text:
\section{Introduction}\label{sec:intro}
In quantum communications, the goal is to send as much quantum
information as possible over a noisy quantum channel using a fixed
number of quantum bits (\textit{qubits}) or higher-dimensional systems (\textit{qudits}). One aims at optimising the
transmission rate so that it approaches the channel capacity
asymptotically.  The communicating parties are assumed to be
physically separated, but they might have access to additional
resources, which may include access to classical communication
channels, pre-shared randomness, and pre-shared entanglement. We focus on the use of entanglement in the design of quantum error-correcting codes (QECCs) to boost either their communication
rates or error-control capabilities.

\subsection{Quantum Codes}\label{subsec:QC}

\begin{figure}
	\centering
	\begin{adjustbox}{width=\hsize}
		\begin{quantikz}
			\lstick{$\ket{\varphi}$} & [1.8cm]\gate[wires=3]{U}
			\qwbundle
			{k} & \gate[3,style={starburst,fill=yellow,draw=red,line
				width=1pt,inner xsep=-6pt,inner ysep=-14pt},
			label style=black]{\text{noise}} & \gate[wires=4,disable auto height]
			{\begin{array}{c} \text{Error} \\ \text{diagnosis} \\ \text{and} \\ \text{recovery} \end{array}} \qw\\
			\lstick{$\ket{0}$} & \qwbundle{n-k-c} & \qw & \qw  \\
			\makeebit{$\ket{\Phi}_{AB}$} & \qwbundle{c} & \qw & \qw \\
			& \qw & \qwbundle{c} & \qw \\
		\end{quantikz}
	\end{adjustbox}
	%
	\caption{The basic structure of an entanglement-assisted quantum error-correcting code. Alice and Bob share the maximally entangled state $\ket{\Phi}_{AB}$ of $c$ maximally entangled qudits. Alice uses her half of the maximally entangled state and $n-k-c$ ancilla in a fixed state $\ket{0}$ in the encoding of her quantum information $\ket{\varphi}$ by the operator $U:=U_{\rm enc}$. The resulting $q^n$-dimensional state passes through the noisy channel. Bob's half of the initial $c$ pairs of maximally entangled states are assumed to be error-free. They will be used in error diagnosis and recovery.}
	\label{fig:circuit}
\end{figure}

A general quantum error-correcting code for qubits that does not use additional resources is a $K$-dimensional subspace the complex Hilbert space of $n$ qubits, which has dimension $2^n$. We use the notation $\dsb{n,k}_2$ for a code $\mathcal{Q}$ of dimension $K=2^k$ and say that $\mathcal{Q}$ encodes $k$ logical qubits into $n$ physical qubits. The encoding operation consists of two
steps.  First, the sender appends $n-k$ ancilla qubits in a fixed state, typically $\ket{0}^{\otimes(n-k)}$, to a state $\ket{\varphi}$ of $k$ qubits. Then an encoding unitary $U_{\rm enc}$ is applied:
\begin{alignat}{5}\label{eq:encode}
\ket{\varphi} & \mapsto \ket{\varphi} \otimes \ket{0}^{\otimes (n-k)} \notag\\
              &\mapsto \ket{\Psi_L}:= U_{\rm enc}\left(\ket{\varphi} \otimes \ket{0}^{\otimes (n-k)}\right).
\end{alignat}
The state $\ket{\Psi_L}$ is called the \textit{encoded state} or the
\textit{logical state}. The unitary $U_{\rm enc}$ acts on the space
formed by the input state $\ket{\varphi}$ and the ancillas
$\ket{0}^{\otimes (n-k)}$.

In an \emph{entanglement-assisted setup} \cite{Brun2006}, we replace $c$ of
the ancillas by $c$ \emph{pairs} of maximally entangled qubits (\textit{ebits} in short), before applying $U_{\rm enc}$.  Let the state
$\ket{\Psi^+}_{AB}$ be an EPR pair \cite{Einstein1935} shared
between the sender Alice and the receiver Bob.  The encoding operation
performed on Alice's qubits is given by
\begin{alignat}{5
\ket{\varphi}& \mapsto \ket{\varphi} \otimes \ket{0} ^{\otimes (n-k-c)} \otimes \ket{\Psi^+}_{AB}^{\otimes c} \notag \\
 & \mapsto\left(U_{\rm enc} \otimes I_{B}\right) \left(\ket{\varphi} \otimes \ket{0}^{\otimes (n-k-c)}  \otimes \ket{\Psi^+}_{AB}^{\otimes c}\right).\label{eq:entencode2}
\end{alignat}
Here $I_B$ denotes the identity operator on the receiver's half of the $c$
maximally entangled pairs, \textit{i.\,e.}, the encoded state
consists of $n+c$ qubits in total. The notation
$\ket{\Psi^+}_{AB}^{\otimes c}$ should be understood as reordering
the qubits such that the first $c$ qubits are with the sender Alice.
We assume that the noise only affects the first $n$ qubits sent over the channel, while the half of the shared $c$ ebits that is with the receiver Bob is not affected.

The ebits are prepared \emph{ahead of time} and are assumed to be
error-free, using, \textit{e.\,g.}, entanglement distillation or a
similar procedure. For more details on quantum entanglement, including
its creation and delivery across quantum networks, one can consult
the survey in \cite{Horodecki2009}, the experiment reported in
\cite{Humphreys2018}, or a recent scheme to generate genuine
multipartite entanglement of a large number of qubits in
\cite{Zhang2021}.

In \textit{teleportation}, one perfect ebit is used in tandem with noiseless classical communication to perfectly transmit one qubit. Entanglement-assisted quantum error-correcting codes (EAQECCs) use some noiseless ebits, without classical communication, to transmit quantum information over a noisy quantum channel.

In \textit{superdense coding}, the sender can apply an operation to her half of a pair of maximally entangled states such that, after sending this qubit, the receiver can decode \emph{two} classical bits of information.  An ancilla in a standard QECC can be interpreted as a
\emph{placeholder} for one bit of classical information about any error that has occurred.  Replacing an ancilla with one half of an ebit can, in theory, enable the receiver to extract \emph{two} bits of classical information regarding the errors. This enhances the error-handling capabilities of EAQECCs over the standard counterparts.

One can execute some communication tasks with fewer total resources or better error control by using an EAQECC instead of a combination of a standard QECC and teleportation.  An asymptotic analysis on the
benefits of pre-shared entanglement in quantum communication is available in \cite{Devetak2008}. We view an EAQECC as a finite-length realization. It is in principle possible to approach the
entanglement-assisted quantum capacity by building larger code blocks as shown in \cite{Bowen2002}.

Shared entanglement does not come for free. The cost of
sharing and purifying ebits means that EAQECCs \emph{do not automatically outperform} standard quantum codes in all circumstances. One measure to assess the advantage is the \textit{net rate}, which subtracts the number of ebits required from the number of qubits
transmitted.  In terms of construction via classical error-correcting codes, however, EAQECCs have fewer restrictions, allowing us to use larger families of classical codes.

The notation $\dsb{n,\kappa,\delta; c}_q$ signifies that the quantum
code $\mathcal{Q}$ is a $q$-ary EAQECC that encodes $\kappa$ logical
\textit{qudits} (quantum systems of dimension $q$) into $n$ physical
qudits, with the help of $n-\kappa-c$ ancillas and $c$ pairs of
maximally entangled qudits.  A quantum code with minimum distance $\delta$ can correct up to $\lfloor{(\delta-1)/2\rfloor}$
single-qudit errors. As shown in Figure \ref{fig:circuit}, Alice transmits the $n$ qudits to Bob. He then performs a syndrome measurement on them together with his half of the $c$ pairs of maximally entangled qudits to correct errors and
to retrieve the $\kappa$ logical qudits. The \textit{rate} $\rho$ and the \textit{net rate} $\Bar{\rho}$ of $\mathcal{Q}$ are, respectively,
\begin{equation}\label{eq:rate}
\rho:= \frac{\kappa}{n} \quad\text{and}\quad
\Bar{\rho}:= \frac{\kappa - c}{n}.
\end{equation}
The abbreviated form $\dsb{n,\kappa,\delta}_q$ is used when $c=0$.

\subsection{Quantum Codes from Classical Codes}\label{subsec:construct}

Let $p$ be a prime and let $s$ be positive integer. Let $q$ be a prime
power $q=p^{m}$ and let $\mathbb{F}_q$ be the finite field with $q$
elements. The multiplicative group of the nonzero elements of
$\mathbb{F}_q$ is denoted by $\mathbb{F}_q^{*}$. For a positive
integer $m$, we denote by $[m]$ the set $\{1,2,\ldots,m\}$. A code
$\mathcal{C}$ of length $n$ is a nonempty subset of
$\mathbb{F}_q^n$. Its \textit{codewords} are vectors of length $n$
with entries from $\mathbb{F}_q$. The {\it weight} of a vector is the
number of its nonzero entries. Given a nonempty $\mathcal{S} \subseteq \mathbb{F}_q^n$, we denote by ${\rm wt} (\mathcal{S})$ the number
$\min\{ {\rm wt}(\mathbf{v})\colon \mathbf{v} \in \mathcal{S}, \mathbf{v}
\neq \mathbf{0}\}$.

A code $\mathcal{C}$ is {\it linear} with parameters $[n,k,d]_q$ if it
is a $k$-dimensional subspace of $\mathbb{F}_q^n$ and its {\it minimum distance}, defined to be the smallest of the weights of its nonzero
codewords, is $d$. A $k \times n$ matrix $G$ whose rows form a basis
for $\mathcal{C}$ is a \textit{generator matrix} of $\mathcal{C}$. If
$G = \begin{pmatrix} I_k\; A \end{pmatrix}$, where $I_k$ is the $k
\times k$ identity matrix, we say that $G$ is \textit{in the standard form}.

We equip $\mathbb{F}_q^n$ and $\mathbb{F}_{q^2}^n$ with the Euclidean
and the Hermitian inner products, respectively. Given an arbitrary vector $\mathbf{x}=(x_1,\ldots,x_n)$ and a codeword $\mathbf{c}=(c_1,\ldots,c_n)$ in $\mathcal{C}$, the {\it Euclidean dual} $\mathcal{C}^{\perp}$ of $\mathcal{C}$ is
\begin{equation}\label{eq:Euclidean}
\mathcal{C}^{\perp} = \left\{\mathbf{x} \in \mathbb{F}_q^n\colon \sum_{i=1}^n x_ic_i=0,
\text{for all $\mathbf{c} \in \mathcal{C}$} \right\}.
\end{equation}
Analogously, the {\it Hermitian dual} $\mathcal{C}^{\perp_{\rm H}}$ of $\mathcal{C}$ is
\begin{equation}\label{eq:Hermitian}
\mathcal{C}^{\perp_{\rm H}} = \left\{\mathbf{x} \in \mathbb{F}_{q^2}^n\colon \sum_{i=1}^n x_i c_i^q = 0,
\text{for all $\mathbf{c} \in \mathcal{C}$} \right\}.
\end{equation}

An $[n,k,d]_q$-code with $k \leq
\lfloor{\frac{n}{2}}\rfloor$ is \textit{self-orthogonal} if it is
contained in its dual. If, moreover, $n=2k$, then the code is
\textit{self-dual}. The notion of the \textit{hull} of a code was
introduced in \cite{Assmus1992} to define the intersection of the code
with its dual. Hence, the \textit{Hermitian hull} of $\mathcal{C}$ is
the code ${\rm Hull}_{\rm H}(\mathcal{C}) = \mathcal{C} \cap
\mathcal{C}^{\perp_{\rm H}}$. A code whose hull is $\{\mathbf{0}\}$
intersects trivially with its dual and is called a \textit{linear complementary dual} (LCD) code. Readers interested to know more
about classical codes may consult \cite{Huffman2021}.

Gottesman formulated the \emph{stabilizer formalism} for QECCs in \cite{Gottesman1997}. It was subsequently expressed in the language of classical coding theory in \cite{Calderbank1998}, triggering fruitful cross-pollination of ideas and results between quantum error control and classical coding theory. A general treatment over any finite field followed in \cite{AK01}. A survey can be found in \cite{Ketkar2006}. The main ingredients are self-orthogonal classical (additive) codes under the (trace) Hermitian inner product. The orthogonality condition imposes constraints on the parameters of the
corresponding quantum codes. Entanglement-assisted schemes enlarge the pool of ingredients to include codes which are not self-orthogonal, but require maximally entangled states as an additional resource.

We recall a general construction route of EAQECCs via the
\emph{non-commuting stabilizers} as explained, with illustrations, in
\cite{BrunHsieh}. For the qubit case, a formal treatment is given in
\cite{Brun2014}. It links arbitrary classical codes over
$\mathbb{F}_4$ as well as pairs of codes over $\mathbb{F}_2$ to qubit
EAQECCs. Extensions to the qudit case, where $q>2$, are given in
\cite{Galindo2019,Galindo2021}. Using $\mathbb{F}_{q^2}$-linear
codes, we have the following construction (see
\cite[Theorem~3]{Galindo2019}).

\begin{prop}[Hermitian construction]\label{prop:two}
Let $\mathcal{C}$ be an $[n,k]_{q^2}$-code, and let
$\mathcal{C}^{\perp_{\rm H}}$ denote its Hermitian dual. Then there
exists an $\dsb{n,\kappa, \delta; c}_q$-code $\mathcal{Q}$ with
\begin{align*}
c &= k - \dim_{\mathbb{F}_{q^2}}
\left(\mathcal{C} \cap \mathcal{C}^{\perp_{\rm H}}\right), \quad
\kappa = n-2k+c \mbox{, and}\\
\delta &= {\rm wt}\left(\mathcal{C}^{\perp_{\rm H}} \setminus \left(\mathcal{C} \cap \mathcal{C}^{\perp_{\rm H}}\right)\right).
\end{align*}
\end{prop}
We note that the construction includes the case that $\mathcal{C}$ is contained in its Hermitian dual $\mathcal{C}^{\perp_{\rm H}}$, which implies $c=0$, \textit{i.\,e.}, the quantum codes do not require entanglement assistance.

Another construction uses a pair of $\mathbb{F}_q$-linear codes of
equal length, yielding the so-called CSS-like family of EAQECCs
(see \cite[Theorem~4]{Galindo2019}).
\begin{prop}[CSS-like construction]\label{prop:one}
Let $\mathcal{C}_i$ be an $[n,k_i,d_i]_q$-code for $i = 1,2$. Then there is an $\dsb{n,\kappa,\delta; c}_q$-code $\mathcal{Q}$ with
\begin{align*}
c &= k_1 - \dim\bigr(C_1 \cap C_2^{\perp}\bigr), ~
\kappa = n-(k_1+k_2)+c \mbox{, and}\\
\delta &= \min\Bigl\{{\rm wt}\bigl(C_1^{\perp} \setminus (C_2 \cap C_1^{\perp})\bigr),
{\rm wt}\bigr(C_2^{\perp} \setminus (C_1 \cap C_2^{\perp})\bigr) \Bigr\}.
\end{align*}
\end{prop}
Again, when $C_2^\bot \subseteq C_1$, we have $c=0$ and the resulting code does not require entanglement assistance.

The code $\mathcal{Q}$ in Proposition \ref{prop:two} is {\it pure} or
{\it nondegenerate} if $\delta = {\rm wt}(\mathcal{C}^{\perp_{\rm H}}) =
d(\mathcal{C}^{\perp_{\rm H}})$. The code $\mathcal{Q}$ in Proposition
\ref{prop:one} is pure whenever $\delta = \min\{d_1,d_2\}$.

\subsection{Our Contributions}

\begin{enumerate}
    \item We establish three propagation rules

    The first rule, given as Theorem \ref{thm:more}, increases $c$, signifying that more entanglement is required. The derived quantum code can send more information without losing anything in terms of error handling.

    Theorem \ref{thm:same} gives the second rule. It keeps $c$ fixed while lengthening the code, reducing its size. If some conditions are met, then the quantum distance may increase.

    The third rule is in Theorem \ref{thm:less}. It decreases $c$ while lengthening the code. There may be a price to pay in terms of smaller distances in some occasions.

    For the last two rules, we have less theoretical control over the quantum distances and, therefore, searches are the next best option.

\item Our propagation rules are applicable to \emph{both} stabilizer QECCs and EAQECCs. Most prior propagation rules were designed for stabilizer QECCs whereas our propagation rules works on nontrivial EAQECCs as well. It is in conducting searches for EAQECCs with excellent parameters that the main advantage of our propagation rules come to the fore. They allow us to control either the distance or the number of ebits.

\item In the realm of classical coding theory, Theorem \ref{thm:hull} provides a simple proof on the equivalence of $\mathbb{F}_{q^2}$-linear codes with diverse Hermitian hull dimensions for $q>2$. For any $[n,k,d]_{q^2}$-code $\mathcal{C}$ with $\dim({\rm Hull}_{\rm H}(\mathcal{C})) = \ell$, there exists an equivalent $[n,k,d]_{q^2}$-code $\mathcal{C}^{\prime}$ with $\dim({\rm Hull}_{\rm H}(\mathcal{C}^{\prime})) = \ell^{\prime}$ for each $\ell^{\prime} \in \{0,1,\ldots,\ell\}$. This generalises the result for Hermitian LCD codes in \cite[Section V]{Carlet2018} that considered only the case of $\ell^{\prime}=0$.
\end{enumerate}

Given an $[n,k,d]_{q^2}$-code $\mathcal{C}$, Section \ref{sec:propagation} discusses three linear
algebraic approaches that derive codes whose dimensions of Hermitian hulls vary. In the first two approaches, the derived codes have fixed parameters $[n,k,d]_{q^2}$, while the dimension of the hull decreases. In the third approach, $k$ is fixed, while both $n$ and the dimension of the hull increase by $1$, and the distance $d$ is either fixed or improved by $1$. Section \ref{sec:bound} discusses upper bounds on the parameters of EAQECCs. They are subsequently used collectively in Section \ref{sec:compute} as a measure of goodness to motivate our computational process and results. The parameters of the resulting qubit and qutrit EAQECCs are listed in the tables after the concluding remarks in Section \ref{sec:conclu}.

\section{Three New Propagation Rules}\label{sec:propagation}

This section presents three new propagation rules based on their effects on $c$, which quantifies the amount of entanglement. We start by devising tools from the classical ingredients.

\subsection{Tools from Classical Coding Theory}\label{subsec:tools}

For any vector $\mathbf{v}:=(v_1,\ldots,v_{n}) \in \mathbb{F}_{q^2}^{n}$, we denote by $\mathbf{v}^q$ the vector $(v_1^q,\ldots,v_{n}^q)$. Let
$\mathcal{C}$ be an $[n,k,d]_{q^2}$-code with generator
matrix $G$, and let $\mathbf{v}_1,\ldots,\mathbf{v}_k$ be the rows of
$G$. We use $G^{\dagger}$ to denote the $n \times k$ matrix whose
columns are $\mathbf{v}_1^q,\ldots,\mathbf{v}_k^q$. We call
$G^{\dagger}$ the {\it Hermitian transpose} of $G$.

Let us recall the relation between a code's generator matrix and its Hermitian hull.
\begin{lem}\label{lem:dimHull}
The dimension of the Hermitian hull is
\begin{alignat}{5}
\dim({\rm Hull}_{\rm H}(\mathcal{C})) &= k-\rank(GG^{\dagger}).\label{eq:dimHull}
\end{alignat}
\end{lem}
\begin{proof}
  A vector $\mathbf{v}$ is an element of ${\rm Hull}_{\rm H}(\mathcal{C})$
  if it is both a codeword of $\mathcal{C}$ and
  $\mathcal{C}^{\perp_{\rm H}}$. The first condition requires
  that $\mathbf{v}$ is in the row span of $G$, that is,
  $\mathbf{v}=\mathbf{u}G$ for some $\mathbf{u}\in\mathbb{F}_{q^2}^k$.
  The second condition requires that $\mathbf{v}$ is in the kernel of
  $G^\dagger$, \textit{i.\,e.}, $\mathbf{v}G^\dagger=\mathbf{0}$. In combination we have
  \begin{alignat}{5}
    {\rm Hull}_{\rm H}(\mathcal{C})=\{\mathbf{v}=\mathbf{u}G\colon
    \mathbf{u}\in\mathbb{F}_q^k\mid \mathbf{u}GG^\dagger=\mathbf{0}\}.
  \end{alignat}
  This implies \eqref{eq:dimHull}.
\end{proof}

A {\it monomial matrix} is a square matrix with exactly one nonzero
entry in each row and each column and zeros elsewhere. The matrix is a
{\it permutation matrix} if all of its nonzero entries are $1$.  Based
on these two families of matrices, two equivalence relations among linear
codes can be defined.

\begin{defn}\label{def1}
Let two linear codes $\mathcal{C}_1$ and $\mathcal{C}_2$ with
respective generator matrices $G_1$ and $G_2$ be given. Then the
following statements hold.
\begin{enumerate}
  \item The codes $\mathcal{C}_1$ and $\mathcal{C}_2$ are permutation
    equivalent if there exists a permutation matrix $P$ such that $G_1
    P$ is a generator matrix of $\mathcal{C}_2$.
  \item The codes $\mathcal{C}_1$ and $\mathcal{C}_2$ are monomially
    equivalent if there exists a monomial matrix $M$ such that $G_1M$
    is a generator matrix of $\mathcal{C}_2$.
\end{enumerate}
\end{defn}

Equivalent codes have the same length, dimension, and minimum
distance. It can be shown (see, \textit{e.\,g.}, \cite[Theorem
  1.6.2]{Huffman2005}) that any linear code is permutation equivalent
to a linear code whose generator matrix is in the standard form. The next
result shows that the respective Hermitian hulls of two permutation
equivalent codes have the same dimension.

Regarding monomially equivalent codes, Carlet {\it et al.} in
\cite{Carlet2018} demonstrated that LCD codes over $\mathbb{F}_q$
exist for all possible parameters when $q>3$.  We will prove that a
self-orthogonal linear code is monomially equivalent to a linear code
with Hermitian hulls of arbitrary dimensions.

\begin{lem}\label{lem1H}
Any two permutation equivalent $\mathbb{F}_{q^2}$-linear codes
$\mathcal{C}_1$ and $\mathcal{C}_2$ have
\[
\dim({\rm Hull}_{\rm H}(\mathcal{C}_1)) = \dim({\rm Hull}_{\rm H}(\mathcal{C}_2)).
\]
\end{lem}

\begin{proof}
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be permutation equivalent
codes with parameters $[n,k,d]_{q^2}$. Let $G_1$ be a generator matrix of $\mathcal{C}_1$. Hence, there is a permutation matrix $P$ such that $G_2 = G_1 P$ is a generator matrix of $\mathcal{C}_2$. This implies that
\[
G_2G_2^\dagger=G_1P(G_1P)^\dagger = G_1PP^{\rm t}G_1^\dagger =
G_1G_1^\dagger.
\]
In combination with Lemma \ref{lem:dimHull}, the
conclusion follows.
\end{proof}

The next result is a generalisation of the results for Hermitian LCD codes in \cite[Section V]{Carlet2018}. In our notation, only the case of $\ell^{\prime}=0$ was considered in the said reference.
\begin{thm}\label{thm:hull}
Let $q>2$ be a prime power and let $\mathcal{C}$ be an
$[n,k,d]_{q^2}$-code with $\dim({\rm Hull}_{\rm H}(\mathcal{C})) =
\ell$. Then there exists an equivalent $[n,k,d]_{q^2}$-code
$\mathcal{C}^{\prime}$ with $\dim({\rm Hull}_{\rm
  H}(\mathcal{C}^{\prime})) = \ell^{\prime}$ for each $\ell^{\prime}
\in \{0,1,\ldots,\ell\}$.
\end{thm}
\begin{proof}
Without loss of generality, we can assume that ${\rm Hull}_{\rm H} (C)$ is an $[n,\ell,d']_{q^2}$-code generated in the standard form by $G_1 =\begin{pmatrix} I_{\ell}\; A \end{pmatrix}$. Moreover, we can choose a generator
matrix for $\mathcal{C}$ of the form
\begin{equation}\label{eq:GenG}
G=\begin{pmatrix}
I_{\ell} & A \\
\mathbf{0} & B
\end{pmatrix}
\end{equation}
and derive
\begin{equation}\label{eq:matprod}
G G^{\dagger} =
\begin{pmatrix}
I_{\ell} & A \\
\mathbf{0} & B
\end{pmatrix}
\begin{pmatrix}
I_{\ell} & \mathbf{0} \\
A^{\dagger} & B^{\dagger}
\end{pmatrix}
=
\begin{pmatrix}
I_{\ell}+ A A^{\dagger} & A B^{\dagger} \\
B A^{\dagger} & B B^{\dagger}
\end{pmatrix}.
\end{equation}
The submatrices $I_{\ell}+ A A^{\dagger}$ and $A B^{\dagger}$ are zero
since $G_1$ generates the Hermitian hull, which is contained in the
Hermitian dual of $\mathcal{C}$. We have $\rank({B B^{\dagger}}) = k -
\ell$. We now consider the generator matrix
\begin{equation}
G^{\prime} = G \;  \Diag\left(a_1,a_2,\ldots,a_{\ell - \ell^{\prime}},1,\ldots,1\right)
\end{equation}
with $a_j\in\mathbb{F}_{q^2}^*$ and $a_j^{q+1} \neq 1$ for $1 \leq j
\leq \ell - \ell^{\prime}$. These conditions can always be met for $q>2$.
Let
\[
T = \begin{pmatrix}
a_1^{q+1}-1 & 0 & \cdots & 0 \\
0  & a_2^{q+1}-1 & \cdots & 0 \\
\vdots  & \vdots & \ddots & \vdots \\
0 & 0  & \cdots & a_{\ell - \ell^{\prime}}^{q+1}-1\\
\end{pmatrix}.
\]
We verify that $G^{\prime} G^{\prime \dagger}$ is the block-diagonal matrix
\begin{multline}{}
G^{\prime} G^{\prime \dagger}  =
\begin{pmatrix}
T & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0}  & B B^{\dagger}\\
\end{pmatrix} = \\
\Diag \left(a_1^{q+1}-1,\cdots,a_{\ell - \ell^{\prime}}^{q+1}-1, 0, \cdots,0,
B B^{\dagger}\right)
\end{multline}
and that $\rank\left(G^{\prime} G^{\prime \dagger}\right) = k -
\ell^{\prime}$. This completes the proof.
\end{proof}

We will need the following lemma (see \cite[Theorem 6.32]{Hou}) later
\begin{lem}\label{lem42}
Let $A$ be an $n\times n$ matrix with rank $s$ over $\mathbb{F}_{q^2}$
such that $A= A^{\dagger}$. Then $A$ is Hermitian congruent to
\[
\Diag \Big(\underbrace{1,\cdots,1}_s,0,\cdots,0\Big).
\]
\end{lem}
Here two matrices $A$ and $B$ over $\mathbb{F}_{q^2}$ are {\it Hermitian congruent} if there exists a nonsingular matrix $D$ such that $B = D A D^{\dagger}$.

Using Lemma \ref{lem42}, we derive the following result. It enables an
$[n,k]_{q^2}$-code with $\ell$-dimensional Hermitian hull to generate
an $[n+1,k]_{q^2}$-code with $(\ell+1)$-dimensional Hermitian hull.
\begin{prop}\label{cor:extension}
Let $0\leq \ell<\min\{k,n-k\}$. Given an $[n,k,d]_{q^2}$-code $\mathcal{C}$ with
$\dim({\rm Hull}_{\rm H}(\mathcal{C}))=\ell$ and generator matrix
$G$, one can add one column to $G$ such that the Hermitian hull of the
extended $[n+1,k,d^{\prime}]_{q^2}$-code $\mathcal{C}^{\prime}$ has
dimension $\dim\left({\rm Hull}_{\rm H}(\mathcal{C}^{\prime})\right) = \ell+1$ and minimum distance $d'$, with $d\le d'\le d+1$.
\end{prop}
\begin{proof}
Let $q=p^m$. Let $G$ be a generator matrix of $\mathcal{C}$. Since
$\dim({\rm Hull}_{\rm H}(\mathcal{C}))=\ell$, by Lemma \ref{lem:dimHull}, $\rank\left(GG^{\dagger}\right)=s$, where $s = k-\ell\ge 1$. By Lemma
\ref{lem42}, there exists a nonsingular $k \times k$ matrix $D$ over
$\mathbb{F}_{q^2}$ such that
\begin{equation}\label{eq:thm41H}
DGG^{\dagger}D^{\dagger}= \Diag \Big(\underbrace{1,\cdots,1}_s,0,\cdots,0 \Big).
\end{equation}
Since $s\ge 1$, the first diagonal entry of the $k\times k$ diagonal
matrix in \eqref{eq:thm41H} must be $1$. Because $D$ is nonsingular,
$DG$ is also a generator matrix of $\mathcal{C}$. Let $\alpha \in
\mathbb{F}_{q^2}$ be such that $\alpha^{q+1} = -1$. Such an $\alpha$
always exists since $\alpha^{q+1}$ runs through $\mathbb{F}_q$ when
$\alpha$ runs through $\mathbb{F}_{q^2}$. Let $G^{\prime}$ be the
$k\times(n+1)$ matrix defined by
\begin{equation}
G^{\prime}=\left(\begin{array}{@{}c|c@{}}
DG & \begin{matrix}
\alpha  \\
0   \\
\vdots  \\
0 \\
\end{matrix}
\end{array}\right).\label{eq:extGenMat}
\end{equation}
Then $G^{\prime}$ generates an $[n+1,k,d^{\prime}]_q$-code
$\mathcal{C}^{\prime}$ with $d\leq d^{\prime}\leq d+1$. One then
verifies that
\begin{alignat}{5}
  G^{\prime}G^{\prime\dagger}
  &=DGG^\dagger G^\dagger+\Diag(\alpha^{q+1},0,\cdots,0)\\
  &= \Diag \Big(0,\underbrace{1,\cdots,1}_{s-1},0,\cdots,0 \Big),
\end{alignat}
and, hence, $\rank(G^{\prime} \, G^{\prime\dagger}) = s-1$.
The claim about the dimension of the hull follows from Lemma
\ref{lem:dimHull}. As $D$ is invertible, the matrix
\begin{alignat}{5}
    D^{-1}G'=\left(\begin{array}{@{}c|c@{}}
          G & D^{-1}\begin{pmatrix}
               \alpha  \\
               0   \\
               \vdots  \\
                0 \\
          \end{pmatrix}
    \end{array}\right)
\end{alignat}
is a generator matrix for $\mathcal{C}'$ in the desired form.
\end{proof}

We note that the matrix $D$ in \eqref{eq:thm41H} is not unique. Moreover, there are $q+1$ choices for the element $\alpha$, which can be at any of the first $s$ positions. Hence, there are many choices for the additional column in Proposition \ref{cor:extension}.

In \cite{lisonvek2014quantum}, Lison{\v{e}}k and Singh proposed a modified construction of quantum codes by relaxing the self-orthogonality requirement. From a linear code $\mathcal{C}$ that is not Hermitian self-orthogonal, one can obtain a new linear code which is Hermitian self-orthogonal by adding some rows and columns to a generator matrix of $\mathcal{C}$. Inspired by this result, we show that an
$[n,k]_{q^2}$-code with $\ell$-dimensional Hermitian hull gives rise to an $[n+1,k+1]_{q^2}$-code with $(\ell+1)$-dimensional Hermitian hull.

\begin{prop}\label{pextendrule}
Let $\mathcal{C}$ be an $[n,k,d]_{q^2}$-code with basis $\{\mathbf{a}_1,\cdots,\mathbf{a}_k\}$ and $\dim({\rm Hull}_{\rm H}(\mathcal{C}))=\ell$, with $0\le\ell<\min\{k,n-k\}$. Let $\mathbf{c}$ be a chosen codeword of $\mathcal{C}^{\perp_{\rm H}}\setminus{\rm Hull}_{\rm H}(\mathcal{C})$ such that $\mathbf{c}\mathbf{c}^{\dagger}\neq 0$. Then there exists an $[n+1,k+1,d^{\prime}]_{q^2}$-code $\mathcal{C}^{\prime}$ with
$\dim\left({\rm Hull}_{\rm H}(\mathcal{C}^{\prime})\right) = \ell+1$ and $d^{\prime}=\min\{d,d_0+1\}$, where $d_0$ is the minimum distance of the code generated by $\{\mathbf{a}_1,\cdots,\mathbf{a}_k,\mathbf{c}\}$.
\end{prop}

\begin{proof}
Such a codeword $\mathbf{c}\in\mathcal{C}^{\perp_{\rm H}} \setminus{\rm Hull}_{\rm H}(\mathcal{C})$ with $\mathbf{c}\mathbf{c}^{\dagger}\neq 0$ always exists since $\dim({\rm Hull}_{\rm H}(\mathcal{C}))=\ell$ with $0\le\ell<\min\{k,n-k\}$. Let $G$ be a generator matrix of $\mathcal{C}$ whose rows are $\{\mathbf{a}_1,\cdots,\mathbf{a}_k\}$. Let $\mathbf{c}\mathbf{c}^{\dagger}=\alpha$. Since $\alpha\in\mathbb{F}_q^*$, there exists $\beta\in\mathbb{F}_{q^2}^*$ such that $\beta^{q+1}=-\alpha$. Let $G^{\prime}$ be the
$(k+1)\times(n+1)$ matrix defined by
\begin{equation}\label{eq26}
G^{\prime} := \left(
\begin{matrix}G&\mathbf{0}_{k\times 1}\\
\mathbf{c}&\beta \end{matrix}\right).
\end{equation}
Then we obtain the code
\[
\mathcal{C}^{\prime}=\{(x_1,\cdots,x_{k+1})\cdot G:x_i\in\mathbb{F}_{q^2},i\in[1,\cdots,k+1]\}.
\]
Putting $x_{k+1}$ to be either zero or nonzero, we deduce that the minimum distance of $\mathcal{C}^{\prime}$ is $d^{\prime}=\min\{d,d_0+1\}$. It follows from (\ref{eq26}) that
\begin{equation}
G^{\prime} G^{\prime\dagger} =
\begin{pmatrix}
G G^{\dagger} & G \mathbf{c}^{\dagger} \\
\mathbf{c} G^{\dagger} & 0
\end{pmatrix}.
\end{equation}
Since $\mathbf{c} \in \mathcal{C}^{\perp_{\rm H}}$, the column  $G \mathbf{c}^{\dagger}$ is a zero vector. Thus, $\dim\left({\rm Hull}_{\rm H}(\mathcal{C}^{\prime})\right) = k+1-\rank(GG^{\dagger})=\ell+1$.
\end{proof}

\subsection{Propagation Rules}\label{subsec:rules}

The new propagation rules are presented based on how they affect the variable $c$.

\begin{thm}[More Entanglement]\label{thm:more}
For $q>2$, the existence of a pure $\dsb{n,\kappa,\delta;c}_q$-code $\mathcal{Q}$, constructed by Proposition \ref{prop:two}, implies the existence of a pure $\dsb{n, \kappa+i, \delta; c+i}_q$-code $\mathcal{Q}^{\prime}$ for
each $i \in\{1,\ldots,\ell\}$, where $\ell$ is the dimension of the
Hermitian hull of the $\mathbb{F}_{q^2}$-linear code
$\mathcal{C}$ that corresponds to $\mathcal{Q}$.
\end{thm}
\begin{proof}
To confirm the assertion, we apply the Hermitian construction in Proposition \ref{prop:two} on the codes from Theorem \ref{thm:hull} and use Lemma \ref{lem:dimHull} to cover the stated range of parameters.
\end{proof}

\begin{example}
Let $\omega$ be a root of $x^2 + 2x + 2 \in \mathbb{F}_3[x]$ and let
$\mathbb{F}_9= \mathbb{F}_3(\omega)$. The $[29,14,12]_9$-code
$\mathcal{C}$ generated by $G=\begin{pmatrix}I_{14}\;
A\end{pmatrix}$, with $A$ being the matrix
\begin{small}
\[\let\w\omega
\setlength\arraycolsep{2.5pt}
\begin{pmatrix}
2 &  \w  &  \w^5 & \w^7 & \w^7 & \w^2 & \w & 0 & 0 & \w^5 & \w^6 & \w^3 & \w^3 &  \w & \w^5 \\
\w^5 & \w & 2 & \w^3 & \w & 0 & \w^6 & \w & 0 & \w^6 & 2 & \w^3 & \w^5 & 2 & 2\\
2 & 0 & 0 & \w^2 & 0 & \w^3 & \w & \w^6 & \w & \w^5 & \w^2 & \w^5 & \w^7 & 0 & \w^6 \\
\w^6 & \w^5 & \w^7 & \w & \w^3 & 2 & \w^7 & \w & \w^6 & \w^6 & \w^3 & 1 & \w & 0 & \w^7 \\
\w^7 & \w^3 & \w^3 & \w^5 & \w^3 & \w^2 & 1 & \w^7 & \w & \w^5 & 2 & \w & \w^5 &  \w^7 & 1 \\
1 & 2 & 1 & \w^7 & \w^2 & \w & 1 & 1 & \w^7 & \w^5 & 1 & \w^2 & \w^6 & \w &  \w^6 \\
\w^6 & \w^6 & \w^2 & \w^2 & \w^6 & \w & 1 & 1 & 1 & \w^3 & \w^3 & \w^3 & 1 & \w & \w^6 \\
\w^6 & \w & 1 & \w^3 & \w^3 & \w^3 & 1 & 1 & 1 & \w & \w^6 & \w^2 & \w^2 & \w^6 & \w^6 \\
\w^6 & \w & \w^6 & \w^2 & 1 & \w^5 & \w^7 & 1 & 1 & \w & \w^2 & \w^7 & 1 & 2 & 1 \\
1 & \w^7 & \w^5 & \w & 2 & \w^5 & \w & \w^7 & 1 & \w^2 & \w^3 & \w^5 & \w^3 & \w^3 & \w^7 \\
\w^7 & 0 & \w & 1 & \w^3 & \w^6 & \w^6 & \w & \w^7 & 2 & \w^3 & \w & \w^7 & \w^5 & \w^6 \\
\w^6 & 0 & \w^7 & \w^5 & \w^2 & \w^5 & \w & \w^6 & \w & \w^3 & 0 & \w^2 & 0 & 0 & 2 \\
2 & 2 & \w^5 & \w^3 & 2 & \w^6 & 0 & \w & \w^6 & 0 & \w &  \w^3 & 2 & \w & \w^5\\
\w^5 & \w & \w^3 & \w^3 & \w^6 & \w^5 & 0 & 0 & \w & \w^2 & \w^7 & \w^7 & \w^5 & \w & 2
\end{pmatrix},
\]
\end{small}%
is Hermitian self-orthogonal. The dual has parameters $[29,15,11]_9$. For their respective $(n,k)$ values, both $\mathcal{C}$ and $\mathcal{C}^{\perp_{\rm H}}$ have the best-known minimum distances. We get a $\dsb{29,1,11;0}_3$-code by Proposition \ref{prop:two}. The existence of a $\dsb{29,1+i,11;i}_3$-code for each $1 \leq i \leq 14$ is guaranteed by Theorem \ref{thm:more}.
\end{example}

Theorem \ref{thm:more} allows for the transmission of a larger number of qudits when more pairs of maximally entangled qudits are available, while preserving the minimum distance $\delta$, the total number $n$ of qudits to be sent, as well as the net rate. The main idea is to multiply the columns of the generator matrix by an invertible diagonal matrix to \emph{decrease} the dimension of the Hermitian hull.

We can use the same approach to try to \emph{increase} the dimension of the Hermitian hull. This yields the following generalisation of the Hermitian construction in Proposition \ref{prop:two}.

\begin{thm}\label{thm:minEntanglement}
Let $\mathcal{C}$ be an $[n,k]_{q^2}$-code whose Hermitian dual is $\mathcal{C}^{\perp_{\rm H}}$. Then there
exists an $\dsb{n,\kappa, \delta; c}_q$-code $\mathcal{Q}$ with
\begin{align}\label{eq:min_entanglement}
c &=\min\left\{\rank\bigl(G\Diag(b_1,\cdots,b_n) G^{\dagger}\bigr)\colon b_i\in\mathbb{F}_q^*\right\}, \\
\kappa &= n-2k+c \mbox{, and }
\delta = {\rm wt}\left(\mathcal{C}^{\perp_{\rm H}} \setminus \left(\mathcal{C}
\cap \mathcal{C}^{\perp_{\rm H}}\right)\right). \notag
\end{align}
\end{thm}
\begin{proof}
Consider the code $\mathcal{C}'$ generated by a matrix $G' := G \Diag(a_1,\cdots,a_n)$, with $a_i\in\mathbb{F}_{q^2}^*$. By Lemma \ref{lem:dimHull},
$\dim({\rm Hull}_{\rm H}(\mathcal{C}')) = k-\rank(G'{G'}^\dagger)$, where
\begin{multline}\label{eq:iso_rank}
\rank(G'{G'}^\dagger)= \\
\rank\left(G \, \Diag(a_1^{q+1},\cdots,a_n^{q+1}) \, G^\dagger\right).
\end{multline}
As $a_i^{q+1}\in\mathbb{F}_q$, it suffices to minimize \eqref{eq:iso_rank} over all invertible diagonal matrices over $\mathbb{F}_q$.  By the surjectivity of the norm, given
$b_i\in\mathbb{F}_q$, there exists $b_i\in\mathbb{F}_{q^2}$ with $b_i^{q+1}=a_i$.
\end{proof}
We do not have an efficient method to determine an equivalent code $\mathcal{C}^{\prime}$ that minimises \eqref{eq:min_entanglement}.

In the extremal case of $c=0$, we obtain a Hermitian self-orthogonal
code by finding particular solutions to a linear system.
\begin{thm}
A given $[n,k,d]_{q^2}$-code $\mathcal{C}$ is equivalent to a Hermitian self-orthogonal code if and only if there is a vector $\mathbf{b}\in\mathbb{F}_q^n$, with $b_i \ne 0$, such that
\begin{equation}\label{eq:punctureCode}
\sum_{i=1}^n b_i x_i y_i^q = 0 \mbox{ for all } \mathbf{x},\mathbf{y}\in\mathcal{C}.
\end{equation}
\end{thm}
\begin{proof}
Let $\mathcal{C}$ be equivalent to a Hermitian self-orthogonal code $\mathcal{C}^{\prime}$. Without loss of generality, we can assume that $\mathcal{C}^{\prime}$ is obtained from $\mathcal{C}$ via multiplication of the coordinates by $a_i \in\mathbb{F}_{q^2}^*$. Setting $b_i=a_i^{q+1}\in\mathbb{F}_q^*$ implies \eqref{eq:punctureCode}.
On the other hand, if \eqref{eq:punctureCode} has a solution, then, by the surjectivity of the norm, we find $a_i\in\mathbb{F}_{q^2}^*$ with $b_i=a_i^{q+1}$. Multiplying the coordinates of $\mathcal{C}$ by $b_i$ yields an equivalent Hermitian self-orthogonal code, provided that $a_i$ and $b_i$ are nonzero for all $i$.
\end{proof}
The linear space of all solutions $\mathbf{b}\in\mathbb{F}_q^n$ of
\eqref{eq:punctureCode} (\textit{i.\,e.}, allowing zero-coordinates as well) is known as a \emph{punctured code} \cite{Rains99}.

Swapping the roles of $\mathcal{C}$ and $\mathcal{C}^{\perp_{\rm H}}$ in Proposition \ref{prop:two} enables us to better control the quantum distance. We use this approach in the proof of the next result.

\begin{thm}[Same Entanglement]\label{thm:same}
If there exists a pure $\dsb{n,\kappa,\delta;c}_q$-code $\mathcal{Q}$ obtained by the Hermitian construction in Proposition \ref{prop:two}, then there exists a pure $\dsb{n+1, \kappa-1, \delta'; c}_q$-code $\mathcal{Q}^{\prime}$ with $\delta\le\delta'\le\delta+1$.
\end{thm}
\begin{proof}
Let $\mathcal{C}^{\perp_{\rm H}}$ be the linear code with parameters $[n,n-k,\delta]_{q^2}$ used in the Hermitian construction of $\mathcal{Q}$. The dimension of the Hermitian hull is
\[
\dim(\mathcal{C}\cap\mathcal{C}^{\perp_{\rm H}})=\ell=n-k-c.
\]
Applying Proposition \ref{cor:extension} to $\mathcal{C}^{\perp_{\rm H}}$ yields an $[n+1,n-k,\delta']_{q^2}$-code ${\mathcal{C}^{\prime}}^{\perp_{\rm H}}$. The dimension of its Hermitian hull is $\ell+1$. Applying the Hermitian construction to $\mathcal{C}^{\prime}$ gives us an $\dsb{n+1,\kappa-1,\delta';c}_q$-code $\mathcal{Q}^{\prime}$.
\end{proof}

Applying the Hermitian construction on the $[n+1,k,d']_{q^2}$-code $\mathcal{C}^{\prime}$ from Proposition \ref{cor:extension} produces an $\dsb{n+1,\kappa-1,\delta';c}_q$-code $\mathcal{Q}^{\prime}$. On the original classical code
$\mathcal{C}$, the outcome is an $\dsb{n,\kappa,\delta;c}_q$-code $\mathcal{Q}$. The minimum distance $\delta'$ of $\mathcal{Q}^{\prime}$ depends on how the extended code $\mathcal{C}^{\prime}$ is built, as illustrated in the following example.
\begin{example}\label{ex:thm10}
Let $\omega$ be a root of $x^2 + 2x +2 \in \mathbb{F}_3[x]$ and let
$\mathbb{F}_9=\mathbb{F}_3(\omega)$. Let $\mathcal{C}$ be the
$[5,4,2]_9$-code generated by $G=\begin{pmatrix} I_4\;B \end{pmatrix}$
with $B=\begin{pmatrix} 2\; 2\;2\;2 \end{pmatrix}^{\rm t}$. Extending
the matrix $G$ by the column
$\begin{pmatrix}1\; w^3\; w^2\; w^7\end{pmatrix}^{\rm t}$, we obtain a $[6,4,3]_9$-code $\mathcal{C}^{\prime}$ with
$\dim{\rm Hull} (\mathcal{C}^{\prime})=1$. Since $\mathcal{C}^{\prime}$ is an MDS code, its Hermitian dual
${\mathcal{C}^{\prime}}^{\perp_{\rm H}}$ has parameters $[6,2,5]_9$.
Using the code $\mathcal{C}^{\prime}$ in Proposition \ref{prop:two} results in a pure $\dsb{6,1,5;3}_3$-code which is optimal by the bounds in the next section. Its net rate is $-1/3$ and it improves on the distance of the distance-optimal quantum code $\dsb{6,1,3}_3$ by $2$.
\end{example}

Based on the derived $[n+1,k+1,d']_{q^2}$-code in Proposition \ref{pextendrule}, we have the following result which allows for the transmission of the same amount of quantum information using a smaller number of pairs of maximally entangled qudits.

\begin{thm}[Less Entanglement]\label{thm:less} The existence of a pure $\dsb{n,\kappa,\delta;c}_q$-code $\mathcal{Q}$, constructed from an $\mathbb{F}_{q^2}$-linear code $\mathcal{C}$ based on Proposition \ref{prop:two}, implies the existence of a pure $\dsb{n+1, \kappa,\delta^{\prime}; c-1}_q$-code $\mathcal{Q}^{\prime}$ with $\delta^{\prime} \leq \delta$.
\end{thm}

\begin{proof}
The desired result follows from Proposition \ref{pextendrule} by a method analogous to the one in the proof of Theorem \ref{thm:same}.
\end{proof}

In Theorem \ref{thm:less}, the minimum distance of the resulting EAQECC is determined by the choice of the codeword $\mathbf{c}$, which was defined earlier in Proposition \ref{pextendrule}. To construct a new EAQECC with good minimum distance, one can try all such codewords and then select an EAQECC with the largest minimum distance from all resulting EAQECCs. The next example illustrates such an implementation.

\begin{example}
Let $\omega$ be a root of $x^2 + 2x +2 \in \mathbb{F}_3[x]$ and let $\mathbb{F}_9=\mathbb{F}_3(\omega)$. Let $\mathcal{C}$ be the $[16, 5, 8]_9$-code with generator matrix $G$ given by
\begin{small}
\[\let\w\omega
\setlength\arraycolsep{3pt}
\begin{pmatrix}
1 & 0 & 0 & 0 & \w^6 & 1 & \w^6 & \w^5 & 0 & \w^6 & \w^6 & 1 & \w^7 & 1 & \w & 1 \\
0 & 1 & 0 & 0 & \w^7 & 2 & \w & \w^2 & 0 & \w & 0 & 0 & 1 & \w^5 & \w^2 & \w^3 \\
0 & 0 & 1 & 0 & 2 & 1 & 1 & 1 & 0 & \w^5 & \w^3 & \w & 2 & \w^2 & \w^2 & \w^3 \\
0 & 0 & 0 & 1 & \w^2 & 1 & \w^7 & \w^2 & 0 & 2 & \w & \w^2 & \w & \w^6 & \w & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 1 & 2 & 1 & 2 & 1 & 2
\end{pmatrix}.
\]
\end{small}
The Hermitian dual code $\mathcal{C}^{\perp_{\rm H}}$ has parameters $[16,11,5]_9$ and the Hermitian hull ${\rm Hull}_{\rm H}(\mathcal{C})$ is a $[16,3,12]_9$-code. Following the proof of Proposition \ref{pextendrule}, we select a codeword
\begin{equation}\label{eq:wordc}
\let\w\omega
\setlength\arraycolsep{1.6pt}
\mathbf{c}:=
\begin{pmatrix}
1 & \w^7 & \w & \w^5 & \w^6 & \w^2 & 0 & \w^5 & \w^3 & 2 & \w^2 & 1 & \w^2 & \w^5 & \w^3 & \w^2
\end{pmatrix}
\end{equation}
of weight $15$ in $\mathcal{C}^{\perp_{\rm H}} \setminus
{\rm Hull}_{\rm H} (\mathcal{C})$ such that $\mathbf{c}\mathbf{c}^{\dagger} = 1$ to obtain the $[17,6,8]$-code $\mathcal{C}^{\prime}$ whose generator matrix is
\begin{equation*}
G^{\prime}=\left(
\begin{matrix}G & \mathbf{0}_{5\times 1}\\
\mathbf{c}& \omega \end{matrix}\right).
\end{equation*}
The Hermitian dual $\mathcal{C}^{\prime \perp_{\rm H}}$ has parameters $[17,11,5]_9$ and ${\rm Hull}_{\rm H} (\mathcal{C}^{\prime})$ is a $[17,4,10]_9$-code.

We now switch perspective and use the $[16,5,8]_9$-code as the $\mathcal{C}^{\perp_{\rm H}}$, instead of as the $\mathcal{C}$, in Proposition \ref{prop:two} to construct a pure $\dsb{16,2,8;8}_3$-code $\mathcal{Q}$. Using the derived $[17,6,8]$-code $\mathcal{C}^{\prime}$ as the $\mathcal{C}^{\perp_{\rm H}}$ in Proposition \ref{prop:two} leads to a pure $\dsb{17,2,8;7}_3$-code $\mathcal{Q}^{\prime}$.

If we have chosen as our codeword $\mathbf{c}$ the vector
\[
\let\w\omega
\setlength\arraycolsep{1.6pt}
\mathbf{c}:=
\begin{pmatrix}
\w^7 & \w^5 & \w & 0 & \w^5 & \w^5 & 0 & 2 & \w^7 & \w^6 & \w^2 & 2 & \w & 2 & \w^2 & \w^3
\end{pmatrix}
\]
of weight $14$, instead of the one in (\ref{eq:wordc}), then the resulting $\mathcal{C}^{\prime}$ would have parameters $[17,6,7]_9$. The constructed pure quantum codes $\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ would have parameters $\dsb{16,2,7;8}_3$ and $\dsb{17,2,7;7}_3$, respectively. This highlights the importance of choosing $\mathbf{c}$ such that $d^{\prime}=d$, that is, $d_0 \geq d-1$, in Proposition \ref{pextendrule}.
\end{example}

The idea of Lison{\v{e}}k and Singh in \cite{lisonvek2014quantum} is to start with a length $n$ classical code with a large hull. One then carefully selects a codeword so that it can be used to extend the length by $1$ and prevent the quantum distance from deteriorating. Our idea here is similar. The advantage is that we have more freedom in choosing the codeword that may lead to a better quantum distance. The two approaches coincide when the classical ingredient $\mathcal{C}$ is $k$-dimensional over $\mathbb{F}_{q^2}$ and its hull has dimension $k-1$.

\section{Upper Bounds}\label{sec:bound}

There is a vast literature on EAQECCs constructed via classical maximum distance separable (MDS) code and (Hermitian) LCD codes. Their parameters and excellent properties allow for a straightforward derivation of the parameters of the corresponding quantum codes. The length of MDS
codes, however, are constrained by the cardinality of the underlying finite fields. Using classical MDS codes over $\mathbb{F}_4$ for the qubit case and $\mathbb{F}_9$ for the qutrit case offer limited insights beyond very small lengths.

The Singleton bound for an $\dsb{n, \kappa, \delta; c}_q$-code $\mathcal{Q}$ in
\cite[Corollary 9]{GraHubWin2022} reads
\begin{align}
  \kappa &\le c+\max\{0,n - 2 \delta + 2\},\label{eq:QMDS_small_distance}\\
  \kappa &\le n-\delta+1,\label{eq:QMDS_trivial}\\
  \kappa
  &\le\frac{(n-\delta+1)(c+2\delta-2-n)}{3\delta-3-n} \mbox{, if } \delta-1 \ge \frac{n}{2}. \label{eq:QMDS_large_distance}
\end{align}
Codes attaining the bound \eqref{eq:QMDS_small_distance} for
$\delta-1 \le\frac{n}{2}$ or the bound \eqref{eq:QMDS_large_distance} for
$\delta-1 \ge\frac{n}{2}$ with equality are called MDS EAQECCs. We note that without the bound \eqref{eq:QMDS_trivial}, the upper bound on the dimension
$\kappa$ would be linear in the \textit{a priori} unbounded number $c$
of maximally entangled pairs of qudits.

To our knowledge, most known families of MDS EAQECCs, \textit{e.\,g.}, those presented in \cite{Guenda2017,Chen2021, Chen2021a,Fang2020,Luo2019}, were built by applying Propositions \ref{prop:two} and \ref{prop:one} on suitably chosen classical MDS codes. In \cite{Guenda2017}, an $[n,k,d]_{q^2}$-code
$\mathcal{C}$, whose Hermitian dual is an $[n,n-k,d^{\prime}]_q$-code, is used in Proposition \ref{prop:two} to yield two EAQECCs with parameters
\begin{alignat}{5}
&\dsb{n, k-\dim({\rm Hull}_{\rm H} (\mathcal{C})),d;
n-k-\dim({\rm Hull}_{\rm H}(\mathcal{C}))}_q, \\
&\dsb{n,n-k-\dim({\rm Hull}_{\rm H}(\mathcal{C})),d^{\prime};
k-\dim({\rm Hull}_{\rm H}(\mathcal{C}))}_q.
\end{alignat}
From an $[n,k,n-k+1]_{q^2}$-MDS code $\mathcal{C}$ and its $[n,n-k,k+1]_{q^2}$-Hermitian
dual $\mathcal{C}^{\perp_{\rm H}}$ with
$\dim({\rm  Hull}_{\rm H}(\mathcal{C})) =
\dim({\rm  Hull}_{\rm H}(\mathcal{C}^{\perp_{\rm H}}))=\ell$, one obtains EAQECCs with parameters
\begin{alignat}{5}
 &\dsb{n,k-\ell,n-k+1;n-k-\ell}_q \mbox{ and }\\
 &\dsb{n,n-k-\ell,k+1;k-\ell}_q.
\end{alignat}
In general, only the code with distance $d\le\frac{n}{2}$ is an
MDS EAQECC, whereas, for $d>\frac{n}{2}$ and $\kappa\le c <n-k$, the bound \eqref{eq:QMDS_large_distance} cannot be achieved with equality.

As shown by Grassl, Huber, and Winter in \cite[Theorem 7]{GraHubWin2022}, any pure $\dsb{n,\kappa,\delta;c}_q$-code obeys the bounds
\begin{equation}\label{eq:old_QMDS}
2 \delta \le n+c-\kappa+2.
\end{equation}
We show that this bound also applies to EAQECCs that can be obtained by Propositions \ref{prop:two} and \ref{prop:one}.

\begin{thm}\label{thm:bound55}
For any $\dsb{n, \kappa, \delta; c}_q$-code $\mathcal{Q}$ obtained by the Hermitian construction in Proposition~\ref{prop:two}, we have
\begin{alignat}{5}
  2 \delta \leq n + c - \kappa +2.
\end{alignat}
\end{thm}
\begin{proof}
Corresponding to the $\dsb{n, \kappa, \delta; c}_q$-code $\mathcal{Q}$, there exists an
$\left[n, n-\kappa-\ell \right]_{q^2}$-code $\mathcal{C}$ such that $\dim_{\mathbf{F}_{q^2}}\left({\rm Hull}_{\rm H}\left(\mathcal{C}\right)\right) =\ell$. If ${\rm Hull}_{\rm H}(\mathcal{C})$ has generator matrix $\begin{pmatrix} I_\ell & R
\end{pmatrix}$, then $\mathcal{C}^{\perp_{\rm H}}$ has generator matrix
\[
\begin{pmatrix}
	I_{\ell} & R\\
	\mathbf{O}_{\kappa \times \ell} & A
\end{pmatrix}.
\]
The code generated by
\[
\begin{pmatrix}
	\mathbf{O}_{ \kappa  \times \ell} & A
\end{pmatrix}
\]
is a subset of $\mathcal{C}^{\perp_{\rm H}} \setminus {\rm Hull}_{\rm H}(\mathcal{C})$ and
has parameters $[n, \kappa, \geq \delta]_{q^2}$. Hence, the linear code
generated by the matrix $A$ has parameters
\[
\left[n- \ell, \kappa , \geq \delta \right]_{q^2}.
\]
By the classical Singleton bound, we arrive at
\[
\delta \leq n - \ell -\kappa+ 1.
\]
Since $c=n-\kappa- 2\ell$, we have
\[
\delta \leq n - \frac{n-\kappa-c}{2} -\kappa+ 1 \iff 2 \delta \leq n+c-\kappa+2.
\]
\end{proof}

The codes in the CSS-like subfamily obeys the bound
\eqref{eq:old_QMDS} as well.
\begin{thm}\label{thms}
For $i=1,2$, let $\mathcal{C}_i$ be an $[n,k_i]_q$-code. Let
$\kappa=n-(k_1+k_2)+c$. For any $\dsb{n ,\kappa , \delta; c}_q$-code $\mathcal{Q}$ obtained by the CSS-like construction in Proposition~\ref{prop:one}, we have
\begin{alignat}{5}
 2 \delta \leq n + c - \kappa + 2.
\end{alignat}
\end{thm}

\begin{proof}
By Proposition \ref{prop:one}, there exists two linear codes
$\mathcal{C}_1$ and $\mathcal{C}_2$ with respective parameters $[n,k_1]_q$ and
$[n,k_2]_q$, where $k_2 = n- \kappa + c- k_1$. We denote by $\Delta$ the code $\mathcal{C}_1 \cap \mathcal{C}_2^{\perp}$ and
let $\ell=\dim_{\mathbf{F}_q}(\Delta)$. Let $\Delta$ be generated by $\begin{pmatrix}
I_\ell & R \end{pmatrix}$. Let
\[
\begin{pmatrix}
I_{\ell} & R\\
\mathbf{O}_{(n-\kappa_2-\ell) \times \ell} & A
\end{pmatrix}
\]
generate $\mathcal{C}_2^{\perp}$. The $[n, n-k_2-\ell, \geq \delta]_q$-code generated by
$\begin{pmatrix}
\mathbf{O}_{(n-k_2-\ell) \times \ell} & A
\end{pmatrix}$ is a subset of $\mathcal{C}_2^{\perp} \setminus
\Delta$. Hence, there exists an $[n- \ell, n-k_2-\ell, \geq
  \delta]_q$-code generated by the matrix $A$. By the
Singleton bound, we infer that $\delta \leq k_2+1$.  In a similar manner, starting from the $\beta$-dimensional code $\Gamma = \mathcal{C}_2 \cap \mathcal{C}_1^{\perp}$, one derives an $[n-\beta, n-k_1-\beta,
  \geq \delta]_q$-code, with $\delta \leq k_1+1$.  Combining the two
inequalities gives us $2 \delta \leq k_1 + k_2 + 2 = n+ c- \kappa +
2$, as promised.
\end{proof}

All these Singleton-type bounds are independent of the alphabet size
$q$.  The classical bound of Griesmer from \cite{Griesmer1960} leads
to a sharper upper bound for lengths $n > q^2+1$.

\begin{thm}
For any $\dsb{n, \kappa, \delta; c}_q$-code $\mathcal{Q}$ obtained by the CSS-like construction in Proposition~\ref{prop:one}, we have
\[
\frac{n+ \kappa + c}{2} \geq \sum_{i=0}^{\kappa-1} \left\lceil\frac{\delta}{q^i}\right\rceil.
\]
\end{thm}
\begin{proof}
Let $k_2 = n-\kappa+ c- k_1$. By the proof of Theorem \ref{thms},
there exist two linear codes $\mathcal{A}$ and $\mathcal{B}$ with
respective parameters $[n- \ell, n-k_2-\ell, \geq \delta]_q$ and
$ [n-\beta,n-k_1-\beta, \geq \delta]_q$, where $\ell$ and $\beta$ are the $\mathbb{F}_q$-dimensions of $\mathcal{C}_1 \cap \mathcal{C}_2^{\perp}$ and $\mathcal{C}_2 \cap \mathcal{C}_1^{\perp}$. Let $G_1$ and $G_2$ be
generator matrices of $\mathcal{C}_1$ and $\mathcal{C}_2$,
respectively. Then the dimension of the solution space of $G_1
G_2^{\rm t} \mathbf{x}^{\rm t} = \mathbf{0}$ is $k_2-
\rank(G_1G_2^{\rm t})$. Since
\[
G_1 G_2^{\rm t} \mathbf{x}^{\rm t} = G_1 (\mathbf{x}G_2)^{\rm t} = \mathbf{0},
\]
we have $k_2-\rank(G_1G_2^{\rm t})=\beta$. Employing the method
analogous to the one we have just used, we arrive at
\[
k_1-\rank(G_2G_1^{\rm t})= \ell \implies k_2-\beta=k_1-\ell.
\]
We note that
\begin{equation}\label{eq:EA_10}
	c=\rank(G_1G_2^{\rm t})=k_1-\ell=k_2-\beta.
\end{equation}
Applying the Griesmer bound to $\mathcal{A}$ and $\mathcal{B}$ gives
us
\begin{align}
n- \ell & \geq \sum_{i=0}^{n-k_2-\ell-1} \left\lceil\frac{\delta}{q^i}\right\rceil = \sum_{i=0}^{n-(k_1+k_2)+c-1} \left\lceil\frac{\delta}{q^i}\right\rceil \mbox{ and}  \label{eq:EA_11}\\
n- \beta & \geq \sum_{i=0}^{n-k_1-\beta-1} \left\lceil\frac{\delta}{q^i}\right\rceil = \sum_{i=0}^{n-(k_1+k_2)+c-1} \left\lceil\frac{\delta}{q^i}\right\rceil. \label{eq:EA_12}
\end{align}
Since $\ell=k_1-c$ and $\beta=k_2-c$, it follows from (\ref{eq:EA_11})
and (\ref{eq:EA_12}) that
\begin{multline*}
2n-\ell-\beta =
2n-(k_1+k_2)+2c=  \\
\geq 2 \sum_{i=0}^{n-(k_1+k_2)+c-1} \left\lceil\frac{\delta}{q^i} \right\rceil.
\end{multline*}
The conclusion follows from $k_2=n-\kappa+c-k_1$.
\end{proof}

\begin{thm}{\rm \cite{Li2015}}\label{thm:Griesmer}
For any $\dsb{n,\kappa, \delta; c}_q$-EAQECC obtained by the Hermitian
construction of Proposition~\ref{prop:two}, we have
\begin{equation}\label{eq:GriesmerH}
\frac{n+\kappa+c}{2} \geq \sum_{i=0}^{\kappa-1} \left\lceil\frac{\delta}{q^{2i}}\right\rceil.
\end{equation}
\end{thm}
\begin{proof}
We have an
$[n-\ell,\kappa, \geq \delta]_{q^2}$-code with $\ell=\frac{n-\kappa-c}{2}$ from the proof of Theorem
\ref{thm:bound55}. By the Griesmer bound,
\[
n-\ell =\frac{n+\kappa+c}{2} \geq \sum_{i=0}^{k-1}\left\lceil\frac{\delta}{q^{2i}}\right\rceil.
\]
\end{proof}

\section{Computational Results}\label{sec:compute}

The results we have derived as well as previously available tools can now be used to search for good entanglement-assisted (EA) qubits and qutrit.

The simplest approach would have been to apply Proposition \ref{prop:two} on $\mathbb{F}_4$ and $\mathbb{F}_9$-linear codes in the current {\tt MAGMA BKLC} database of codes with best-known minimum distances \cite{Bosma1997,Grassl:codetables}. Most codes in the database are LCD codes or codes with small Hermitian hulls. For qutrit codes, in light of Theorems \ref{thm:hull} and \ref{thm:more}, we prefer codes with large Hermitian hulls, \textit{e.\,g.}, specially crafted quasi-cyclic codes with large Hermitian hulls based on the construction method in \cite[Section III]{ELOS19}. Such classical codes yield EAQECCs with a wider range of parameters. The parameters also depend on the minimum distances of their respective dual codes. The choice of classical codes to record in the said database does not take the above into consideration. One can switch the role of the code and its dual in Proposition \ref{prop:two} so that the code from the database provide information on the minimum distance.

Theorem \ref{thm:same} yields good codes on numerous occasions. Determining the matrix $D$, however, is time consuming and the resulting parameters are often already covered by the other construction approaches. Computational evidences indicate that the benefit from applying Theorem \ref{thm:same} occurs when $d' = d+1$. Replacing the diagonal matrix on the right hand side of (\ref{eq:thm41H}) by a matrix of rank $s$ sometimes allows for a more efficient randomized procedure to find a suitable matrix $D$ that eventually leads to a good qutrit code.

For qutrit codes, we use Theorem
\ref{thm:minEntanglement} to determine the minimum number of maximally entangled pairs $c_{\text{min}}$, \textit{e.\,g.}, by exhaustive search, or use a randomised search to find a smaller value.

\begin{table*}
\caption{A Concise Version of the Parameters of Good Qubit EA-QECCs with $3 \leq n \leq 64$. To obtain the full table, one applies the propagation rules in Section \ref{sec:compute}. Interested readers may contact the authors for data of lengths up to $n=128$. We exclude the entries with $c=0$ since a large database for such codes is already available in \cite{Grassl:codetables}.}
\label{table:qubit}
	\renewcommand{\arraystretch}{1.2}
	\centering
		\begin{tabular}{llllll}
			\toprule
			$\dsb{n,\kappa,\delta;c}_2 $ & $\dsb{n,\kappa,\delta;c}_2 $ & $\dsb{n,\kappa,\delta;c}_2 $ & $\dsb{n,\kappa,\delta;c}_2 $ & $\dsb{n,\kappa,\delta;c}_2 $ & $\dsb{n,\kappa,\delta;c}_2 $ \\
			\midrule
			$ \dsb{3, 1, 3; 2}_2$ &
			$ \dsb{5, 4, 2; 1}_2$ &
			$ \dsb{5, 0, 4; 1}_2$ &
			$ \dsb{6, 0, 6; 4}_2$ &
			$ \dsb{7, 6, 2; 1}_2$ &
			$ \dsb{7, 4, 3; 3}_2$ \\

			$ \dsb{8, 5, 3; 3}_2$ &
			$ \dsb{8, 4, 4; 4}_2$ &
			$ \dsb{8, 2, 5; 4}_2$ &
			$ \dsb{8, 0, 8; 6}_2$ &
			$ \dsb{9, 8, 2; 1}_2$ &
			$ \dsb{9, 6, 3; 3}_2$ \\

			$ \dsb{9, 4, 5; 5}_2$ &
			$ \dsb{9, 2, 6; 5}_2$ &
			$ \dsb{10, 6, 4; 4}_2$ &
			$ \dsb{10, 4, 5; 4}_2$ &
			$ \dsb{10, 4, 6; 6}_2$ &
			$ \dsb{10, 0, 10; 8}_2$ \\

			$ \dsb{11, 10, 2; 1}_2$ &
			$ \dsb{11, 7, 3; 2}_2$ &
			$ \dsb{11, 6, 4; 3}_2$ &
			$ \dsb{11, 5, 6; 6}_2$ &
			$ \dsb{12, 9, 3; 3}_2$ &
			$ \dsb{12, 8, 4; 4}_2$ \\

			$ \dsb{12, 5, 6; 5}_2$ &
			$ \dsb{12, 0, 12; 10}_2$ &
			$ \dsb{13, 12, 2; 1}_2$ &
			$ \dsb{13, 10, 3; 3}_2$ &
			$ \dsb{13, 9, 4; 4}_2$ &
			$ \dsb{14, 0, 14; 12}_2$ \\

			$ \dsb{15, 14, 2; 1}_2$ &
			$ \dsb{15, 9, 5; 6}_2$ &
			$ \dsb{16, 13, 3; 3}_2$ &
			$ \dsb{16, 9, 4; 1}_2$ &
			$ \dsb{16, 9, 6; 7}_2$ &
			$ \dsb{16, 0, 16; 14}_2$ \\

			$ \dsb{17, 16, 2; 1}_2$ &
			$ \dsb{17, 13, 3; 2}_2$ &
			$ \dsb{17, 9, 6; 6}_2$ &
			$ \dsb{17, 0, 8; 1}_2$ &
			$ \dsb{17, 0, 12; 9}_2$ &
			$ \dsb{18, 15, 3; 3}_2$ \\

			$ \dsb{18, 10, 5; 4}_2$ &
			$ \dsb{18, 7, 9; 11}_2$ &
			$ \dsb{18, 0, 10; 6}_2$ &
			$ \dsb{18, 0, 18; 16}_2$ &
			$ \dsb{19, 18, 2; 1}_2$ &
			$ \dsb{19, 13, 4; 4}_2$ \\

			$ \dsb{20, 15, 3; 1}_2$ &
			$ \dsb{20, 15, 4; 5}_2$ &
			$ \dsb{20, 14, 5; 6}_2$ &
			$ \dsb{20, 10, 6; 4}_2$ &
			$ \dsb{20, 9, 8; 9}_2$ &
			$ \dsb{20, 0, 20; 18}_2$ \\

			$ \dsb{21, 20, 2; 1}_2$ &
			$ \dsb{21, 16, 4; 5}_2$ &
			$ \dsb{21, 15, 5; 6}_2$ &
			$ \dsb{21, 9, 7; 6}_2$ &
			$ \dsb{22, 16, 4; 4}_2$ &
			$ \dsb{22, 12, 5; 4}_2$ \\

			$ \dsb{22, 0, 22; 20}_2$ &
			$ \dsb{23, 22, 2; 1}_2$ &
			$ \dsb{23, 18, 4; 5}_2$ &
			$ \dsb{23, 14, 5; 5}_2$ &
			$ \dsb{23, 14, 6; 7}_2$ &
			$ \dsb{24, 18, 4; 4}_2$ \\

			$ \dsb{24, 16, 5; 6}_2$ &
			$ \dsb{24, 16, 6; 8}_2$ &
			$ \dsb{24, 0, 24; 22}_2$ &
			$ \dsb{25, 24, 2; 1}_2$ &
			$ \dsb{25, 20, 3; 3}_2$ &
			$ \dsb{25, 19, 4; 4}_2$ \\

			$ \dsb{25, 11, 7; 6}_2$ &
			$ \dsb{25, 0, 14; 11}_2$ &
			$ \dsb{26, 22, 3; 4}_2$ &
			$ \dsb{26, 21, 4; 5}_2$ &
			$ \dsb{26, 13, 7; 7}_2$ &
			$ \dsb{26, 0, 26; 24}_2$ \\

			$ \dsb{27, 26, 2; 1}_2$ &
			$ \dsb{27, 23, 3; 4}_2$ &
			$ \dsb{27, 22, 4; 5}_2$ &
			$ \dsb{28, 23, 3; 3}_2$ &
			$ \dsb{28, 23, 4; 5}_2$ &
			$ \dsb{28, 0, 28; 26}_2$ \\

			$ \dsb{29, 28, 2; 1}_2$ &
			$ \dsb{29, 25, 3; 4}_2$ &
			$ \dsb{29, 23, 4; 4}_2$ &
			$ \dsb{29, 0, 12; 1}_2$ &
			$ \dsb{30, 26, 3; 4}_2$ &
			$ \dsb{30, 25, 4; 5}_2$ \\

			$ \dsb{30, 0, 16; 12}_2$ &
			$ \dsb{30, 0, 30; 28}_2$ &
			$ \dsb{31, 30, 2; 1}_2$ &
			$ \dsb{31, 26, 4; 5}_2$ &
			$ \dsb{31, 22, 5; 5}_2$ &
			$ \dsb{31, 16, 6; 1}_2$ \\

			$ \dsb{31, 16, 9; 13}_2$ &
			$ \dsb{31, 9, 13; 16}_2$ &
			$ \dsb{31, 2, 14; 11}_2$ &
			$ \dsb{32, 27, 3; 3}_2$ &
			$ \dsb{32, 26, 4; 4}_2$ &
			$ \dsb{32, 24, 5; 6}_2$ \\

			$ \dsb{32, 0, 32; 30}_2$ &
			$ \dsb{33, 32, 2; 1}_2$ &
			$ \dsb{33, 28, 4; 5}_2$ &
			$ \dsb{33, 26, 5; 7}_2$ &
			$ \dsb{34, 29, 3; 3}_2$ &
			$ \dsb{34, 28, 4; 4}_2$ \\

			$ \dsb{34, 26, 5; 6}_2$ &
			$ \dsb{34, 24, 6; 8}_2$ &
			$ \dsb{34, 0, 14; 8}_2$ &
			$ \dsb{34, 0, 34; 32}_2$ &
			$ \dsb{35, 34, 2; 1}_2$ &
			$ \dsb{35, 30, 4; 5}_2$ \\

			$ \dsb{35, 28, 5; 7}_2$ &
			$ \dsb{35, 26, 6; 9}_2$ &
			$ \dsb{35, 12, 8; 1}_2$ &
			$ \dsb{35, 4, 14; 11}_2$ &
			$ \dsb{36, 32, 3; 4}_2$ &
			$ \dsb{36, 30, 4; 4}_2$ \\

			$ \dsb{36, 28, 5; 6}_2$ &
			$ \dsb{36, 26, 6; 8}_2$ &
			$ \dsb{36, 17, 10; 13}_2$ &
			$ \dsb{36, 9, 17; 23}_2$ &
			$ \dsb{36, 0, 36; 34}_2$ &
			$ \dsb{37, 36, 2; 1}_2$ \\

			$ \dsb{37, 30, 5; 7}_2$ &
			$ \dsb{37, 28, 6; 9}_2$ &
			$ \dsb{37, 14, 13; 19}_2$ &
			$ \dsb{38, 28, 6; 8}_2$ &
			$ \dsb{38, 16, 13; 20}_2$ &
			$ \dsb{38, 16, 14; 22}_2$ \\

			$ \dsb{38, 14, 8; 2}_2$ &
			$ \dsb{38, 0, 38; 36}_2$ &
			$ \dsb{39, 38, 2; 1}_2$ &
			$ \dsb{39, 30, 6; 9}_2$ &
			$ \dsb{39, 22, 10; 17}_2$ &
			$ \dsb{39, 16, 8; 3}_2$ \\

			$ \dsb{39, 15, 11; 12}_2$ &
			$ \dsb{39, 13, 17; 26}_2$ &
			$ \dsb{39, 12, 9; 3}_2$ &
			$ \dsb{39, 0, 18; 15}_2$ &
			$ \dsb{40, 36, 3; 4}_2$ &
			$ \dsb{40, 31, 4; 1}_2$ \\

			$ \dsb{40, 30, 5; 4}_2$ &
			$ \dsb{40, 30, 6; 8}_2$ &
			$ \dsb{40, 22, 10; 16}_2$ &
			$ \dsb{40, 20, 12; 20}_2$ &
			$ \dsb{40, 18, 13; 20}_2$ &
			$ \dsb{40, 16, 14; 20}_2$ \\

			$ \dsb{40, 0, 40; 38}_2$ &
			$ \dsb{41, 40, 2; 1}_2$ &
			$ \dsb{41, 36, 3; 3}_2$ &
			$ \dsb{41, 32, 5; 5}_2$ &
			$ \dsb{41, 32, 6; 9}_2$ &
			$ \dsb{41, 20, 7; 1}_2$ \\

			$ \dsb{41, 20, 13; 21}_2$ &
			$ \dsb{41, 18, 14; 21}_2$ &
			$ \dsb{42, 38, 3; 4}_2$ &
			$ \dsb{42, 34, 5; 6}_2$ &
			$ \dsb{42, 33, 6; 9}_2$ &
			$ \dsb{42, 28, 8; 14}_2$ \\

			$ \dsb{42, 22, 12; 20}_2$ &
			$ \dsb{42, 20, 11; 16}_2$ &
			$ \dsb{42, 20, 13; 20}_2$ &
			$ \dsb{42, 20, 14; 22}_2$ &
			$ \dsb{42, 16, 9; 6}_2$ &
			$ \dsb{42, 0, 16; 8}_2$ \\

			$ \dsb{42, 0, 42; 40}_2$ &
			$ \dsb{43, 42, 2; 1}_2$ &
			$ \dsb{43, 39, 3; 4}_2$ &
			$ \dsb{43, 36, 5; 7}_2$ &
			$ \dsb{43, 34, 4; 3}_2$ &
			$ \dsb{43, 34, 6; 9}_2$ \\

			$ \dsb{43, 31, 6; 8}_2$ &
			$ \dsb{43, 29, 8; 14}_2$ &
			$ \dsb{43, 21, 14; 22}_2$ &
			$ \dsb{43, 0, 18; 13}_2$ &
			$ \dsb{44, 39, 3; 3}_2$ &
			$ \dsb{44, 36, 4; 4}_2$ \\

			$ \dsb{44, 29, 8; 13}_2$ &
			$ \dsb{44, 21, 14; 21}_2$ &
			$ \dsb{44, 0, 44; 42}_2$ &
			$ \dsb{45, 44, 2; 1}_2$ &
			$ \dsb{45, 41, 3; 4}_2$ &
			$ \dsb{45, 31, 8; 14}_2$ \\

			$ \dsb{45, 24, 12; 21}_2$ &
			$ \dsb{45, 22, 7; 1}_2$ &
			$ \dsb{46, 41, 3; 3}_2$ &
			$ \dsb{46, 38, 4; 4}_2$ &
			$ \dsb{46, 38, 5; 8}_2$ &
			$ \dsb{46, 34, 6; 6}_2$ \\

			$ \dsb{46, 28, 8; 12}_2$ &
			$ \dsb{46, 32, 8; 14}_2$ &
			$ \dsb{46, 19, 9; 5}_2$ &
			$ \dsb{46, 17, 10; 7}_2$ &
			$ \dsb{46, 25, 11; 19}_2$ &
			$ \dsb{46, 24, 12; 20}_2$ \\

			$ \dsb{46, 1, 15; 7}_2$ &
			$ \dsb{46, 1, 19; 17}_2$ &
			$ \dsb{46, 0, 46; 44}_2$ &
			$ \dsb{47, 46, 2; 1}_2$ &
			$ \dsb{47, 41, 3; 2}_2$ &
			$ \dsb{47, 40, 4; 5}_2$ \\

			$ \dsb{47, 33, 8; 14}_2$ &
			$ \dsb{48, 43, 3; 3}_2$ &
			$ \dsb{48, 42, 4; 6}_2$ &
			$ \dsb{48, 35, 7; 13}_2$ &
			$ \dsb{48, 33, 8; 13}_2$ &
			$ \dsb{48, 23, 13; 21}_2$ \\

			$ \dsb{48, 16, 17; 26}_2$ &
			$ \dsb{48, 0, 20; 16}_2$ &
			$ \dsb{48, 0, 48; 46}_2$ &
			$ \dsb{49, 48, 2; 1}_2$ &
			$ \dsb{49, 45, 3; 4}_2$ &
			$ \dsb{49, 43, 4; 6}_2$ \\

			$ \dsb{49, 35, 8; 14}_2$ &
			$ \dsb{50, 46, 3; 4}_2$ &
			$ \dsb{50, 43, 4; 5}_2$ &
			$ \dsb{50, 36, 7; 12}_2$ &
			$ \dsb{50, 35, 8; 13}_2$ &
			$ \dsb{50, 19, 9; 1}_2$ \\

			$ \dsb{50, 0, 50; 48}_2$ &
			$ \dsb{51, 50, 2; 1}_2$ &
			$ \dsb{51, 46, 3; 3}_2$ &
			$ \dsb{51, 45, 4; 6}_2$ &
			$ \dsb{51, 43, 5; 8}_2$ &
			$ \dsb{51, 34, 6; 1}_2$ \\

			$ \dsb{51, 38, 7; 13}_2$ &
			$ \dsb{51, 37, 8; 14}_2$ &
			$ \dsb{51, 9, 12; 2}_2$ &
			$ \dsb{52, 48, 3; 4}_2$ &
			$ \dsb{52, 46, 4; 6}_2$ &
			$ \dsb{52, 44, 5; 8}_2$ \\

			$ \dsb{52, 17, 10; 1}_2$ &
			$ \dsb{52, 0, 22; 18}_2$ &
			$ \dsb{52, 0, 52; 50}_2$ &
			$ \dsb{53, 52, 2; 1}_2$ &
			$ \dsb{53, 48, 3; 3}_2$ &
			$ \dsb{53, 47, 4; 6}_2$ \\

			$ \dsb{53, 44, 5; 7}_2$ &
			$ \dsb{53, 19, 10; 2}_2$ &
			$ \dsb{53, 0, 16; 1}_2$ &
			$ \dsb{54, 50, 3; 4}_2$ &
			$ \dsb{54, 46, 5; 8}_2$ &
			$ \dsb{54, 44, 6; 10}_2$ \\

			$ \dsb{54, 0, 54; 52}_2$ &
			$ \dsb{55, 54, 2; 1}_2$ &
			$ \dsb{55, 51, 3; 4}_2$ &
			$ \dsb{55, 47, 4; 4}_2$ &
			$ \dsb{55, 47, 5; 8}_2$ &
			$ \dsb{56, 52, 3; 4}_2$ \\

			$ \dsb{56, 48, 5; 8}_2$ &
			$ \dsb{56, 0, 24; 20}_2$ &
			$ \dsb{56, 0, 56; 54}_2$ &
			$ \dsb{57, 56, 2; 1}_2$ &
			$ \dsb{57, 52, 3; 3}_2$ &
			$ \dsb{57, 49, 4; 4}_2$ \\

			$ \dsb{57, 49, 5; 8}_2$ &
			$ \dsb{58, 54, 3; 4}_2$ &
			$ \dsb{58, 0, 58; 56}_2$ &
			$ \dsb{59, 58, 2; 1}_2$ &
			$ \dsb{59, 55, 3; 4}_2$ &
			$ \dsb{60, 55, 3; 3}_2$ \\

			$ \dsb{60, 0, 60; 58}_2$ &
			$ \dsb{61, 60, 2; 1}_2$ &
			$ \dsb{61, 57, 3; 4}_2$ &
			$ \dsb{61, 0, 18; 1}_2$ &
			$ \dsb{62, 58, 3; 4}_2$ &
			$ \dsb{62, 51, 4; 1}_2$ \\

			$ \dsb{62, 0, 62; 60}_2$ &
			$ \dsb{63, 62, 2; 1}_2$ &
			$ \dsb{63, 58, 3; 3}_2$ &
			$ \dsb{63, 44, 6; 1}_2$ &
			$ \dsb{63, 17, 13; 4}_2$ &
			$ \dsb{63, 16, 14; 5}_2$ \\

			$ \dsb{64, 60, 3; 4}_2$ &
			$ \dsb{64, 53, 4; 1}_2$ &
			$ \dsb{64, 49, 5; 1}_2$ &
			$ \dsb{64, 40, 10; 14}_2$ &
			$ \dsb{64, 37, 8; 1}_2$ &
			$ \dsb{64, 31, 9; 1}_2$ \\

			$ \dsb{64, 25, 11; 1}_2$ &
			$ \dsb{64, 24, 12; 2}_2$ &
			$ \dsb{64, 21, 16; 17}_2$ &
			$ \dsb{64, 17, 15; 9}_2$ &
			$ \dsb{64, 16, 16; 14}_2$ &
			$ \dsb{64, 12, 14; 2}_2$ \\

			$ \dsb{64, 2, 22; 12}_2$ &
			$ \dsb{64, 1, 20; 9}_2$ &
			$ \dsb{64, 1, 23; 13}_2$ &
			$ \dsb{64, 1, 27; 25}_2$ &
			$ \dsb{64, 0, 24; 14}_2$ &
			$ \dsb{64, 0, 64; 62}_2$ \\
			\bottomrule
		\end{tabular}
	\end{table*}

\begin{table*}
\caption{A Concise Version of the Parameters of Good Qutrit EA-QECCs with $3 \leq n \leq 36$. To obtain the full table, one applies the propagation rules in Section \ref{sec:compute}. Interested readers may contact the authors for data of lengths up to $n=50$. We include entries with $c=0$ since a large database for such codes is not yet currently available online.}
\label{table:qutrit}
\renewcommand{\arraystretch}{1.2}
\centering
\begin{tabular}{llllll}
		\toprule
		$\dsb{n,\kappa,\delta;c}_3 $ & $\dsb{n,\kappa,\delta;c}_3 $ & $\dsb{n,\kappa,\delta;c}_3 $ & $\dsb{n,\kappa,\delta;c}_3 $ & $\dsb{n,\kappa,\delta;c}_3 $ & $\dsb{n,\kappa,\delta;c}_3 $ \\
		\midrule
		$ \dsb{3, 0, 3; 1 }_3$ &
		$ \dsb{5, 4, 2; 1 }_3$ &
		$ \dsb{5, 2, 3; 1 }_3$ &
		$ \dsb{5, 0, 4; 1 }_3$ &
		$ \dsb{5, 0, 5; 3 }_3$ &
		$ \dsb{6, 2, 4; 2 }_3$ \\

		$ \dsb{7, 3, 3; 0 }_3$ &
		$ \dsb{7, 0, 6; 3 }_3$ &
		$ \dsb{8, 4, 3; 0 }_3$ &
		$ \dsb{8, 0, 7; 4 }_3$ &
		$ \dsb{10, 6, 3; 0 }_3$ &
		$ \dsb{10, 4, 4; 0 }_3$ \\

		$ \dsb{10, 4, 5; 2 }_3$ &
		$ \dsb{10, 1, 6; 1 }_3$ &
		$ \dsb{10, 2, 7; 4 }_3$ &
		$ \dsb{10, 0, 8; 4 }_3$ &
		$ \dsb{10, 0, 9; 6 }_3$ &
		$ \dsb{10, 0, 10; 8 }_3$ \\

		$ \dsb{11, 0, 11; 9 }_3$ &
		$ \dsb{12, 0, 12; 10 }_3$ &
		$ \dsb{13, 4, 7; 5 }_3$ &
		$ \dsb{13, 1, 10; 8 }_3$ &
		$ \dsb{13, 0, 13; 11 }_3$ &
		$ \dsb{14, 8, 3; 0 }_3$ \\

		$ \dsb{14, 5, 6; 3 }_3$ &
		$ \dsb{14, 2, 9; 6 }_3$ &
		$ \dsb{14, 0, 14; 12 }_3$ &
		$ \dsb{15, 9, 3; 0 }_3$ &
		$ \dsb{15, 5, 5; 0 }_3$ &
		$ \dsb{15, 5, 7; 4 }_3$ \\

		$ \dsb{15, 4, 8; 5 }_3$ &
		$ \dsb{15, 0, 12; 9 }_3$ &
		$ \dsb{15, 0, 15; 13 }_3$ &
		$ \dsb{16, 10, 3; 0 }_3$ &
		$ \dsb{16, 9, 4; 1 }_3$ &
		$ \dsb{16, 7, 5; 1 }_3$ \\

		$ \dsb{16, 6, 6; 2 }_3$ &
		$ \dsb{16, 6, 7; 4 }_3$ &
		$ \dsb{16, 5, 8; 5 }_3$ &
		$ \dsb{16, 4, 9; 6 }_3$ &
		$ \dsb{16, 2, 10; 6 }_3$ &
		$ \dsb{16, 1, 11; 7 }_3$ \\

		$ \dsb{16, 1, 12; 9 }_3$ &
		$ \dsb{16, 0, 13; 10 }_3$ &
		$ \dsb{16, 0, 16; 14 }_3$ &
		$ \dsb{17, 11, 3; 0 }_3$ &
		$ \dsb{17, 10, 4; 1 }_3$ &
		$ \dsb{17, 9, 5; 2 }_3$ \\

		$ \dsb{17, 8, 6; 3 }_3$ &
		$ \dsb{17, 0, 14; 11 }_3$ &
		$ \dsb{17, 0, 17; 15 }_3$ &
		$ \dsb{18, 12, 3; 0 }_3$ &
		$ \dsb{18, 11, 4; 1 }_3$ &
		$ \dsb{18, 10, 5; 2 }_3$ \\

		$ \dsb{18, 6, 8; 4 }_3$ &
		$ \dsb{18, 6, 9; 6 }_3$ &
		$ \dsb{18, 4, 10; 6 }_3$ &
		$ \dsb{18, 0, 18; 16 }_3$ &
		$ \dsb{19, 13, 3; 0 }_3$ &
		$ \dsb{19, 12, 4; 1 }_3$ \\

		$ \dsb{19, 11, 5; 2 }_3$ &
		$ \dsb{19, 2, 13; 11 }_3$ &
		$ \dsb{19, 0, 19; 17 }_3$ &
		$ \dsb{19, 0, 20, 19 }_3$ &
		$ \dsb{20, 12, 5; 2 }_3$ &
		$ \dsb{20, 10, 6; 4 }_3$ \\

		$ \dsb{20, 9, 7; 5 }_3$ &
		$ \dsb{20, 6, 10; 6 }_3$ &
		$ \dsb{20, 5, 11; 9 }_3$ &
		$ \dsb{20, 3, 12; 9 }_3$ &
		$ \dsb{20, 0, 15; 12 }_3$ &
		$ \dsb{20, 0, 20; 18 }_3$ \\

		$ \dsb{21, 15, 3; 0 }_3$ &
		$ \dsb{21, 13, 4; 0 }_3$ &
		$ \dsb{21, 11, 6; 4 }_3$ &
		$ \dsb{21, 10, 7; 5 }_3$ &
		$ \dsb{21, 2, 14; 11 }_3$ &
		$ \dsb{21, 2, 15; 13 }_3$ \\

		$ \dsb{21, 1, 16; 14 }_3$ &
		$ \dsb{21, 0, 21; 19 }_3$ &
		$ \dsb{22, 16, 3; 0 }_3$ &
		$ \dsb{22, 15, 4; 1 }_3$ &
		$ \dsb{22, 12, 6; 4 }_3$ &
		$ \dsb{22, 11, 7; 5 }_3$ \\

		$ \dsb{22, 1, 17; 15 }_3$ &
		$ \dsb{22, 0, 22; 20 }_3$ &
		$ \dsb{23, 17, 3; 0 }_3$ &
		$ \dsb{23, 16, 4; 1 }_3$ &
		$ \dsb{23, 9, 8; 6 }_3$ &
		$ \dsb{23, 4, 13; 11 }_3$ \\

		$ \dsb{23, 0, 23; 21 }_3$ &
		$ \dsb{24, 13, 6; 3 }_3$ &
		$ \dsb{24, 11, 8; 7 }_3$ &
		$ \dsb{24, 6, 11; 8 }_3$ &
		$ \dsb{24, 6, 12; 10 }_3$ &
		$ \dsb{24, 0, 19; 16 }_3$ \\

		$ \dsb{24, 0, 24; 22 }_3$ &
		$ \dsb{25, 17, 4; 0 }_3$ &
		$ \dsb{25, 15, 5; 2 }_3$ &
		$ \dsb{25, 9, 10; 8 }_3$ &
		$ \dsb{25, 5, 14; 12 }_3$ &
		$ \dsb{25, 3, 16; 14 }_3$ \\

		$ \dsb{25, 2, 17; 15 }_3$ &
		$ \dsb{25, 0, 25; 23 }_3$ &
		$ \dsb{26, 18, 4; 0 }_3$ &
		$ \dsb{26, 13, 7; 5 }_3$ &
		$ \dsb{26, 12, 8; 6 }_3$ &
		$ \dsb{26, 11, 9; 7 }_3$ \\

		$ \dsb{26, 10, 10; 8 }_3$ &
		$ \dsb{26, 7, 12; 9 }_3$ &
		$ \dsb{26, 6, 14; 12 }_3$ &
		$ \dsb{26, 3, 17; 15 }_3$ &
		$ \dsb{26, 2, 18; 16 }_3$ &
		$ \dsb{26, 0, 26; 24 }_3$ \\

		$ \dsb{27, 19, 4; 0 }_3$ &
		$ \dsb{27, 17, 5; 2 }_3$ &
		$ \dsb{27, 15, 6; 2 }_3$ &
		$ \dsb{27, 14, 7; 5 }_3$ &
		$ \dsb{27, 13, 8; 6 }_3$ &
		$ \dsb{27, 12, 9; 7 }_3$ \\

		$ \dsb{27, 11, 10; 8 }_3$ &
		$ \dsb{27, 8, 12; 9 }_3$ &
		$ \dsb{27, 8, 13; 11 }_3$ &
		$ \dsb{27, 7, 14; 12 }_3$ &
		$ \dsb{27, 4, 16; 13 }_3$ &
		$ \dsb{27, 3, 18; 16 }_3$ \\

		$ \dsb{27, 2, 19; 17 }_3$ &
		$ \dsb{27, 1, 20; 18 }_3$ &
		$ \dsb{27, 0, 27, 25 }_3$ &
		$ \dsb{28, 20, 4; 0 }_3$ &
		$ \dsb{28, 14, 8; 6 }_3$ &
		$ \dsb{28, 13, 9; 7 }_3$ \\

		$ \dsb{28, 12, 10; 8 }_3$ &
		$ \dsb{28, 11, 11; 9 }_3$ &
		$ \dsb{28, 10, 13; 12 }_3$ &
		$ \dsb{28, 9, 12; 9 }_3$ &
		$ \dsb{28, 8, 14; 12 }_3$ &
		$ \dsb{28, 5, 16; 13 }_3$ \\

		$ \dsb{28, 4, 17; 14 }_3$ &
		$ \dsb{28, 4, 18; 16 }_3$ &
		$ \dsb{28, 4, 19; 18 }_3$ &
		$ \dsb{28, 2, 20; 18 }_3$ &
		$ \dsb{28, 1, 21; 19 }_3$ &
		$ \dsb{28, 0, 28; 26 }_3$ \\

		$ \dsb{29, 23, 3; 0 }_3$ &
		$ \dsb{29, 21, 4; 0 }_3$ &
		$ \dsb{29, 19, 5; 2 }_3$ &
		$ \dsb{29, 17, 6; 2 }_3$ &
		$ \dsb{29, 16, 7; 5 }_3$ &
		$ \dsb{29, 0, 29; 27 }_3$ \\

		$ \dsb{30, 23, 4; 1 }_3$ &
		$ \dsb{30, 21, 6; 5 }_3$ &
		$ \dsb{30, 18, 7; 6 }_3$ &
		$ \dsb{30, 10, 15; 16 }_3$ &
		$ \dsb{30, 0, 12; 0 }_3$ &
		$ \dsb{30, 0, 30; 28 }_3$ \\

		$ \dsb{31, 25, 3; 0 }_3$ &
		$ \dsb{31, 25, 4; 2 }_3$ &
		$ \dsb{31, 22, 5; 3 }_3$ &
		$ \dsb{31, 22, 6; 5 }_3$ &
		$ \dsb{31, 0, 31; 29 }_3$ &
		$ \dsb{32, 26, 4; 2 }_3$ \\

		$ \dsb{32, 23, 5; 3 }_3$ &
		$ \dsb{32, 23, 6; 5 }_3$ &
		$ \dsb{32, 0, 32; 30 }_3$ &
		$ \dsb{33, 27, 3; 0 }_3$ &
		$ \dsb{33, 27, 4; 2 }_3$ &
		$ \dsb{33, 25, 5; 4 }_3$ \\

		$ \dsb{33, 24, 6; 5 }_3$ &
		$ \dsb{33, 14, 10; 9 }_3$ &
		$ \dsb{33, 13, 13; 14 }_3$ &
		$ \dsb{33, 13, 14; 16 }_3$ &
		$ \dsb{33, 12, 15; 17 }_3$ &
		$ \dsb{33, 11, 16; 18 }_3$ \\

		$ \dsb{33, 10, 17; 19 }_3$ &
		$ \dsb{33, 8, 18; 19 }_3$ &
		$ \dsb{33, 8, 19; 21 }_3$ &
		$ \dsb{33, 7, 20; 22 }_3$ &
		$ \dsb{33, 0, 33; 31 }_3$ &
		$ \dsb{34, 28, 3; 0 }_3$ \\

		$ \dsb{34, 26, 5; 4 }_3$ &
		$ \dsb{34, 25, 6; 5 }_3$ &
		$ \dsb{34, 12, 12; 10 }_3$ &
		$ \dsb{34, 0, 22; 18 }_3$ &
		$ \dsb{34, 4, 23; 24 }_3$ &
		$ \dsb{34, 0, 34; 32 }_3$ \\

		$ \dsb{35, 30, 3; 1 }_3$ &
		$ \dsb{35, 28, 4; 1 }_3$ &
		$ \dsb{35, 26, 6; 5 }_3$ &
		$ \dsb{35, 15, 10; 8 }_3$ &
		$ \dsb{35, 0, 35; 33 }_3$ &
		$ \dsb{36, 31, 3; 1 }_3$ \\

		$ \dsb{36, 30, 4; 2 }_3$ &
		$ \dsb{36, 27, 5; 3 }_3$ &
		$ \dsb{36, 27, 6; 5 }_3$ &
		$ \dsb{36, 20, 7; 2 }_3$ &
		$ \dsb{36, 19, 10; 11 }_3$ &
		$ \dsb{36, 18, 8; 4 }_3$ \\

		$ \dsb{36, 17, 9; 5 }_3$ &
		$ \dsb{36, 15, 11; 9 }_3$ &
		$ \dsb{36, 15, 14; 17 }_3$ &
		$ \dsb{36, 14, 13; 14 }_3$ &
		$ \dsb{36, 14, 15; 18 }_3$ &
		$ \dsb{36, 13, 12; 9 }_3$ \\

		$ \dsb{36, 11, 16; 17 }_3$ &
		$ \dsb{36, 11, 18; 21 }_3$ &
		$ \dsb{36, 10, 17; 18 }_3$ &
		$ \dsb{36, 10, 19; 22 }_3$ &
		$ \dsb{36, 8, 20; 22 }_3$ &
		$ \dsb{36, 8, 21; 24 }_3$ \\

		$ \dsb{36, 0, 36; 34 }_3$ &
		&&&&\\
		\bottomrule
	\end{tabular}
\end{table*}

We provide tables of parameters of the best-performing qubit and qutrit EAQECCs that we can explicitly construct in Tables \ref{table:qubit} and \ref{table:qutrit}. Among the parameters $\dsb{n,\kappa,\delta;c}_q$, all other parameters being equal, we record the smallest $n$, the largest $\kappa$, the largest $\delta$, and the smallest $c$ for $q \in \{2,3\}$. The tables are compressed using the propagation rules in this paper and those given in \cite{Galindo2019,Galindo2021,GraHubWin2022}. For ease of reference we list the following eight propagation rules. The first four rules are trivial. Rules $(6)$ to $(8)$ come from Theorem \ref{thm:more}, \cite[Theorem 7]{Luo2022}, and \cite[Theorem 8]{Luo2022}, respectively.
\begin{enumerate}
\item[(1)] Length extension: $[\![n,k,d;c]\!]_q \longrightarrow
      [\![n+1,k,d;c]\!]_q$.
\item[(2)] Subcode: $[\![n,k,d;c]\!]_q \longrightarrow
      [\![n,k-1,d;c]\!]_q$.
\item[(3)] Less distance: $[\![n,k,d;c]\!]_q \longrightarrow
      [\![n,k,d-1;c]\!]_q$.
\item[(4)] Requiring more entanglement:\\
$[\![n,k,d;c]\!]_q \longrightarrow
      [\![n,k,d;c+1]\!]_q$.
\item[(5)] Puncturing, assuming $d>1$ and $c<n-k$:\\ $[\![n,k,d;c]\!]_q\longrightarrow
      [\![n-1,k,d-1;c]\!]_q$.
\item[(6)] Increasing the dimension of pure $q$-ary quantum code with $q>2$ by using just enough extra entanglement: $[\![n,k,d;c]\!]_q\longrightarrow
      [\![n,k+1,d;c+1]\!]_q$.
\item[(7)] Reducing the length by using just enough extra entanglement: $[\![n,k,d;c]\!]_q \longrightarrow
      [\![n-1,k,d;c+1]\!]_q$.
\item[(8)] Shortening pure quantum code:\\
$[\![n,k,d;c]\!]_q \longrightarrow [\![n-1,k+1,d-1;c]\!]_q$.
\end{enumerate}

\section{Concluding Remarks}\label{sec:conclu}

The use of pre-shared entanglement in quantum error control raises
questions. How do entanglement-assisted QECCs compare to other QECCs that draw on different resources? In what setups can they be more useful than the others? The enhanced rate or better error-handling capability offered by EAQECCs must be paid for by the additional cost of pre-shared entanglement. On the more practical front, one asks how to best share ebits and how many of them to share.

It is possible for the net rate $\Bar{\rho}(Q)$ to be zero or
negative.
Can such a code be useful in practice? Since shared entanglement $\ket{\phi}_{AB}$ is independent of the message $\ket{\varphi}$, it can be prepared ahead of time and stored to be used as and when needed. In a quantum network, where usage varies over time, Alice and Bob can use periods of low usage to accumulate ebits. These can then be utilized to increase the transmission rate without trading off on the error-correcting power when the network usage grows higher.

Codes with positive net rate can be used as building blocks in the
construction of \textit{catalytic quantum codes}, leading to the
quantum analogue of highly-efficient classical codes such as Turbo and LDPC codes \cite{Wilde2014}.

The quantum setup provides a rich ground for coding theorists of the more classical mould to venture into topics hitherto less explored. Instead of focusing on quantum codes that meet the analogue of the Singleton bound, for example, constructing qubit and qutrit codes that have better chances of being implemented in actual quantum devices and networks could take a more focal position.

We identify the following open directions for further investigation.
\begin{enumerate}
    \item Establish sharper lower and upper bounds on the parameters of best EAQECCs, especially for qubit and qutrit codes.
    \item Find the quantum code with the largest rate for a specified quantum distance and hull dimension. The duality can be chosen among suitable choices of inner products, depending on the construction routes.
    \item In the classical setting, given a length $n$ and dimension $k$, construct a code with the largest hull and optimal dual distance.
\end{enumerate}

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text:
Q: Error evaluationScheme recommenderlab in R I am newbie in R, actually I have mining movielens-M data to provide recommendation system, but when I run my code following below, I faced some error like this:
> ##read the rating data for all users
> readData<-function(){
+   ratingDF <- read.delim("ratings.dat", sep=':',header=F)
+   ratingDF <- subset(ratingDF, select = c("V1","V3","V5","V7"))
+   names(ratingDF) <- c("userID","movieID","rating","timestamp")
+
+   moviesDF <- readLines("movies.dat")
+   moviesDF <- as.data.frame(do.call("rbind",strsplit(moviesDF,"::")),stringsAsFactors = FALSE)
+   names(moviesDF) <- c("movieID","Title","Genre")
+
+   return(list(ratingDF=ratingDF, movieDF=moviesDF))
+ }
>
>
> ##data cleansing and processing
> preProcess= function(ratingDF,moviesDF){
+
+   ratingDF[,2] <- dataList$movieDF$Title[as.numeric(ratingDF[,2])]
+   #remove duplicate
+   ratingDF <- ratingDF[!duplicated(ratingDF[,1:2]),]
+ }
>
>
> createRatingMatrix <- function(ratingDF){
+
+   #converting the ratingData data frame into rating matrix
+   ratingDF_tmp <- dcast(ratingDF, userID ~ movieID, value.var = "rating", index ="userID")
+   ratingDF <- ratingDF_tmp[,2:ncol(ratingDF_tmp)]
+
+   ratingMat<-as(ratingDF,"matrix")
+   movieRatingMat<-as(ratingMat,"realRatingMatrix")
+
+   #setting up the dimnames
+   dimnames(movieRatingMat)[[1]]<-row.names(ratingDF)
+   return(movieRatingMat)
+
+ }
>
> #create recommender model
> evaluateModels<-function(movieRatingMat){
+
+   #find out and anlyze available recommendation algorithm option for realRatingMatrix data
+   recommenderRegistry$get_entries(dataType="realRatingMatrix")
+   scheme <- evaluationScheme(movieRatingMat, method="split", train=.9, k=1, given=10, goodRating=4)
+   algorithms<-list(
+     RANDOM = list(name="RANDOM", param=NULL),
+     POPULAR = list(name="POPULAR", param=NULL),
+     UBCF = list(name="UBCF", param=NULL),
+     IBCF = list(name="IBCF", param=NULL)
+   )
+
+   #run algorithms, predict next n movie
+   results<-evaluate(scheme, algorithms, n=c(1,3,5,10,15,20))
+
+   #select the first results
+   return(results)
+
+ }
>
>
> ##load movie lens data
> dataList<-readData()
>
> ratingDF<-preProcess(dataList$ratingDF, dataList$movieDF)
>
>
> movieRatingMat<-createRatingMatrix(ratingDF)
>
> evalList<-evaluateModels(movieRatingMat)
 Show Traceback

 Rerun with Debug
 Error in .local(data, ...) : Some observations have size<given!
>

In here, I have known the problem, it is because given parameter, but I dont know why it can happened.

A: In recommender lab, given parameter is single number of items given for evaluation or a vector of length of data giving the number of items given for each observation. So why you faced an error is because the vector size of your data is less that the vector of given size.

A: @Kavipriya: Some of your users in your movieRatingMat matrix (that is some rows) might have ratings for less than 10 movies say 4 or 6, and in that case, you are asking recommenderlab to use 10 ratings anyways which it can't use because they are missing right. The workaround is to use a lower value for given = or input that matrix which has a sufficient number of ratings for every user. In case you don't know maths behind it: package calculates the distance between 2 vectors and both the vectors need to be of the same length. Hope that helps.

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text:
Edward Henry Richard Harcourt-Smith conocido como Ed Harcourt, es un músico y compositor británico nacido el 14 de agosto de 1977 en Lewes (Sussex Oriental, Inglaterra).  Hasta la fecha ha publicado seis álbumes de estudio, dos EP, y trece sencillos. Su álbum debut, Here Be Monsters, fue nominado en 2001 a los Mercury Prize. Su música está influenciada por artistas como Tom Waits, Nick Cave o Jeff Buckley, entre otros.

Vida personal
Harcourt es el menor de los tres hermanos, su padre es un oficial del Ejército Británico de ascendencia aristocrática. Es sobrino-nieto de la gastrónoma Elizabeth David y de Nicholas Ridley, Baron Ridley de Liddesdale. Su hermano es el reconocido paleontólogo William Harcourt-Smith. Está casado con Gita Harcourt-Smith, cantante y compositora de The Langley Sisters. La pareja tiene dos hijos, una niña llamada Roxy, y un niño llamado Franklyn.

Carrera musical
Antes de comenzar su carrera como solista, Harcourt tocó el bajo y los teclados con "Snug", una banda formada a mediados de los 90 por él mismo, James Deane, Ed Groves, y Johnny Lewsley. La banda grabó dos álbumes y un puñado de sencillos juntos antes de disolverse.

En 2000, Harcourt grabó su debut en solitario con el mini-álbum Maplewood directamente con una grabadora multipista en la mansión de su abuela en Sussex. Tras firmar con Heavenly Records y publicar Maplewood en noviembre de 2000, Harcourt grabó su álbum de estudio del debut Here Be Monsterscon los productores Gil Norton y Tim Holmes. El álbum fue lanzado en junio de 2001, y alcanzó la posición 84 en el UK Albums Chart. Un mes después de su publicación, el álbum fue nominado los Mercury Prize de 2001.

En octubre de 2002, Harcourt publica una versión del tema de Brian Wilson, "Still I Dream of It", coincidiendo con este lanzamiento, graba su segundo álbum From Every Sphere que se publica en febrero de 2003. El álbum se convirtió en su mayor éxito de ventas en el Reino Unido, alcanzando el puesto número 39 de las listas británicas. En el resto de Europa también tuvo muy buena acogida, llegando al puesto número 6 en Suecia, el 25 en Noruega, y el 103 en Francia.  Del álbum From Every Sphere se extrajeron dos sencillos, "All of Your Days Will Be Blessed" que alcanzó el puesto 35 de las listas de éxitos y "Watching the Sun Come Up", que llegó al puesto 79. Tras la gira por el Reino Unido, Harcourt publicó su tercer disco, Strangers, en septiembre de 2004. El álbum llegó al número 57 de las listas británicas y al número 7 en las suecas. Se extrajeron los sencillos "This One's for You", "Born in the '70s", y "Loneliness."

Durante 2005, Harcourt interpretó algunos conciertos con un proyecto paralelo llamado Wild Boar. En agosto de 2005 publicó una recopilación de caras B y rarezas bajo el título de Elephant's Graveyard.

El cuarto álbum de Harcourt, The Beautiful Lie, fue publicado en junio de 2006, recibiendo una cálida acogida por parte de la crítica.  Allmusic publicó; "una aventura estimulante y con frecuencia magnífica, esencial para los viejos fans y un buen lugar de comienzo para los recién llegados." A pesar de las buenas críticas, el álbum no llegó a se un éxito en las listas británicas, alcanzando un modesto puesto 97. The Beautiful Lie fue coproducido por Jari Haapalainen, y contó con la colaboración de The Magic Numbers,  que cantan coros en "Revolution in the Heart".Graham Coxon, tocó la guitarra en "Visit from the Dead Dog y Gita, la esposa de Harcourt tocó el violín en "Braille." A principios de 2007 colaboró con el trompetista francés de jazz Erik Truffaz en Arkhangelsk. En octubre de ese mismo año publicó el álbum recopilatorio Until Tomorrow Then: The Best of Ed Harcourt.

Tras finalizar su contrato con Heavenly/EMI, Harcourt firmó con Dovecote Records para lanzar en Estados Unidos The Beautiful Lie, al mismo tiempo que publicaba el EP Russian Roulette en mayo de 2009. También en 2009, Harcourt compuso la música para la banda sonora original de la secuela de Donnie Darko, S. Darko. A comienzos de 2010, escribió la canción "Isabel" para Aldeas Infantiles SOS y el Fondo de Ayuda de Emergencia a Haití.

Su quinto álbum de estudio, Lustre, fue publicado el 14 de junio de 2010, fue el primer álbum publicado con su propio sello discográfico "Piano Wolf Recordings" (distribuido por Essential Music Marketing). Ed declaró en su blog oficial de MySpaces que grabó el álbum con el productor Ryan Hadlock en el Condado de Snohomish en el Estado de Washington

En 2013, Harcourt anunció en su perfil de Facebook el lanzamiento de dos nuevos álbumes, uno de los cuales había sido compuesto apenas en un mes y grabado por él mismo en seis horas.  El álbum llevó por título Back into The Woods y fue lanzado el 25 de febrero de 2013 via CCCLX, el nuevo sello discográfico de Sean Adams, fundador de "Drowned in Sound". En enero de 2014 vio la luz "Time of Dust", mini-álbum de seis canciones, publicado también con CCCLX.

En 2013, Harcourt comenzó a trabajar con la artista pop británica Sophie Ellis-Bextor para quién produjo y coescribió su quinto álbum de estudio, Wanderlust.

En 2015, Harcourt prestó su voz para el tema "Villain" del álbum 8:58 de Paul Hartnoll.

Directos

Harcourt ha actuado como telonero de numerosas bandas de prestigio como R.E.M., Snow Patrol, Wilco, Beth Orton, The Divine Comedy, Supergrass, Norah Jones, Brian Jonestown Massacre y Neil Finn. Ha actuado también junto a bandas y artistas como Magnet, The Magic Numbers, The Tiny, Sondre Lerche, Regina Spektor, Leslie Feist, Patti Smith y Marianne Faithfull.

Discografía
 Here Be Monsters (2001)
 From Every Sphere (2003)
 Strangers (2004)
 Elephant's Graveyard (álbum recopilatorio, 2005)
 The Beautiful Lie (2006)
 Until Tomorrow Then: The Best of Ed Harcourt (2007) recopilatorio
 Lustre (2010)
 Back into The Woods (2013)
 Time of Dust (2014)
 Furnaces (2016)

Referencias

Enlaces externos

Artistas de Capitol Records
Cantantes masculinos de Inglaterra
Pianistas de Inglaterra
Cantautores de Inglaterra
Productores discográficos de Reino Unido
Productores de música rock
Productores de pop

redpajama_set_name: RedPajamaWikipedia


The company has announced a new "greenfield" investment for an integrated plant (tissue production and converting facilities) in Oklahoma. The new production site will be located in Inola, Rogers County, near the town of Tulsa, Oklahoma.
"I am glad to announce this new greenfield investment – the second one, after the first made in Ohio. This new investment will support our growth in the US significantly, being the American market is one of the crucial ones for our Group" commented Luigi Lazzareschi, CEO of Sofidel Group.
"Sofidel will build here a new, modern integrated plant to serve the South-Central and Central-Western areas of this great country, where we have been operating since 2012 and we have been growing year after year at a very fast pace. I would like to thank all the partners that have supported us in making this new project, they have played an integral role in helping us achieve this goal".
The investment refers to the installation of two tissue paper machines for an overall production capacity of 120,000 tonnes per year each, with related converting lines. One machine will be dedicated mainly to toilet tissue production, while the second one will be used mostly to produce towels.
As a whole, Sofidel is committed to investing $ 360 million over three years (2018 – 2019 – 2020).
The facilities building is expected to start in March 2018 and will be completed by Q2 2020, while the converting lines and the warehouse functions will be in place by the end of Q2 2019. The new production plant, once completed, will employ 300 full-time workers.


text:
accent on the individual: bulletin of florida technological university, 1972 - 1973 (1)
university of central florida catalog, 1985 - 1986 (1)
university of central florida graduate catalog, 1987 - 1988 (1)
florida technological university -- catalogs (1)
florida technological university -- curricula -- catalogs (1)
florida technological university -- curricula -- periodicals (1)
Description: 166
This catalog includes general introductions to the University of Central Florida, its policies, programs and degree requirements, colleges and departments, and course offerings. It also contains pictures of the campus (cover; p. 18, Higher...
University of Central Florida Catalog, 1985 - 1986
This catalog includes general introductions to the University of Central Florida, its policies, programs and degree requirements, colleges and departments, and course offerings. It also contains pictures of the campus (cover; p. 8, university sign;...
Accent on the individual: Bulletin of Florida Technological University, 1972 - 1973
Florida Technological University -- Catalogs; Florida Technological University -- Curricula -- Catalogs; Florida Technological University -- Curricula -- Periodicals; Universities and colleges -- Curricula -- Florida -- Orlando -- Catalogs; ...
This catalog includes general introductions to Florida Technological University, its policies, programs and degree requirements, colleges and departments, and course offerings. It also contains pictures of campus buildings and facilities (cover,...
University of Central Florida Graduate Catalog, 1990 - 1991
This catalog includes general introductions to the University of Central Florida, its policies, programs and degree requirements, colleges and departments, and course offerings. It also contains pictures of the campus (p. 7, Reflection Pond and the...
This catalog includes general introductions to the University of Central Florida, its policies, programs and degree requirements, colleges and departments, and course offerings. It also contains pictures of the campus life (p. 10; p. 32; p. 40,...

redpajama_set_name: RedPajamaCommonCrawl


text:
Vitabu Reads: From the Land of Diamonds to the Isle of Spice
When young Christopher Williams ventured out of Freetown on a merchant ship bound for Matadi in 1925, little did he know that he would leave footprints in the sand that an unborn grandson would track.
Using the figurative milestones planted by Christopher almost a hundred years ago, Sigismond Henry Tucker creates an intricate picture of his family's migration story of love, endurance, and hope through the pain of separation.
Sigismond's autobiography "From the Land of Diamonds to the Isle of Spice" is a complex lattice of intersecting interests.
On the one hand are black Africans like Sigismond's grandfather, arriving by ship at trading ports dotted all over the continent's big coast; people native to the Congo, Malinke merchants from Senegal carrying out trade on routes throughout west and central Africa; and the bold Americans, English, and Welsh exploring opportunities in old ports of slavery.
Sigismond doesn't say what set his grandfather's travels in motion from Freetown in 1925.
But it probably has to do with the unchanging story of young black men looking for work with big European employers like merchant vessels, which offered a half decent wage mixed in with some adventure.
Labor strikes happening in the colony of Freetown could have been another reason.
Angry railway workers were demanding thousands of pounds in unpaid wages and redundancy money from the colonial government in Freetown. The resulting stoppages would have made life difficult for young people like Christopher who were just starting out in the workforce.
Incidentally, one of the two seafarers with Sigismond's grandfather was Isaac Theophilus Akunna Wallace-Johnson, a workers' leader, journalist, and activist.
ITA's story is well-known.
Born into a poor family in Sierra Leone, I.T.A. Wallace-Johnson emerged as a leader in school. After being demobilized in 1920, Wallace-Johnson moved from job to job, before settling as a clerk in the Freetown municipal government. After being fired from this job in 1926, he left Sierra Leone and became a sailor.
There's some confusion as to the dates in Sigismond's book however. In one section, he says the three seamen left around 1925. In another chapter, he wrote that Wallace-Johnson lost his job as a customs officer in 1926 and left Sierra Leone to become a sailor.
Once in Matadi, the author tells us the three seamen, including Christopher, Sigismond's grandfather, ran into a rich American who offered them a job. Christopher, we are told, later moved to a mining town inland.
The story doesn't end there.
Christopher marries a woman originally from Senegal and the couple have two daughters. He grows rich but after a near fatal encounter with a dying buffalo, Christopher thought of reconnecting with his roots by sending one of his daughters home to Kossoh Town in Hastings, Freetown, to see the sister he had left behind.
His wish came true when Mr. Thompson, the third seaman who had traveled with Christopher and Wallace Johnson, decides to return home and take Christopher's daughter, 12-year-old Lillian, who could only speak Lingala and Bakongo, back to Freetown.
Time passes. Lillian grows up and settles down with Sigismond's father, who had moved to Hastings in Freetown from Sherbo Island, home of the Tuckers descended from an English trader and agent, John Tucker, and a Sherbro princess.
By 1953, Sigismond's parents had separated and he doesn't see his father again until 1968 when word came that he had died in Kenema, about 300 miles east of Freetown.
A few years later Sigismond leaves Freetown for the United Kingdom in September of 1971; he doesn't return until 1990.
But not before going to Brazzaville in December of 1989. By then he had met and married his wife from Grenada in the Caribbean and started his own family in Manchester, England.
Private inquiries had led Sigismond and his mother to the family she left behind when she was only 12-years old.
The reunion and restoration of Christopher's family is probably one of the most fascinating parts in this intriguing story of human voyages and migration in Sigismond's DNA. #seafarersmatter
Vitabu Reads: From the Land of Diamonds to the Isl...


text:
Jared Kushner (ur. 10 stycznia 1981 w Livingston) – amerykański inwestor i przedsiębiorca żydowskiego pochodzenia.

Życiorys
Jared Kushner z zawodu jest inwestorem i biznesmenem. W 2017 został mianowany na doradcę ds. umów handlowych i Bliskiego Wschodu w gabinecie Donalda Trumpa.

Odznaczenia

 Wstęga Orderu Orła Azteckiego (Meksyk, 2018)

Życie prywatne
Od 2009 jego żoną jest Ivanka Trump, z którą ma trójkę dzieci.

Przypisy

Amerykańscy przedsiębiorcy
Odznaczeni Orderem Orła Azteckiego
Urodzeni w 1981


text:
Die Kirche St. Moritz in Speyer war eine gotische Pfarreikirche, die auf dem heutigen Königplatz im Zentrum der Altstadt von Speyer stand. Sie galt damals als die bedeutendste gotische Kirche in der Vorderpfalz.

Geschichte
Die Kirche St. Moritz wurde im 11. Jahrhundert als Hauptpfarrei der Bürgerstadt gegründet. Moderne Forschungen gehen von einer Gründung der Kirche im ausgehenden 9. oder 10. Jahrhundert aus. Im 13. Jahrhundert wurde die Kirche im spätromanischen Stil neu erbaut. Die Kirche besaß einen spitzhelmigen Westturm von 60 Meter Höhe und war ca. 40 Meter lang, 25 Meter breit und bis zum Dach etwa 20 m hoch. Im 15. Jahrhundert, zwischen 1422 und 1468, wurde das vor den Mauern der Stadt auf dem Germansberg befindliche Stift St. German zur St. Moritz-Kirche verlegt und fortan als Stift St. German und Moritz oder St. Germanus und Mauritius bezeichnet. Die Kirche erhielt dabei ein spätgotisches Querhaus und einen Chor. Sie war zudem von einer etwa 2 Meter hohen Ringmauer umgeben.

Beim Speyerer Stadtbrand von 1689 wurde die Kirche bis auf Turm und Umfassungsmauern niedergebrannt. Man setzte sie offenbar teilweise wieder für den Gottesdienst instand, denn es sind zwischen 1729 und 1798 Kirchenbücher der Moritzpfarrei vorhanden, für 1764 ist dort außerdem die Priesterweihe von Damian Hugo Philipp von Lehrbach durch den Speyerer Weihbischof Johann Adam Buckel belegt. Auch das Stift bestand bis zur Säkularisation weiter. Danach wurden seine Güter verpachtet und 1803 verkauft.

Um 1806 wurde das Gotteshaus gänzlich abgebrochen und an der Stelle der heutige Königsplatz angelegt, unter dem bis heute die Fundamente der Kirche liegen.

Die Erinnerungen an dieses ehemalige Wahrzeichen der Bürgerstadt von Speyer sind heute im Historischen Museum der Pfalz ausgestellt. Auch sind noch Zeichnungen von dem einstigen Sakralbau erhalten.

Literatur

Einzelnachweise

Moritz
Mauritiuskirche
Gotisches Bauwerk in Rheinland-Pfalz
Romanische Kirche
Bauwerk der Romanik in Rheinland-Pfalz
Gotische Kirche
Moritz
Zerstört in den 1800er Jahren