|^2)- \min_{\boldsymbol{x}\in\mathcal{X}^{(i,1)}}(\|\boldsymbol{y}-\boldsymbol{x}\|^2)\right),\label{eq:LLR_awgn}
\end{align}
where only the nearest constellation point with $b_{i}=0$ (or $b_{i}=1$) to $\boldsymbol{y}$ are considered \cite[Eq.~(6)]{viterbi98}. Analogously with \eqref{eq:approxLLR}, we further approximate \eqref{eq:LLR_awgn} as
\begin{align}
    &\text{LLR}(b_{i}|\boldsymbol{y})\approx \notag \\ &-\frac{1}{2\sigma^2/n}\left (\min_{\boldsymbol{x}\in\mathcal{I}^{(i,0)}(\boldsymbol{y})}(\|\boldsymbol{y}-\boldsymbol{x}\|^2)- \min_{\boldsymbol{x}\in\mathcal{I}^{(i,1)}(\boldsymbol{y})}(\|\boldsymbol{y}-\boldsymbol{x}\|^2)\right),\label{eq:LLR_awgn_approx}
\end{align}

For our designed VCs with a cubic coding lattice, the important set is defined as in \eqref{eq:I} with a radius $R$. This parameter provides a trade-off between low computational complexity (small $R$) and good decoding performance (high $R$). If ${\mathcal{I}^{(i,0)}(\boldsymbol{y})=\varnothing}$, then setting ${\max_{\boldsymbol{x}\in \mathcal{I}^{(i,0)}(\boldsymbol{y})}f_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})}$ to a small probability is equivalent to setting ${\min_{\boldsymbol{x}\in \mathcal{I}^{(i,0)}(\boldsymbol{y})}\|\boldsymbol{y}-\boldsymbol{x}\|^2}$ to a large default value $q$ which is larger than $R^2$, and the same rule applies to $\mathcal{I}^{(i,1)}(\boldsymbol{y})$. A default value close to the boundary of the important region (e.g., $q=R^2+1$) is usually not a good choice, neither a very large $q$ (e.g., $q=100 R^2$). The decoding performance can be roughly optimized by testing different $q$ for a given $R$.

Similarly, for very large scaled VCs, the important set can be generalized as
\begin{align}
    \mathcal{I}(\boldsymbol{y})\triangleq \{\boldsymbol{x}:\|\boldsymbol{x}+\boldsymbol{a}-\mathcal{Q}_{\Lambda}(\boldsymbol{y}+\boldsymbol{a})\| \leq R,\boldsymbol{x} \in \Gamma\}.
\end{align}
Then \eqref{eq:approxLLR} can be used to estimate the LLR for the scaled VCs.

\subsection{Results}
 Fig. \ref{Fig:MI} shows the estimated MI for multidimensional VCs using our proposed estimation method in \ref{subsec:MIest_method} and parameters are chosen as suggested in \ref{subsec:MIest_forVC}. To validate the correctness of our estimation, the MI simulated with the exact $f_{\boldsymbol{Y}}(\boldsymbol{y})$ in \eqref{eq:fy} for some moderate-size VCs are included as benchmarks. It shows that our estimated MI is consistent with the benchmark MIs (markers). With $L_{32}$ as the shaping lattice, at SNR $=45$ dB, we can observe the maximum $0.935$ dB shaping gain. At SNR $=25$ dB, the gap to capacity is reduced from $1.33$ dB to $0.48$ dB compared with the QAM constellation.

\begin{figure}[tbp]
\centering
\begin{tikzpicture}

	\begin{axis}[
		xmin=5,
		xmax=48,
		ymin=2, ymax=16,
		xlabel={SNR [dB]},
		ylabel={MI [bits/symbol/dimension-pair]},
		ylabel style={at={(axis description cs:-0.01,0.5)}, anchor=north},
		cycle list name=myCycleList,
	    legend pos=north west,
		legend cell align=left,
		legend style={fill=white,fill opacity=0.4, draw opacity=1,text opacity=1},

		ylabel style={yshift=.1cm},
		xlabel style={xshift=-.05cm},
		height =0.4\textwidth,
		width=0.5\textwidth,
	]
	\addplot+[black,mark=none] table[
		x=SNR,
		y=C,
	] {./figures/MI/Capacity.txt};\addlegendentry{AWGN\\Capacity}
	\addplot+[black, dashed ,mark=none] table[
		x=SNR,
		y=I,
	] {./figures/MI/I_qam.txt};\addlegendentry{QAM}
	\addplot+[lines-1,mark=none] table[
		x=SNR,
		y=I,
	] {./figures/MI/I_D4.txt};\addlegendentry{$\mathbb{Z}^4/2^4D_{4}$}

	\addplot+[lines-2,mark=none] table[
		x=SNR,
		y=I,
	] {./figures/MI/I_E8.txt};\addlegendentry{$\mathbb{Z}^8/2^3E_{8}$}
	\addplot+[lines-3,mark=none] table[
		x=SNR,
		y=I,
	] {./figures/MI/I_L32_r4.txt};\addlegendentry{$\mathbb{Z}^{32}/2^4L_{32}$}
	\addplot+[lines-4,mark=none] table[
		x=SNR,
		y=I,
	] {./figures/MI/I_L32_r5.txt};\addlegendentry{$\mathbb{Z}^{16}/2^5L_{32}$}
	\addplot+[lines-5,mark=none] table[
		x=SNR,
		y=I,
	] {./figures/MI/I_L32_r6.txt};\addlegendentry{$\mathbb{Z}^{32}/2^6L_{32}$}
	\addplot+[lines-6,mark=none] table[
		x=SNR,
		y=I,
	] {./figures/MI/I_L32_r7.txt};\addlegendentry{$\mathbb{Z}^{32}/2^7L_{32}$}

	\addplot+[only marks,color=lines-1,mark=|] table[
		x=SNR,
		y=I,
	] {./figures/MI/Ireal_D4.txt};
	\addplot+[only marks,color=lines-2] table[
		x=SNR,
		y=I,
	] {./figures/MI/Ireal_E8.txt};

    \node[anchor=east] (source1) at (axis cs:40,14){};
      \node[] (destination1) at (axis cs:42.9,14){};
      \node[] (top1) at (axis cs:42.2,14.55){};
      \node[] (bot1) at (axis cs:42.2,13.45){};
      \draw[line width=0.2mm,-](top1)--(bot1);
      \draw[line width=0.3mm,dashed,dash pattern=on 0.8mm off 0.4mm,->](source1)--(destination1);

    \node[anchor=west] (source2) at (axis cs:46,14){};
      \node[] (destination2) at (axis cs:43.2,14){};
      \node[] (top2) at (axis cs:43.8,14.55){};
      \node[] (bot2) at (axis cs:43.8,13.45){};
      \draw[line width=0.2mm,-](top2)--(bot2);
       \draw[line width=0.3mm,dashed,dash pattern=on 0.8mm off 0.4mm,->](source2)--(destination2);
    \node[shape=rectangle] (b) at (37,14.5) {\footnotesize{$1.53$ dB}};

    \node[anchor=east] (source3) at (axis cs:22.5,8.15){};
      \node[] (destination3) at (axis cs:25.2,8.15){};
     \node[] (top3) at (axis cs:24.4,8.66){};
      \node[] (bot3) at (axis cs:24.4,7.64){};
      \draw[line width=0.2mm,-](top3)--(bot3);
       \draw[line width=0.3mm,dashed,dash pattern=on 0.8mm off 0.4mm,->](source3)--(destination3);
    \node[anchor=west] (source4) at (axis cs:27,8.15){};
      \node[] (destination4) at (axis cs:24.5,8.15){};
      \node[] (top4) at (axis cs:25.1,8.66){};
      \node[] (bot4) at (axis cs:25.1,7.64){};
      \draw[line width=0.2mm,-](top4)--(bot4);
       \draw[line width=0.3mm,dashed,dash pattern=on 0.8mm off 0.4mm,->](source4)--(destination4);
    \node[shape=rectangle] (b) at (23,8.8) {\footnotesize{$0.48$ dB}}{};

    \node[anchor=east] (source5) at (axis cs:41.5,14.5){};
      \node[] (destination5) at (axis cs:44.9,14.5){};
     \node[] (top5) at (axis cs:44.15,15.05){};
      \node[] (bot5) at (axis cs:44.15,13.95){};
      \draw[line width=0.2mm,-](top5)--(bot5);
       \draw[line width=0.3mm,dashed,dash pattern=on 0.8mm off 0.4mm,->](source5)--(destination5);
    \node[anchor=west] (source6) at (axis cs:48,14.5){};
      \node[] (destination6) at (axis cs:44.8,14.5){};
      \node[] (top6) at (axis cs:45.5,15.05){};
      \node[] (bot6) at (axis cs:45.5,13.95){};
      \draw[line width=0.2mm,-](top6)--(bot6);
       \draw[line width=0.3mm,dashed,dash pattern=on 0.8mm off 0.4mm,->](source6)--(destination6);
    \node[shape=rectangle] (b) at (42.8,15.2) {\footnotesize{$0.935$ dB}}{};
 	\end{axis}
\end{tikzpicture}
\caption{The estimated MI as a function of the SNR for multidimensional VCs (solid curves without markers). The markers are the MI estimated using the exact $f_{\boldsymbol{Y}}(\boldsymbol{y})$ by \eqref{eq:fy}.}
\label{Fig:MI}
\end{figure}

 We also investigate the performance of our VCs in coded systems. An LDPC code\footnote{The codeword length is $64800$ and $50$ decoding iterations are used.} from the digital video broadcasting (DVB-S2) standard \cite{dvbs2} with multiple code rates are applied to our designed VCs. Our proposed LLR approximation in \eqref{eq:LLR_awgn_approx} is applied. Fig.~\ref{Fig:LDPC} shows the estimated required SNRs to achieve a BER below $10^{-4}$ after LDPC decoding for the VCs based on the lattice partition $\mathbb{Z}^4/2^6D_{4}$ ($R^2=20,\;q=50$), $\mathbb{Z}^8/2^6E_{8}$ ($R^2=6,\;q=20$), $\mathbb{Z}^{16}/2^6\Lambda_{16}$ ($R=3,\;q=20$), and $\mathbb{Z}^{32}/2^6L_{32}$ ($R^2=2,\;q=13$). The default values $q$ are not globally optimized for different VCs, since our goal is not to design an LDPC code to maximize the AIRs. For comparison, we present results for the scaled VC based on the lattice partition $D_{4}/2^4D_{4}$ and our VC based on the lattice partition $\mathbb{Z}^{4}/2^4D_{4}$ simulated using the LLR in \eqref{eq:LLR_awgn}, i.e., without the further approximation in \eqref{eq:LLR_awgn_approx}. The result shows that our VC with a cubic coding lattice can have slightly higher AIRs than the scaled VC. In this case, the loss of coding gain due to the cubic coding lattice is more than compensated by the usage of a lower-rate LDPC code to obtain the same net AIR, which is consistent with the BER improvements observed in \cite[Fig.~4]{ourISIT}. The better AIR performance might come from that a pseudo-Gray labeling is more efficient for the cubic coding lattice, since our VC $\mathbb{Z}^{4}/2^4D_{4}$ has a Gray penalty of $1.01$, which is half of that for the scaled VC $D_{4}/2^4D_{4}$ ($1.99$).
\begin{figure}[htbp]
\centering
\begin{tikzpicture}

	\begin{axis}[
		xmin=15,
		xmax=50,
		ymin=0, ymax=14,
		xlabel={SNR [dB]},
		ylabel={AIR [bits/symbol/dimension-pair]},
		ylabel style={at={(axis description cs:-0.01,0.5)}, anchor=north},
		cycle list name=myCycleList,
	    legend pos=south east,
		legend cell align=left,
		legend style={fill=white, fill opacity=0.4, draw opacity=1,text opacity=1},
		ylabel style={yshift=.1cm},
		xlabel style={xshift=-.05cm},
		height =0.4\textwidth,
		width=0.5\textwidth,
	]
	\addplot+[black,mark=none] table[
		x=SNR,
		y=C,
	] {./figures/MI/Capacity.txt};\addlegendentry{AWGN Capacity}
	\addplot+[black,dashed,mark=none] table[
		x=SNR,
		y=I,
	] {./figures/MI/I_qam.txt};\addlegendentry{MI QAM}
	\addplot+[lines-1] table[
		x=SNR,
		y=I,
	] {./figures/LDPC/Icoded_D4_cubic_whole.txt};\addlegendentry{LDPC $\mathbb{Z}^4/2^4D_{4}$}
	\addplot+[dashed,lines-1,mark=star,mark options=solid] table[
		x=SNR,
		y=I,
	] {./figures/LDPC/Icoded_D4_fine_whole.txt};\addlegendentry{LDPC $D_{4}/2^4D_{4}$}
	\addplot+[lines-2] table[
		x=SNR,
		y=I,
	] {./figures/LDPC/Icoded_D4_cubic.txt};\addlegendentry{LDPC $\mathbb{Z}^4/2^6D_{4}$}
	\addplot+[lines-3] table[
		x=SNR,
		y=I,
	] {./figures/LDPC/Icoded_E8_cubic.txt};\addlegendentry{LDPC $\mathbb{Z}^8/2^6E_{8}$}
	\addplot+[lines-4,mark=diamond] table[
		x=SNR,
		y=I,
	] {./figures/LDPC/Icoded_L16_cubic.txt};\addlegendentry{LDPC $\mathbb{Z}^{16}/2^6\Lambda_{16}$}
	\addplot+[lines-5] table[
		x=SNR,
		y=I,
	] {./figures/LDPC/Icoded_L32_cubic.txt};\addlegendentry{LDPC $\mathbb{Z}^{32}/2^6L_{32}$}

 	\end{axis}
\end{tikzpicture}
\caption{The estimated AIR as a function of the SNR for multidimensional VCs with the DVB-S2 LDPC codes. The code rate $R_{\text{c}}\in \{1/3, 1/2, 3/5, 2/3, 3/4, 5/6, 9/10\}$. }
\label{Fig:LDPC}
\end{figure}

\section{Conclusion}
We proposed Voronoi constellations with a cubic coding lattice, for which we follow Kurkoski's encoding and decoding algorithms and apply pseudo-Gray labeling to minimize the BER. As a structured geometric shaping method, this class of VC has high shaping gains of up to $1.03$ dB and low complexity. Thanks to its cubic coding lattice, an MI estimation method for very large constellation size based on importance sampling is proposed for the first time, which enables us to observe an up to $0.85$ dB shaping gain for medium SNR values. Our proposed LLR approximation method makes the analysis of AIRs of our VCs in coded systems at high spectral efficiencies possible, which is infeasible for the scaled VCs. Compared with the conventional scaled VCs, our designed VCs have the advantages that 1) the decoding algorithm is simpler, 2) the spectral efficiency for mapping integers to bits is improved to $1$ [bit/symbol/dimension-pair] realized by rotating and scaling the shaping lattices, 3) the AIRs after combining with an LDPC code can be higher due to our better pseudo-Gray labeling, and 4) analysing the MI and AIR becomes feasible for very large constellation sizes.

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
People on the move, 10/16
By The Denver Post | newsroom@denverpost.com
SCIENTIFIC ASSEMBLY OF THE AMERICAN ACADEMY OF FAMILY PHYSICIANS: Recently honored Dr. Jeffrey J. Cain, chief of family medicine at The Children's Hospital, as the first-place winner for best research presentation overall.
THE WIENS GROUP: Appointed Eric R. Jonsen as general counsel. Jonsen has been in private practice since 1985, specializing in trials involving commercial disputes, complex litigation, insurance and real estate.
THE INSTITUTE OF MEDICINE OF THE NATIONAL ACADEMIES: Recently inducted 64 new members, including Dr. Richard D. Krugman, dean of the School of Medicine at the University of Colorado at Denver and Health Sciences Center. Krugman was nominated for his international prominence in the field of child-abuse prevention.
SPYDER ACTIVE SPORTS INC.: Promoted Cees DeWitte to executive vice president of global sales, Christophe Weissenberger to director of sales-Europe and Joe Vernachio to executive vice president of product. New to Spyder is Jeff Sale, as director of North American specialty sales; Peter Valles, creative director; and Mike Hyp, director of graphic design-Venom.
AMERICAN ASSOCIATION OF ORAL AND MAXILLOFACIAL SURGEONS: Re-elected Dr. Steven R. Nelson of Denver to a third one-year term as speaker of its House of Delegates.
PIKES PEAK ASSOCIATION OF REALTORS: Rosalinda Chaney, a team leader for Keller Williams Premier Realty, was recently installed as the new president of the chapter.
WIND CREST: The Highlands Ranch retirement community named Mike Conway as director of general services.
THE SPITFIRE GROUP: Named Christopher Boyer as director of business development.
AZTEK NETWORKS (formerly Aztek Engineering): Announced that Steven M. Bruny has been named chief executive, succeeding company founder Mark Elder, who will continue as chief technical officer. Aztek also named Spencer Punter as vice president of marketing and business development.
SOCIETY OF MARKETING PROFESSIONAL SERVICES: The Colorado Chapter honored Dena Wyatt, principal and owner of Marketing Evolutions Inc., with the Leonardo Award, the highest award given by SMPS-Colorado.
FACILITYLOGIC: Lynette Simmons has joined the firm as director of construction services and project management.
The Denver Post
The Denver Post is a nine-time Pulitzer Prize-winning newspaper that has covered the Rocky Mountain Empire since 1892.
newsroom@denverpost.com
Intel building $20 billion Ohio computer chip facility amid global shortage
From space data to a beer garden, economic development commission approves $9 million in incentives

redpajama_set_name: RedPajamaCommonCrawl


text:
Here's How Sheikh Mo is Helping the Victims of the Sierra Leone Landslide
Shiekh Mo
Sheikh Mo Sierra Leone Landslide
8 October, 2017 by Georgia Bobley
In August, more than 1,000 people died in a landslide in Sierra Leone's capital, Freetown. In addition to the deaths, more than 3,000 people were left homeless. As the African city struggles to regain some sense of normalcy, Sheikh Mo has stepped in to offer aid.
Yesterday, the Dubai royal ordered an Airbus A330 carrying 50 metric tons of humanitarian aid to be sent to Sierra Leone. The plane departed, carrying basic relief items such as food, water, medical supplies, mosquito nets, and blankets, from Al Maktoum Airport.
. @HHShkMohd orders aid airlift to Sierra Leone to deliver core relief items to communities affected by a devastating landslide. #Dubai pic.twitter.com/6cXZn5ACl6
— Dubai Media Office (@DXBMediaOffice) October 7, 2017
"They are struggling to rebuild their lives and critical humanitarian support is still needed," said Jagan Chapagain, International Federation of Red Cross and Red Crescent Societies' (IFRC) Under Secretary General for Programs and Operations.
The supplies were gathered at the International Humanitarian City in Dubai (IHC), which was founded in 2003. According to The National, the organization is chaired by Princess Haya and is also the largest humanitarian hub in the world.
Sheikh Mo regularly steps in to offer aid when disasters strike internationally-- just last month, he sent aid to hurricane victims in the US, after the UAE embassy evacuated Emiratis from Florida before the hurricanes struck.
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text:
Q: Taylor Series and Differentiation with Sigma notation $f(x) = \frac{x}{(2-3x)^2}$ Use Term By Term Differentiation to Find the Taylor Series about $x$=3 for
Give The Open Interval of Convergence and express as sigma notation
 $\sum A_n(x-3)^n$
$f(x) = \frac{x}{(2-3x)^2}$
So I have Found the Taylor Series for 1/(2-3x) to be
$\sum{(-3)^{n}\cdot(x-3)^{n} }\cdot{(-7)^ {n+1}}$
How Do you find the original function taylor series and its interval of convergence and then express in sigma notation of the form  $\sum A_n(x-3)^n$

A: Since you are concerned by the Taylor series of $$f(x) = \frac{x}{(2-3x)^2}$$ around $x=3$, just as able answered, you can start using $x-3=y$, $x=y+3$, $2-3x=-3y-7$ and consider the  Taylor series of $$f(y) = \frac{y+3}{(3y+7)^2}$$ around $y=0$.
May be, a convenient way could then be to write $$y+3=(3y+7)^2\sum_{i=0}^\infty a_i\,y^i$$ and identify the coefficients. Developing the last expression, we then have $$3+y=9\sum_{i=0}^\infty a_i\,y^{i+2}+42\sum_{i=0}^\infty a_i\,y^{i+1}+49\sum_{i=0}^\infty a_i\,y^i$$ and we need to identify the $a_i$ coefficients.
For the terms of degree $0$ and $1$ present in the lhs, we then have $$3=49a_0\implies a_0=\frac 3{49}$$ $$1=42a_0+49a_1\implies a_1=-\frac{11}{343}$$ For all other terms $$49a_n+42a_{n-1}+9a_{n-2}=0$$ So, we have the recursion for which you must notice that the characteristic equation has a double root equal to $-\frac 37$.
Using the initial conditions, the standard method then leads to $$a_n=\frac{1}{147} \left(-\frac 37\right)^n(2n+9)$$ and finally $$f(x) = \frac{x}{(2-3x)^2}=\frac{1}{147}\sum_{i=0}^\infty \left(-\frac 37\right)^n(2n+9)(x-3)^i$$

A: you can make a change of variable $x = 3 + h.$  then we have
$$\begin{align} f(x) &= \frac{x}{(2-3x)^2} = \frac{3+h}{(2 - 9-3h)^2}=\frac 1{49}(3+h)\left(1 + \frac{3h}7\right)^{-2} \\
&=\frac 1{49}(3+h)\left(1 -\frac 21 \frac{3h}7 + \frac{2\cdot3}{1\cdot2}\left(\frac{3h}{7}\right)^2 - \frac{2\cdot3 \cdot 4}{1\cdot2\cdot 3}\left(\frac{3h}{7}\right)^2 + \ldots\right)   \end{align} $$
i will let you multiply out the two factors and collect the like terms.

redpajama_set_name: RedPajamaStackExchange


On behalf of The Ellis Firm, APLC posted in Nursing Home Neglect on Friday, October 4, 2013.
People in San Diego may have seen a recent editorial about the state of politics in Washington, D.C. and Sacramento and that policies that are decided at the state and federal level often come with significant corporate backing by way of campaign contributions and lobbying efforts. While this has been an inescapable reality of politics-as-usual for many decades, the columnist worries that this issue may be directly impacting the quality of care that nursing home patients in California receive. When the policies are influenced by large corporations that have a financial stake in the outcome of health care legislation, oftentimes the losers are those who are the patients of these corporations.
The article noted that certain California lobbying groups have donated a tremendous amount of money to political candidates and that in recent years such groups have enjoyed a number of favorable concessions under California law. For example, severely disabled persons in California should have the option of receiving home-based care as opposed to nursing home care but a majority of these people are shuffled into more expensive nursing homes, wherein the state willingly pays much higher sums to these nursing home operators.
Meanwhile, the cost of nursing home care has skyrocketed over the last 5 years from about $67,000 per year to around $83,000. However, according to the writer, the quality of care does not seem to be improving. One nurse described the nursing home neglect she saw while on the job, claiming that her patients were often denied the reasonable standard of care they deserve in order to seemingly protect the nursing home operator's bottom line.
California disabled people and senior citizens deserve the best care available. Whether due to the issues argued by the columnist or more direct negligence on the part of a care provided, some patients will unfortunately end up being subjected to nursing home abuse. Victims of such abuse or neglect may wish to contact an experienced personal injury attorney who can review the situation and help patients recover the compensation, and dignity, they deserve.


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text:
Tag Archives: data compilations
Some updates
So I have fallen a bit behind in my promised extended discussion of the iiNet case. I'm halfway through the full judgment, but have unfortunately been sidetracked with some other work-related tasks.
There has been no shortage of interesting copyright judgments in the last week. In addition to the iiNet decision, there has been the decision of Larrikin Music Publishing Pty Ltd v EMI Songs Australia Pty Limited [2010] FCA 29 (where the band Men At Work was found to have infringed Larrikin's copyright in the children's song 'Kookaburra Sits in the Old Gum Tree', by using the flute riff in their famous 'Down Under' song), and on Monday, in Telstra Corporation Limited v Phone Directories Company Pty Ltd [2010] FCA 44, a single judge of the Federal Court held that copyright does not subsist in Telstra's Yellow Pages and White Pages directories (choosing to follow IceTV rather than Desktop Marketing).
These are all important decisions, and I have every intention of getting to them (for reading and blogging in detail) as soon as I can. In the meantime, if you are interested in reading some updates and discussions, I'd recommend viewing Nic Suzor's blog on the iiNet decision (1, 2, and 3) and Warwick Rothnie's post on the Telstra decision.
data compilationsiiNet; authorisation; copyright
ANDS guides – copyright and data
October 23, 2009 kylie
To follow on from my copyright and data presentation post –
Professor Anne Fitzgerald and I have produced two short guides for the Australian National Data Service (ANDS): one on Copyright and Data and the other on Creative Commons and Data. The Copyright and Data guide is now available (in html and pdf formats) from the ANDS website, the Creative Commons and Data guide should (hopefully) be available next week.
ANDSdata compilations
Presentation: Copyright and Data
This morning I gave a presentation on Copyright and Data as part of QUT's Division of Technology, Information and Learning Support research seminars.
I have licensed my presentation under a Creative Commons Attribution 2.5 Australia licence. It is available for download here:
Copyright and Data (.pptx) or Copyright and Data (.pdf)
Together with Professor Anne Fitzgerald, I have also authored a short guide on Copyright and Data for the Australian National Data Service (ANDS). It is available from the ANDS website.
data compilations
Copyright protection and data compilations
April 2, 2009 kylie
Over on the Digital Curation Blog, Chris Rushbridge has an interesting post entitled, "Are research data facts and does it matter?"
I have just posted a response, which I am reproducing here:
I would like to take this opportunity to explain some of the research we have undertaken in the OAK Law Project and conclusions we have reached regarding copyright protection of data compilations in Australia. We have two primary publications addressing this area: Building the Infrastructure for Data Access and Reuse in Collaborative Research: An Analysis of the Legal Context and Practical Data Management: A Legal and Policy Guide.
s10(1) of the Australian Copyright Act 1968 defines a literary work to include a "compilation". This is where protection for data compilations under Australian law derives from. Any data that is collected, arranged, organised and presented in a logical fashion will usually be regarded as a compilation.
Chris makes a good point that many data compilations will require a great deal of effort, analysis and creativity. In the US, creativity is a requirement before a data compilation can be protected by copyright. In Australia, creativity is not required. Only that the compilation is a result of the exercise of skill, knowledge or judgment in the arrangement of the data, or the investment of substantial labour or expense in collection the material (Desktop Marketing v Telstra).
It can often be difficult to tell whether a compilation is one that would attract copyright protection. In our work, we have tended to err on the side of caution and assume that most compilations will attract copyright protection. This is because the threshold in Australia is so low. The main case in this area, Desktop Marketing v Telstra, involved the copying of a telephone directory. A telephone directory is merely a compilation of names and numbers listed in alphabetical order. If this is a compilation that attracts copyright, then most other compilations are likely to be protected by copyright under Australian law as well.
Copyright law does not protect mere facts or information. Rather, it protects the expression of facts or information in a material form. This means that generally there would not be a problem with copying some of the basic facts contained in a compilation. For example, if I were to list the names and numbers of a small collection of my colleagues on my website, that would not usually be a problem. I have extracted the data that I need, in a fairly "random" fashion (in that I have not just copied a few pages of names and numbers in alphabetical order directly from the White Pages). I have not copied the way that the data is arranged in the telephone directory (the "expression").
In regards to Science Commons' decision to discontinue advocating the application of Creative Commons licences to data compilations, my understanding is that they came to this decision for two reasons:
(1) It was not always clear in the US whether the relevant compilation attracted copyright. If it did not but a person had put a CC licence on the compilation in the mistaken belief that it did, then restrictions would have been imposed on that dataset (e.g. that it could only be used non-commercially) which actually had no legal basis for being imposed; and
(2) CC licences all contain an attribution requirement and Science Commons were concerned about what they call "attribution stacking" – i.e. where a dataset is compiled from data contributed by many different researchers, it would be extremely difficult for a user to attribute all of those researchers.
At OAK Law, we still believe that CC licences can be applied to datasets in Australia because the concerns noted by Science Commons do not arise to the same degree in Australia. Firstly, we have a lower threshold test for copyright protection, meaning that copyright will more readily attach to datasets in Australia and the first problem noted by Science Commons is less likely to occur. Nevertheless, to be sure, we usually advocate that the widest CC licence – the attribution only licence – be applied to datasets. Secondly, unlike in the US, Australian copyright law includes Moral Rights, meaning that creators have to be attributed anyway, regardless of whether a CC licence is applied or not. We think there are various ways of getting around the "attribution stacking" problem – for example, a group of researchers could agree on a common way to be attributed (e.g. we could be attributed as "the OAK Law Project"), or the data could be attributed using a URL, which an interested party can visit and which can list all the contributors (and this list can be added to over time). The advantage of applying CC licences to data, in our view, is that it provides some certainty to users about what they can and cannot do with that data.
If you are interested in this issue, I would also advise reading these posts by Robin Rice and Rufus Pollock.


Chris Hurn - Entrepreneur, Speaker, Writer/Blogger, Leader. Mercantile Capital Corporation only finances commercial real estate for other entrepreneurs and small business owners all over the U.S. utilizing the SmartChoice Commercial Loan program (aka the SBA 504 loan). Christopher G. Hurn, chief executive officer of Mercantile Capital Corporation in Orlando, is the author of The Entrepreneur's Secret to Creating Wealth: How the Smartest Business Owners Build Their Fortunes, his first book. He also has a special offer for entrepreneurs: buy two or more copies of his new book, and he will donate all proceeds to the NFIB Young Entrepreneur Foundation. The book, which sells for $19.95, offers Hurn's unique perspective on wealth he gained lending money - over $424.2 million in 2012 - to fund the efforts of small business entrepreneurs to acquire or develop their own facilities. Hurn, a frequent guest as a small business commentator on Fox Business News, The Huffington Post, PBS and other media outlets, said he knows how small business entrepreneurs think because he is one.


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Considered one of the best in the area, West Olympia Kindercare is sure to meet all of your wants and needs. First and foremost, they understand how difficult it may be to leave your child with someone else and want to make your child's transition as easy and calm as possible. Putting you and your child at ease, is just the first of many reasons why Kindercare is perfect for you! Offering programs for all ages beginning at infants and going all the way to pre-kindergarten. Each one contributes something entirely different for your child's growing experience. The director is extremely nice, and the teachers are always helpful and nurturing. Making sure your child has a fun, secure place to explore is their number one priority, and their passion for creating such an environment is truly amazing.
This relatively large center, is full of the most positive, and unbelievably friendly staff around! Inside you'll find roughly three classrooms per age group, each one completed with awesome teachers, and eager to learn children. The facility is bright and welcoming, offering an abundance of educational experiences for children of all ages. Serendipity Children's Center is proud of their unique style of teaching, and commitment to each child as an individual. Even more unique are the numerous classes offered for an extra charge. Sign your child up for various sports, ballet, swim lessons, and more! These additional enrichment opportunities will save you hassle of running around town! What are you waiting for!? Check out Serendipity now!


Join Project Manager, Dave Fenity for a Bohn Park Progress Tour, on Tuesday, August 8!

Please help the Town get a sense of who is attending, by filling out the form below.

 All attendees must wear closed-toe shoes (it is a working construction site). Plan for about 45 minutes. Meet by the multi-purpose court (tennis/basketball) off 2nd Ave. Please park at the Wastewater Treatment Plant, across the street.
Join Project Manager, Dave Fenity for a Bohn Park Progress Tour, on Tuesday, August 8!
Please help the Town get a sense of who is attending, by filling out the form below.
All attendees must wear closed-toe shoes (it is a working construction site). Plan for about 45 minutes. Meet by the multi-purpose court (tennis/basketball) off 2nd Ave. Please park at the Wastewater Treatment Plant, across the street.
Please direct questions to Project Manager, Dave Fenity, at dfenity@townoflyons.com.


text:
Q: Who initialize the variables of interface in java? Since there is no constructor in Interface, who initialize the variables of interface in java?

A: Java uses implicit modifiers for interfaces. So the interface:
 public interface I {
     String FIELD = "Some field";

     void method();
 }

Can in fact be written as:
public interface I {
     public static final String FIELD = "Some field";

     public abstract void method();
}

Then you see that the fields in an interface all are public static final and the methods public abstract.
Because the field is static final and static-initializer-blocks are not allowed in interfaces it has to be initialized directly and thus no constructor is used.

A: This btw is no different than a public static final variable inside a class too - it will not be initialized by the constructor, since well, it's a constant - as javap outputs it ConstantValue.
 class Test {
      public static final int x = 5; // ACC_PUBLIC, ACC_STATIC, ACC_FINAL
                                     //   ConstantValue: int 5

      public int y = 3; // putfield , initialized inside default constructor
 }

EDIT Holger is right in his comment that there can be initialization code within a static block. For example having a class:
class C {

}

That has a reference in an interface:
interface Test {
    C c = new C();
}

would result in a static block at compile time that would look something like this:
interface Test {

    C c;

    static {
        c = new C();
    }
}


Sanchez, M. O. and Chang, Anne B. (2009). Congenital rubella pneumonitis complicated by Pneumocystis jiroveci infection with positive long term respiratory outcome: A case report and literature review. Pediatric Pulmonology,44(12):1235-1239.
Rubella remains to be a significant illness in the developing countries because of limited access to immunizations. In congenital rubella syndrome, lung involvement becomes evident within the few months of life, as a manifestation of the late onset rubella syndrome. The lungs and other organs become involved secondary to immunopathologic mechanisms and immunodeficiency predisposes affected patients to opportunistic pathogens. We report the clinical, respiratory and immunologic data of a young boy who developed rubella pneumonitis and concomitant infection with Pneumocystis jiroveci. Despite the complicated clinical course, the child survived. At follow-up he has a normal pulmonary examination, mild hyperinflation only on his chest radiograph, normal immunology and normal respiratory reactance and resistance.