New constructions of entanglement-assisted quantum codes

Abstract

We present two new constructions of entanglement-assisted quantum error-correcting codes using some fundamental properties of (classical) linear codes in an effective way. The main ideas include linear complementary dual codes and related concatenation constructions. Numerical examples in modest lengths show that our constructions perform better than known constructions in the literature. We also give a proof on a generalization of binary Singleton type bound on entanglement-assisted quantum error-correcting codes to arbitrary q-ary entanglement-assisted quantum error-correcting codes.

Appendices

Appendix A: Singleton type bound for entanglement-assisted quantum codes

In Appendix A we prove a Singleton type bound for entanglement-assisted quantum codes using methods from [1, 2, 19, 20].

Recall that ${\mathbb F}_{q}$ is the finite field with q elements. Let $\text {char}({\mathbb F}_{q})=p$ so that ${\mathbb F}_{p} \subseteq {\mathbb F}_{q}$. Let ${\mathbb C}$ denote the complex field. We regard ${\mathbb C}^{q}$ as a Hilbert space and let |x〉 be the (column) vectors of an orthonormal basis of ${\mathbb C}^{q}$, where the labels x are elements of ${\mathbb F}_{q}$. Let $\zeta _{p}=\exp (2\pi \sqrt {-1}/p)$ be a p-th root of unity in ${\mathbb C}$. Let $\text {tr}:{\mathbb F}_{q} \rightarrow {\mathbb F}_{p}$ be the trace map from ${\mathbb F}_{q}$ onto ${\mathbb F}_{p}$.

For $a,b \in {\mathbb F}_{q}$, let $\mathcal {X}(a)$ and $\mathcal {Z}(b)$ be the unitary operations on ${\mathbb C}^{q}$ defined by

$$ \begin{array}{@{}rcl@{}} \mathcal{X}(a)|x\rangle=|x+a\rangle \text{and} \mathcal{Z}(b)|x\rangle=\zeta_p^{\text{tr}(bx)}|x\rangle. \end{array} $$

Let ${\mathbb C}^{q^{n}}=({\mathbb C}^{q})^{\otimes n}$ be the n-th tensor of ${\mathbb C}^{q}$. The coordinate basis is given by

$$ \begin{array}{@{}rcl@{}} |\textbf{x}\rangle=|x_1\rangle \otimes |x_2 \rangle \otimes {\cdots} \otimes |x_n\rangle. \end{array} $$

Consider $\textbf {a}=(a_{1},\ldots , a_{n}), \textbf {b}=(b_{1},\ldots , b_{n}) \in {{\mathbb F}_{q}^{n}}$. Let $\mathcal {X}(\textbf {a})$ and $\mathcal {Z}(\textbf {b})$ be the unitary operations on ${\mathbb C}^{q^{n}}$ defined by

$$ \begin{array}{@{}rcl@{}} \mathcal{X}(\textbf{a})|\textbf{x}\rangle=|\textbf{x+a}\rangle \text{and} \mathcal{Z}(\textbf{b})|\textbf{x}\rangle=\zeta_p^{\text{tr}(\textbf{b} \cdot \textbf{x})}|\textbf{x}\rangle, \end{array} $$

where $\textbf {b} \cdot \textbf {x}={\sum }_{i=1}^{n} b_{i} x_{i} \in {\mathbb F}_{q}$.

The set $\mathcal {E}_{n}=\{{\zeta _{p}^{u}}\mathcal {X}(\textbf {a}) \mathcal {Z}(\textbf {b}): \textbf {a}, \textbf {b} \in {{\mathbb F}_{q}^{n}}, u \in {\mathbb F}_{p}\}$ is a finite group of order pq²ⁿ. Let $\textbf {E}_{n} \subseteq \mathcal {E}_{n}$ be the subset defined as

$$ \begin{array}{@{}rcl@{}} \textbf{E}_n=\{\mathcal{X}(\textbf{a}) \mathcal{Z}(\textbf{b}): \textbf{a}, \textbf{b} \in {\mathbb F}_q^n\}. \end{array} $$

For $\textbf {a}=(a_{1},\ldots , a_{n}), \textbf {b}=(b_{1},\ldots , b_{n}) \in {{\mathbb F}_{q}^{n}}$ and $e=\mathcal {X}(\textbf {a}) \mathcal {Z}(\textbf {b}) \in \textbf {E}_{n}$, let the (quantum) weight w_Q(e) of e be the integer

$$ \begin{array}{@{}rcl@{}} w_Q(e)=\#\{1\le i \le n: a_i \neq 0 \text{or} b_i \neq 0\}. \end{array} $$

For 0 ≤ i ≤ n, let $\mathcal {E}_{n}[i]\subseteq \mathcal {E}_{n}$ be the subset defined as

$$ \begin{array}{@{}rcl@{}} \mathcal{E}_n[i]= \{e \in \mathcal{E}_n: w_Q(e)=i\}. \end{array} $$

Definition A.1

A q-ary quantum code of length n is a subspace Q of ${\mathbb C}^{q^{n}}$ such that $\dim _{{\mathbb C}} Q=K \ge 1$. The minimum distance of Q is the largest integer d such that for any error $e \in \mathcal {E}_{n}[i]$ with i < d we have

$$ \begin{array}{@{}rcl@{}} |v\rangle, |w\rangle \in Q \text{are orthogonal vectors} \Rightarrow \langle v | e | w \rangle=0. \end{array} $$

Such a code Q is called an ((n, K, d))_q code.

Next we formally define q-ary entanglement-assisted quantum error correcting codes. Let c be a positive integer. For $|a \rangle \in {\mathbb C}^{q^{n}}$, $|b \rangle \in {\mathbb C}^{q^{c}}$, and $|\psi \rangle =|a \rangle \otimes |b \rangle \in {\mathbb C}^{q^{n+c}}$ note that |a〉〈a| is a qⁿ × qⁿ matrix, |b〉〈b| is a q^c × q^c matrix, and |ψ〉〈ψ| is a q^n+c × q^n+c matrix. Recall that Tr is the trace operation on square matrices and hence Tr(|a〉〈a|) is a complex number. Let Tr_A(|ψ〉〈ψ|) denote the q^c × q^c over ${\mathbb C}$ defined as

$$ \begin{array}{@{}rcl@{}} \text{Tr}_A(|\psi\rangle \langle \psi|)=\text{Tr}(|a\rangle \langle a|) |b\rangle \langle b|. \end{array} $$

We extend this definition to $C^{q^{n+c}}={\mathbb C}^{q^{n}} \otimes {\mathbb C}^{q^{c}}$ as follows: Let $|\psi \rangle \in {\mathbb C}^{q^{n+c}}$ be an arbitrary vector. There exist an integers s ≥ 1, vectors $|a_{1}\rangle , \ldots , |a_{s}\rangle \in {\mathbb C}^{q^{n}}$, vectors $|b_{1}\rangle , \ldots , |b_{s}\rangle \in {\mathbb C}^{q^{c}}$ and complex numbers $\lambda _{1}, \ldots , \lambda _{s} \in {\mathbb C}$ such that |ψ〉 = λ₁|a₁〉⊗|b₁〉 + ⋯ + λ_s|a_s〉⊗|b_s〉. Put |ψ_i〉 = a_i ⊗ b_i for 1 ≤ i ≤ s. We further assume without loss of generality that $\{|a_{i}\rangle : 1\le i \le s\}\subseteq {\mathbb C}^{q^{n}}$ is an orthonormal set or $\{|b_{i}\rangle : 1\le i \le s\}\subseteq {\mathbb C}^{q^{c}}$ is an orthonormal set. Note that Tr_a(|ψ_i〉〈ψ_i|) is a q^c × q^c matrix defined above. Now we define Tr_A(|ψ〉〈ψ|) for the arbitrarily chosen vector $|\psi \rangle \in {\mathbb C}^{q^{n+c}}$ as the q^c × q^c matrix given by

$$ \begin{array}{@{}rcl@{}} \text{Tr}_A(|\psi\rangle \langle \psi|)= \mid \lambda_1 \mid^2 \text{Tr}_A(|\psi_1\rangle \langle \psi_1|) + {\cdots} + \mid \lambda_s \mid^2 \text{Tr}_A(|\psi_s\rangle \langle \psi_s|). \end{array} $$

Definition A.2

An ((n, K, d; c))_q entanglement-assistedq-ary quantum error correcting code is a subspace Q of ${\mathbb C}^{q^{n+c}}$ such that $\dim _{{\mathbb C}} Q=K \ge 1$ satisfying the following two properties:

• 1.) $\displaystyle |\psi \rangle \in Q \Rightarrow \text {Tr}_{A}(|\psi \rangle \langle \psi |)= \frac {1}{q^{c}} I_{q^{c}}$, where $I_{q^{c}}$ is the identity operation on ${\mathbb C}^{q^{c}}$.

• 2.) For integer 0 ≤ i < d and each error $e \in \mathcal {E}_{n}[i]$ we have:

  $$ \begin{array}{@{}rcl@{}} |v^{\prime}\rangle, |w^{\prime}\rangle \in Q \text{are orthogonal vectors} \Rightarrow \langle v^{\prime} | e^{\prime} | w^{\prime} \rangle=0, \end{array} $$

  where $e^{\prime }$ is the error in E_n+c defined as $e^{\prime }=e \otimes I_{q^{c}}$.

Let {|ψ_i〉 : 1 ≤ i ≤ K} be an orthogonal basis of Q. Let

$$ \begin{array}{@{}rcl@{}} P= \sum\limits_{i=1}^K |\psi_i\rangle \langle \psi_i| \end{array} $$

be the orthogonal projection of ${\mathbb C}^{q^{n+c}}$ onto Q. For 1 ≤ i ≤ n and 1 ≤ j ≤ c be the split weight enumerators of Q defined by

$$ \begin{array}{@{}rcl@{}} B_{i,j}=\frac{1}{K^2} \underset{e^{\prime} \in \textbf{E}_c[j]}{\underset{e \in \textbf{E}_n[i]}{\sum}} \left( \text{Tr}\left( (e \otimes e^{\prime})P\right)\right)^2, \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} B_{i,j}^{\perp}=\frac{1}{K} \underset{e^{\prime} \in \textbf{E}_c[j]}{\underset{e \in \textbf{E}_n[i]}{\sum}} \text{Tr}\left( (e \otimes e^{\prime})P(e \otimes e^{\prime})P \right). \end{array} $$

We need to recall some notions and properties of Krawtchouk polynomials before the next theorem. We refer to [26, Section 5.7] for details. Let m be a positive integer. For 0 ≤ i ≤ m, the i-th q²-ary Krawtchouk polynomial with respect to m is

$$ \begin{array}{@{}rcl@{}} P_{i}(x;m)=\sum\limits_{j=0}^i (-1)^j \left( q^2-1\right)^{i-j} \left( \begin{array}{c}x\\j \end{array}\right) \left( \begin{array}{c}n-x\\i-j \end{array}\right). \end{array} $$

(12)

In this paper we consider q²-ary Krawtchouk polynomials. Note that P_i(x; m) is a polynomial of degree i in x with the leading coefficient (−q²)ⁱ/i! and the constant term $\left (\begin {array}{c}{n}\\{i} \end {array}\right )(q-1)^{i}$. In particular

$$ \begin{array}{@{}rcl@{}} P_{0}(x;m)=1 \text{and} P_{1}(x;m)=-q^2x+1. \end{array} $$

(13)

Let 0 ≤ r, s ≤ m be integers. Recall that the Kronecker symbol δ_{r, s} is given by

$$ \begin{array}{@{}rcl@{}} \delta_{r,s} = \left\{ \begin{array}{cl} 1 & \text{if } r=s, \\ 0 & \text{if } r \neq s. \end{array} \right. \end{array} $$

We have the orthogonality relations (see [26, Section 5.7, Corollary 18]) that

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i=0}^m P_{r}(i;m)=P_{i}(s;m)=q^{2m} \delta_{r,s}. \end{array} $$

(14)

Moreover we also have (see [26, Section 5.7, Exercise 41]) that

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i=0}^m \left( \begin{array}{c}{n-r}\\{n-s} \end{array}\right) P_r(x;m)=q^{2s} \left( \begin{array}{c}{n-x}\\{s} \end{array}\right). \end{array} $$

(15)

Let n, c be positive integers. Let $f(x,y) \in {\mathbb R}[x,y]$ be a polynomial of degree at most n in x and at most c in y. The properties above imply that there exist uniquely determined coefficients $f_{u,v} \in {\mathbb R}$ for 0 ≤ u ≤ n and 0 ≤ v ≤ c such that

$$ \begin{array}{@{}rcl@{}} f(x,y)=\sum\limits_{u=0}^{n} \sum\limits_{v=0}^{c} f_{u,v} P_u(x;n) P_v(y;c). \end{array} $$

(16)

The expansion in (16) is called the Krawtchouk expansion of f(x, y). Using (14) we determine the coefficients in the Krawtchouk expansion as

$$ \begin{array}{@{}rcl@{}} f_{i,j}=\frac{1}{q^{2(n+c)}} \sum\limits_{u=0}^{n} \sum\limits_{v=0}^{c} f(u,v) P_u(i;n)P_v(j;c) \end{array} $$

(17)

for 0 ≤ i ≤ n and 0 ≤ j ≤ c.

Now we are ready for the next theorem. The proof of the following theorem uses the same arguments as in [20, Theorem 2] and hence we skip its proof.

Theorem A.3

Assume that Q is an ((n, K, d; c))_q entanglement-assisted q-ary quantum error correcting code with projector P and split weight enumerators B_{i, j} and $B_{i,j}^{\perp }$ for 0 ≤ i ≤ n, 0 ≤ j ≤ c, respectively. Then the following properties hold:

• 1.) $B_{0,0}=B_{0,0}^{\perp }=1$.

• 2.) $B_{i,j}^{\perp } \ge B_{i,j} \ge 0$.

• 3.) B_0,j = 0 for j = 1,…,c.

• 4.) $B_{i,0}=B_{i,0}^{\perp }$ for i = 1,…,d − 1.

• 5.) $B_{d,0} < B_{d,0}^{\perp }$.

• 6.)

  $$ \begin{array}{@{}rcl@{}} B_{i,j}^{\perp}=\frac{K}{q^{n+c}} \sum\limits_{u=0}^{n} \sum\limits_{v=0}^{c} B_{u,v}P_{i}(u;n)P_j(v;c) \end{array} $$

  and

  $$ \begin{array}{@{}rcl@{}} B_{i,j}=\frac{1}{q^{n+c}K} \sum\limits_{u=0}^{n} \sum\limits_{v=0}^{c} B_{u,v}^{\perp} P_{i}(u;n)P_j(v;c). \end{array} $$

The following theorem is similar to [2, Theorem 4] and [20, Theorem 5].

Theorem A.4

Let Q be an ((n, K, d; c))_q entanglement-assisted quantum error correcting code. For 0 ≤ i ≤ n and 0 ≤ j ≤ c let $f_{i,j} \in {\mathbb R}$ such that

$$ \begin{array}{@{}rcl@{}} f_{i,j} \ge 0 \text{for all } i,j \text{ in this range}, \end{array} $$

(18)

and

$$ \begin{array}{@{}rcl@{}} f_{i,0} > 0 \text{for} 0 \le i \le d-1. \end{array} $$

(19)

Let $f(x,y) \in {\mathbb R}[x,y]$ be the polynomial defined using the coefficients f_i,j above and the Krawtchouk polynomials in x and y variables given by

$$ \begin{array}{@{}rcl@{}} f(x,y)=\sum\limits_{i=0}^n\sum\limits_{j,0}^{c} f_{i,j} P_{i}(x;n) P_j(y;c) \end{array} $$

(20)

Assume further that

$$ \begin{array}{@{}rcl@{}} f(\alpha,\beta) \le 0 \text{for all integers } d \le \alpha \le n \text{and} 1 \le \beta \le c. \end{array} $$

(21)

Then we have

$$ \begin{array}{@{}rcl@{}} K \le \frac{1}{q^{n+c}} \max_{0 \le \ell \le d-1} \frac{f(\ell,0)}{f_{\ell,0}}. \end{array} $$

Proof

Let B_{i, j} and $B_{i,j}^{\perp }$ be the split weight enumerators of Q. Using (18) we obtain

$$ \begin{array}{@{}rcl@{}} LHS:=q^{n+c} K \sum\limits_{i=0}^{d-1} f_{i,0} B_{i,0} \le q^{n+c} K \sum\limits_{i=0}^{d-1} \sum\limits_{j=0}^{c} f_{i,j} B_{i,j}. \end{array} $$

(22)

It follows from Theorem A.3, item 6) that

$$ \begin{array}{@{}rcl@{}} B_{i,j}=\frac{1}{q^{n+c}K} \sum\limits_{u=0}^{n} \sum\limits_{v=0}^{c} B_{u,v}^{\perp} P_{i}(u;n)P_j(v;c) \end{array} $$

(23)

for 0 ≤ i ≤ d − 1 and 0 ≤ j ≤ c. Combining (22) and (23) we obtain that

$$ \begin{array}{@{}rcl@{}} \begin{array}{rcl} LHS & \le & {\sum}_{i=0}^{d-1}{\sum}_{j=0}^{c} f_{i,j} {\sum}_{u=0}^{n} {\sum}_{v=0}^{c} B_{u,v}^{\perp} P_{i}(u;n)P_j(v;c) \\ \\ & = & {\sum}_{u=0}^{n} {\sum}_{v=0}^{c} B_{u,v}^{\perp} f(u,v), \end{array} \end{array} $$

(24)

where we use the definition in (20). Using (24) and (21) we get that

$$ \begin{array}{@{}rcl@{}} LHS \le \sum\limits_{i=0}^{d-1} B_{i,0}^{\perp} f(i,0). \end{array} $$

(25)

Combining (25) and Theorem A.3, item 4 we conclude that

$$ \begin{array}{@{}rcl@{}} q^{n+c}K \sum\limits_{i=0}^{d-1} f_{i,0} B_{i,0} \le \sum\limits_{i=0}^{d-1} f(i,0) B_{i,0}. \end{array} $$

(26)

It follows from (26) that we have

$$ \begin{array}{@{}rcl@{}} K \le \frac{1}{q^{n+c}} \frac{{\sum}_{i=0}^{d-1} f(i,0) B_{i,0}}{{\sum}_{i=0}^{d-1} f_{i,0} B_{i,0}} \le \frac{1}{q^{n+c}} \max_{0 \le i \le d-1} \frac{f(i,0)}{f_{i,0}}. \end{array} $$

□

The main result of this Appendix is the following Singleton type bound for entanglement-assisted quantum error correcting codes.

Theorem A.5

Let Q be an ((n, K, d; c))_q entanglement-assisted quantum error correcting code. If $d \le \frac {n+2}{2}$, then

$$ \begin{array}{@{}rcl@{}} K \le q^{n+c-2(d-1)}. \end{array} $$

Proof

Let $f(x,y)\in {\mathbb R}[x,y]$ be the polynomial

$$ \begin{array}{@{}rcl@{}} f(x,y)=\frac{q^{2(n+c-d+1)}}{(-1)^{c} c!} \prod\limits_{i=d}^{n} \left( 1 - \frac{x}{i} \right) \prod\limits_{j=1}^{c} \left( y-j\right). \end{array} $$

(27)

For integers 0 ≤ α ≤ n and 0 ≤ β ≤ c we have

• f(α, β) = 0 if 1 ≤ β ≤ c, and

• $f(\alpha ,0)=q^{2(b+c-d+1)} \frac {\left (\begin {array}{c}{n-\alpha }\\{n-d+1} \end {array}\right )}{\left (\begin {array}{c}{n}\\{d-1} \end {array}\right )}$

by definition in (27). Consider the Krawtchouk expansion of f(x, y)

$$ \begin{array}{@{}rcl@{}} f(x,y)=\sum\limits_{u=0}^{n} \sum\limits_{v=0}^{c} f_{u,v} P_u(x;n) P_v(y;c), \end{array} $$

with the coefficients $f_{u,v} \in {\mathbb R}$. It follows from (17) that we have

$$ \begin{array}{@{}rcl@{}} f_{u,v}=\frac{1}{q^{2(n+c)}} \sum\limits_{\alpha=0}^{n} \sum\limits_{\beta=0}^{c} f(\alpha,\beta) P_{\alpha}(i;n)P_{\beta}(j;c). \end{array} $$

As f(α, β) = 0 for 0 ≤ α ≤ c and 1 ≤ β ≤ c for the function f(x, y) in (27) and P₀(y, c) = 1 (see (13)) we obtain that

$$ \begin{array}{@{}rcl@{}} f_{u,v} & = & \displaystyle \frac{1}{q^{2(n+c)}} \sum\limits_{\alpha =0}^{n} f(\alpha,0) P_{\alpha}(u;n) \\ & = & \frac{1}{q^{2(n+c)}} \sum\limits_{\alpha =0}^{n} q^{2(n+c-d+1)}\frac{\left( \begin{array}{cc}{n-\alpha}\\{n-d+1} \end{array}\right)}{\left( \begin{array}{cc}{n}\\{d-1} \end{array}\right)}P_{\alpha}(u;n) \\ & = & q^{2(-d+1)} \sum\limits_{\alpha=0}^{n} \frac{\left( \begin{array}{cc}{n-\alpha}\\{n-d+1} \end{array}\right)}{\left( \begin{array}{cc}{n}\\{d-1} \end{array}\right)}P_{\alpha}(u;n). \end{array} $$

(28)

It follows from (16) that we have

$$ \begin{array}{@{}rcl@{}} \sum\limits_{\alpha=0}^{n} \left( \begin{array}{cc}{n-\alpha}\\{n-d+1} \end{array}\right) P_{\alpha}(u;n) = q^{2(d-1)} \left( \begin{array}{c}{n-u}\\{d-1} \end{array}\right). \end{array} $$

(29)

Combining (28) and (29) we get

$$ \begin{array}{@{}rcl@{}} f_{u,v}=\frac{\left( \begin{array}{cc}{n-u}\\{d-1} \end{array}\right)}{\left( \begin{array}{cc}{n}\\{d-1} \end{array}\right)} \text{for } 0 \le u \le n \text{ and } 0 \le v \le c. \end{array} $$

(30)

In particular f_{u, v} is independent from v.

For 0 ≤ u₁ ≤ u₂ ≤ d − 1 we claim that

$$ \begin{array}{@{}rcl@{}} \frac{f(u_1,0)}{f_{u_1,0}} \ge \frac{f(u_2,0)}{f_{u_2,0}}. \end{array} $$

(31)

Indeed using (30) we get

$$ \begin{array}{@{}rcl@{}} \begin{array}{cc} \displaystyle \frac{f(u_1,0)}{f_{u_1,0}}= \frac{ \left( \begin{array}{cc}{n-u_{1}}\\{n-d+1} \end{array}\right) }{\left( \begin{array}{cc}{n-u_{1}}\\{d-1} \end{array}\right)} \ge \frac{f(u_2,0)}{f_{u_2,0}}= \frac{\left( \begin{array}{cc}{n-u_{2}}\\{n-d+1} \end{array}\right)}{\left( \begin{array}{cc}{n-u_{2}}\\{d-1} \end{array}\right)} & \iff \\ \\ \displaystyle \frac{\frac{(n-u_1)!}{(n-d+1)!(n-u_1-n+d-1)!}}{ \frac{(n-u_1)!}{(d-1)!(n-u_1-d+1)!} } \ge \frac{\frac{(n-u_2)!}{(n-d+1)!(n-u_2-n+d-1)!}}{ \frac{(n-u_2)!}{(d-1)!(n-u_2-d+1)!} } & \iff \\ \\ \displaystyle \frac{(n-u_1-d+1)!}{ (d-u_1-1)! } \ge \frac{(n-u_2-d+1)!}{ (d-u_2-1)! } & \iff \\ \\ \frac{(n-u_1-d+1)!}{ (n-u_2-d+1)! } \ge \frac{(d-u_1-1)!}{ (d-u_2-1)! } & \iff \\ \\ \displaystyle (n-u_1-d+1) {\cdots} (n-u_2-d+2) \ge (d-u_1-1) {\cdots} (d-u_2) & \overset{(*)}{\iff} \\ \\ n-u_1-d+1 \ge d-u_1-1 & \iff \\ \\ \displaystyle d \le \frac{n+2}{2}, & \end{array} \end{array} $$

where in the step (*) above we use the observation that both sides are the products of u₂ − u₁ consecutive integers such that the left hand side ends at n − u₁ − d + 1 and the right hand side ends at d − u₁ − 1. As $d \le \frac {n+2}{2}$ by assumption, this completes the proof of the claim in (31).

We apply Theorem A.4 using f(x, y) in (27). Note that the assumptions of Theorem A.4 hold. Using Theorem A.4 and the claim in (31) we conclude that

$$ \begin{array}{@{}rcl@{}} \begin{array}{rcl} K & \le & \displaystyle \frac{1}{q^{n+c}} \frac{f(0,0)}{f_{0,0}} \\ & = & \displaystyle \frac{1}{q^{n+c}} f(0,0) \\ & = & \displaystyle \frac{1}{q^{n+c}} \frac{q^{2(n+c-d+1)}}{(-1)^{c} c!} (-1)^{c} c! \\ & = & \displaystyle q^{n+c-2(d-1)}. \end{array} \end{array} $$

□

Remark A.6

If q = 2, then Theorem A.5 corresponds to [20, Theorem 6]. It is clear from the proof of Theorem A.5 that the factor $\frac {1}{(-1)^{c} c!}$ is necessary in the definition of f(x, y) in (27) in the proof of Theorem A.5. However this factor is forgotten in the proof of [20, Theorem 6] in [20]. Hence we also present this small correction in our detailed proof.

Remark A.7

Let S be an abelian subgroup of $\mathcal {E}_{n}$. If S satisfies some commutation relations given in [24] (see also [9] and [20]), then

$$ \begin{array}{@{}rcl@{}} C=\{|\psi\rangle \in {\mathbb C}^{q^{n+c}}: g|\psi \rangle = | \psi \rangle \text{for all} g \in S\}. \end{array} $$

is called an entanglement-assisted q-ary quantum error correcting stabilizer code. We refer to [9, 24] and [20] for further details on entanglement-assisted quantum stabilizer codes. In particular C is a subspace of ${\mathbb C}^{q^{n+c}}$ and C is an entanglement-assisted q-ary quantum error correcting code in the sense of Definition A.2. If the parameters of C are shown as ((n, K, d; c))_q, then we call C as an [[n, K, d; c]]_q throughout this paper in order to emphasize that C is a stabilizer code. Note that [9, Theorem 4] is a construction of stabilizer codes. As our constructions are based on [9, Theorem 4], the quantum codes in all sections except the appendices are quantum stabilizer codes and denoted using [[⋅]]_q. It is clear that Theorem A.5 holds for quantum stabilizer codes as well.

Appendix B: : The algorithm of Carlet et. al. [6]

In Appendix B, we explain an important step in detail of our Construction 1. Recall that we present Construction 1 in Section 3.

Let q ≥ 4 be a prime power. Let G be a k × n matrix over ${\mathbb F}_{q}$, which is a generator matrix of an [n, k, d]_q code C. We observe that [6] gives an algorithm in order to modify G to a k × n matrix $G^{\prime }$ over ${\mathbb F}_{q}$ such that the linear code $C^{\prime }$ having $G^{\prime }$ as a generator matrix has the following properties: $C^{\prime }$ is an [n, k, d]_q over ${\mathbb F}_{q}$ (same parameters as C) and $\text {Hull}(C^{\prime })=\{0\}$ (while Hull(C) is not necessarily {0}).

In this appendix we explain this algorithm in detail.

First we find a n × n permutation matrix P over ${\mathbb F}_{q}$ such that G₁ = GP is in the systematic form. Namely

$$ G_1=[I_k A], $$

where I_k is the k × k identity matrix and A is a k × (n − k) matrix over ${\mathbb F}_{q}$. For example using Gaussian elimination we can find the row reduced echelon form of G. This indicates the pivot columns in the row reduced echelon form of G. Changing the pivot columns to the first k columns gives a method to find such a permutation matrix P.

Let C₁ be the ${\mathbb F}_{q}$-linear code having G₁ as a generator matrix. Note that C₁ is permutation equivalent to C and hence C₁ is again an [n, k, d]_q code.

Here we need to introduce a notation. For a symmetric k × k matrix M and a subset $I \subseteq \{1,2, \ldots , k\}$ with #I = t, let M_I denote the (k − t) × (k − t) symmetric submatrix of M obtained by removing the i-th column and i-th row of M for each i ∈ I. Note that M_I = M if I = ∅. We denote M_I = 0 if I = {1, 2,…,k} by convention (see Remark B.1).

Note that $G_{1}{G_{1}^{T}}=(GP)(P^{T}G^{T})=GG^{T}$ as PP^T = I_n. Let 0 ≤ tlek be the integer such that rank(GG^T) = k − t. It follows from (2) that let $\dim _{{\mathbb F}_{q}} \text {Hull}(C)=t$. Hence there exists a subset $I \subseteq \{1, \ldots ,k\}$ such that $\left (GG^{T}\right )_{I}$ is invertible. In fact there is a deterministic way to choose such I as follows: Let J be the subset of {1, 2,…,k} such that j ∈ J if and only if the j-th column is a pivot in the reduced row echelon form of GG^T. Put I = {1, 2,…,k}∖ J.

For 1 ≤ i ≤ k, let $\alpha _{i} \in {\mathbb F}_{q}$ be chosen as follows:

$$ \begin{array}{@{}rcl@{}} \alpha_i \in \left\{\begin{array}{cl} {\mathbb F}_q \setminus \{0,1,-1\} & \text{if } i \in I, \\ \{1\} & \text{if } i \not\in I. \end{array} \right. \end{array} $$

(32)

Let D be the n × n diagonal matrix having entries α₁,α₂,…,α_k, 1,…, 1 in order, namely D = Diag(α₁,α₂,…,α_k, 1,…, 1). Let G₂ be the k × n matrix obtained from G₁ as

$$ \begin{array}{@{}rcl@{}} G_2=G_1D. \end{array} $$

Let C₂ be the ${\mathbb F}_{q}$-linear code having G₂ as a generator matrix. Recall that a square matrix M is called a monomial matrix if there exists exactly one nonzero entry in each row and column of M (see, [18, page 24]). Moreover if G is a k × n generator matrix of a code C and M is an n × n monomial matrix, then the code having GM as a generator matrix is called monomially equivalent to C (see, [18, page 469]). Here D is a monomial matrix and C₂ is monomially equivalent to C₁. In particular C₂ is an [n, k, d]_q code, having the same parameters as C₁.

Let D₁ be the k × k diagonal matrix given by D₁ = Diag(α₁,α₂,…,α_k). Note that

$$ \begin{array}{@{}rcl@{}} G_2=[I_k A]\left[\begin{array}{cc} D_1 & 0 \\ 0 & I_{n-k} \end{array} \right]=[D_1 A ] \end{array} $$

and hence

$$ \begin{array}{@{}rcl@{}} \begin{array}{rcl} G_2G_2^T &= &D_1^2 + AA^T = I_k +AA^T + \text{Diag}(\alpha_1^2-1, \ldots, \alpha_k^2-1) \\ & = &G_1G_1^T + \text{Diag}(\alpha_1^2-1, \ldots, \alpha_k^2-1). \end{array} \end{array} $$

(33)

Recall that $\text {rank}(G_{1}{G_{1}^{T}})=k-t$ and $\alpha _{1}, \ldots , \alpha _{k} \in {\mathbb F}_{q}$ are chosen as in (32). Hence using (33) and [6, Lemma 9] we conclude that

$$ \begin{array}{@{}rcl@{}} \det (G_2G_2^T)= \left( \prod\limits_{i \in I} (\alpha_i^2-1) \right) \det(G G^T)_I \neq 0, \end{array} $$

(34)

where $\det (GG^{T})_{I}=1$ if I = {1, 2,…,k} by convention (see Remark B.1). Therefore Hull(C₂) = {0}.

Remark B.1

If G is self-orthogonal, then GG^T = 0 and hence t = k. In this degenerate case it is necessary to choose I = {1, 2,…,k}. We have the conventions that (GG^T)_I = 0 and $\det (GG^{T})_{I}=1$ when I = {1, 2,…,k}. Indeed in this degenerate case (33) becomes

$$ \begin{array}{@{}rcl@{}} G_2G_2^T= 0 + \text{Diag}(\alpha_1^2-1, \ldots, \alpha_k^2-1) \end{array} $$

and hence we use the convention (GG^T)_I = 0. Again in this degenerate case (34) becomes

$$ \begin{array}{@{}rcl@{}} \det (G_2G_2^T)= \left( \prod\limits_{i \in I} (\alpha_i^2-1) \right) \end{array} $$

and hence we use the convention $\det (GG^{T})_{I}=1$. We need to use the convention that $\det (GG^{T})_{I}=1$ if I = {1, 2,…,k}. The convention (GG^T)_I = 0 is not essential as it appears only in the background.

In summary we obtain the following algorithm:

Input:

For integers 1 ≤ k ≤ n, let G is a k × n matrix of rank k over ${\mathbb F}_{q}$. Let C be the ${\mathbb F}_{q}$ linear code having G as a generator matrix. Let C be an [n, k, d]_q code.

Output:

$G^{\prime }$ is a k × n matrix of rank k over ${\mathbb F}_{q}$. Let $C^{\prime }$ be the ${\mathbb F}_{q}$ linear code having $G^{\prime }$ as a generator matrix. We obtain that $C^{\prime }$ is an [n, k, d]_q code with $\text {Hull}(C^{\prime })=\{0\}$.

• i) Find an n × n permutation matrix P such that GP is in the systematic form so that GP = [I_kA].

• ii) Let rank(GG^T) = k − t, Find a subset $I \subseteq \{1, \ldots , k\}$ such that #I = t and (GG^T)_I is invertible.

• iii) For 1 ≤ i ≤ k, let $\alpha _{i} \in {\mathbb F}_{q}$ be chosen as follows:

  $$ \begin{array}{@{}rcl@{}} \alpha_i \in \left\{\begin{array}{cl} {\mathbb F}_q \setminus \{0,1,-1\} & \text{if} i \in I, \\ \{1\} & \text{if} i \not\in I. \end{array} \right. \end{array} $$

  (35)

• iv) Put $G^{\prime }=GP\text {Diag}(\alpha _{1}, \ldots ,\alpha _{k},1,\ldots ,1)$.

Remark B.2

In many cases for a starting code [n, k, d]_q code C we know a generator matrix G in systematic form. In such a case we take P = I_n and we skip step i) in the algorithm.

In the following examples we illustrate this algorithm. In these algorithms the generators matrices in the input are in systematic form and hence we take P = I_n as explained in Remark B.2.

Example B.1

Let q = 4 and $w \in {\mathbb F}_{4}$ be an element satisfying w² + w + 1 = 0. Note that w is a primitive element of ${\mathbb F}_{4}$. Let G be the 7 × 23 matrix over ${\mathbb F}_{4}$ given by

$$ \begin{array}{@{}rcl@{}} G=\left[ \begin{array}{ccccccccccccccccccccccc} 1&0&0&0&0&0& 0& 0& w^2& 0& w^2& w& 1& w& 1& w^2& w^2& 0& 0& w& 1& 0& w \\ 0&1&0&0&0&0& 0& 0& w& 0& 1& w& 0& 0& w^2& w^2& 1& w& 1& w& w^2& 0& w^2 \\ 0&0&1&0&0&0& 0& 0& w^2& w^2& w& w^2& 1& w& w& 0& 0& w^2& 0& w& 0& 1& 1 \\ 0&0&0&1&0&0& 0& 0& 1& 1& 0& 1& w& w& 1& w^2& 1& w^2& 0& 1& 1& w^2& w^2 \\ 0&0&0&0&1&0& 0& 0& w^2& 0& w^2& w& w& 0& 0& w& 0& w^2& 1& w^2& w^2& w^2& w \\ 0&0&0&0&0&1& 0& 0& w& 0& 1& w& 0& w^2& 1& w& 0& 1& w& w^2& w& w& 0 \\ 0&0&0&0&0&0& 1& 0& 0& w& 0& 1& w& 0& w^2& 1& w& 0& 1& w& w^2& w& w \end{array} \right]. \end{array} $$

Let C be the ${\mathbb F}_{4}$-linear code having G as a generator matrix. In fact C is a code in the database of Magma [3]. Using the Magma command $\text {BKLC}({\mathbb F}_{4},23,7)$ (or $\text {BDLC}({\mathbb F}_{4},23,12)$) we immediately obtain C and the matrix G above. Then C is an [23, 7, 12]₄ code. Moreover C is an optimal code in both of the senses in Notation 3.4 and Notation 3.5 below. We refer to the online table [14] for the optimality.

We observe that rank(GG^T) = 6 and hence $\dim _{{\mathbb F}_{4}} \text {Hull}(C)=1$. Using the algorithm explained in this appendix, let $I=\{7\}\subseteq \{1,2, \ldots , 7\}$,

$$ \begin{array}{@{}rcl@{}} D_1=\text{Diag}(\underbrace{1, \ldots,1}_{6 \text{times}},w,\underbrace{1, \ldots,1}_{16 \text{times}}) \end{array} $$

and G₁ = GD₁. Moreover let C₁ be the ${\mathbb F}_{4}$-linear code having G₁ as a generator matrix. We obtain that C₁ is an [23, 7, 12]₄ code and Hull(C₁) = {0}.

Example B.2

Let q = 4 and $\in {\mathbb F}_{4}$ be an element satisfying w² + w + 1 = 0. Let G be the 9 × 19 matrix over ${\mathbb F}_{4}$ given by

$$ \begin{array}{@{}rcl@{}} G=\left[ \begin{array}{ccccccccccccccccccc} 1& 0& 0& 0& 0& 0& 0& 0& 0& 1& w^2& w& w& 0& 1& 0& w^2& w^2& w\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& w& 0& 0& 1& w& w& 1& 1& w& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& w& 0& 0& 1& w& w& 1& 1& w\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& w& 1& 1& w^2& 0& w^2& w& w^2& 0& w\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& w& w^2& w& w& w^2& w& w^2& w^2& w& w^2\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& w^2& 0& w& w^2& w& 0& w& 1& 1& w^2\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& w^2& 1& 1& w^2& w^2& 1& 0& 0& w^2& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& w^2& 1& 1& w^2& w^2& 1& 0& 0& w^2\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& w^2& w& w& 0& 1& 0& w^2& w^2& w& 1 \end{array} \right]. \end{array} $$

Let C be the ${\mathbb F}_{4}$-linear code having G as a generator matrix. Using the Magma command $\text {BDLC}({\mathbb F}_{4},19,8)$ we immediately obtain C and the matrix G above. Then C is an [19, 9, 8]₄ code. Note that C is a best known code in the sense of Notation 4.5 (see [14]).

We observe that rank(GG^T) = 0 and hence $\dim _{{\mathbb F}_{4}} \text {Hull}(C)=9$. Using the algorithm explained in this appendix, let $I=\{1,2, \ldots , 9\}\subseteq \{1,2, \ldots , 9\}$,

$$ \begin{array}{@{}rcl@{}} D_1=\text{Diag}(\underbrace{w, \ldots,w}_{ 9 \text{times}},\underbrace{1, \ldots,1}_{ 10 \text{times}}) \end{array} $$

and G₁ = GD₁. Moreover let C₁ be the ${\mathbb F}_{4}$-linear code having G₁ as a generator matrix. We obtain that C₁ is an [19, 9, 8]₄ code and Hull(C₁) = {0}.

Example B.3

Let q = 4 and $\in {\mathbb F}_{4}$ be an element satisfying w² + w + 1 = 0. Let G be the 7 × 20 matrix over ${\mathbb F}_{4}$ given by

$$ \begin{array}{@{}rcl@{}} G=\left[ \begin{array}{cccccccccccccccccccc} 1& 0& 0& 0& 0& 0& 0& w^2& 0& 0& 1& w& w^2& 1& w& 0& w& 1& 1& 1\\ 0& 1& 0& 0& 0& 0& 0& 0& w& w^2& 1& 0& w^2& w^2& 0& w& 1& w^2& 0& w\\ 0& 0& 1& 0& 0& 0& 0& w^2& w& w^2& 0& 1& 1& w& w^2& 0& w& w& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& w^2& w^2& w& w& w& w& 0& 0& w^2& w& w^2& w^2& 1\\ 0& 0& 0& 0& 1& 0& 0& 0& w& 0& w^2& w^2& w^2& w& 1& 0& w& w^2& w& 0\\ 0& 0& 0& 0& 0& 1& 0& 1& 0& w& w& 0& w& 1& 1& 1& w^2& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1& w& 1& 0& 1& w^2& w& 1& 0& 1& 0& 0& w^2& w \end{array} \right]. \end{array} $$

Let C be the ${\mathbb F}_{4}$-linear code having G as a generator matrix. Using the Magma command $\text {BKLC}({\mathbb F}_{4},20,7)$ we immediately obtain C and the matrix G above. Then C is an [20, 7, 10]₄ code. Note that C is a best known code in the sense of Notation 3.4 (see [14]).

We observe that rank(GG^T) = 6 and hence $\dim _{{\mathbb F}_{4}} \text {Hull}(C)=1$. Using the algorithm explained in this appendix, let $I=\{7\}\subseteq \{1,2, \ldots , 7\}$,

$$ \begin{array}{@{}rcl@{}} D_1=\text{Diag}(\underbrace{1, \ldots,1}_{ 6 \text{times}},w,\underbrace{1, \ldots,1}_{ 13 \text{times}}) \end{array} $$

and G₁ = GD₁. Moreover let C₁ be the ${\mathbb F}_{4}$-linear code having G₁ as a generator matrix. We obtain that C₁ is an [20, 7, 10]₄ code and Hull(C₁) = {0}.