Correction to: Quantum error-correcting codes from the quantum construction X

Correction to: Quantum Inf Process https://doi.org/10.1007/s11128-023-04122-x

The parameters of the codes in Section 2 and Section 3 of our article [1] are not correctly stated. The $e= k -\textrm{dim}_{\mathbb {F}_{q}}(C \cap C^{\perp })$ in Theorem 2.9 should be $e=n- k -\textrm{dim}_{\mathbb {F}_{q}}(C \cap C^{\perp })$. The correct statement of Theorem 2.9 is

FormalPara Theorem 2.9

(Quantum construction X of the Euclidean dual) Let C be a linear code with parameters $[n, k]_{q}$ over $\mathbb {F}_{q}$. Let $e:= n-k -\textrm{dim}_{\mathbb {F}_{q}}(C \cap C^{\perp })$. Then there exists an $[[n+e, n-2k+e, d(Q)]]_q$ QEC code Q, where $d(Q)\ge \textrm{min}\{d(C^{\perp }), d(C +C^{\perp })+1\}$.

This affects the corresponding parameters of the QEC codes in the statement of Theorems 3.2, 3.7, 3.9, Lemma 3.14, Theorems 3.15, 3.16 and 3.20. Moreover, the length and dimension of the QEC codes were erroneously displayed. The correct statements of the above theorems and the lemma are the following:

FormalPara Theorem 3.2

Let q be a prime power and $\textrm{gcd}(n,q)=1$. Let $x^{n}-1$ be factored completely into nonassociated irreducible factors in $\mathbb {F}_{q}[x]$ as

$$\begin{aligned} x^n-1=f_{1}(x)f_{2}(x)\cdots f_{s}(x)h_{1}(x)h_{1}^{*}(x)\cdots h_{t}(x)h_{t}^{*}(x), \end{aligned}$$

where $f_{1}(x),f_{2}(x), \ldots , f_{s}(x)$ are irreducible polynomials that are associates to their own reciprocals, and $h_{1}(x),h_{1}^{*}(x);\ldots ; h_{t}(x),h_{t}^{*}(x)$ are pairs of mutually reciprocal irreducible polynomials. If $C=\langle f_{i_1}(x)\cdots f_{i_l}(x)g_{j_1}(x)\cdots g_{j_m}(x)\rangle$ with $g_{j_e}(x)=h_{j_e}(x)~or~h_{j_e}^{*}(x)$ for $1\le e\le m$, then

• (1) $C^{\perp }=\langle \prod _{i\in \{1,\ldots ,s\}\backslash \{i_1,\ldots ,i_l\}}f_{i}(x) \prod _{j\in \{j_1,\ldots ,j_m\}}g_{j}(x)\prod _{j\in \{1,\ldots ,t\}\backslash \{j_1,\ldots ,j_m\}} h_{j}(x)h_{j}^{*}(x)\rangle .$

• (2) $\textrm{Hull}(C)=\langle \prod _{i\in \{1,\ldots ,s\}}f_{i}(x) \prod _{j\in \{j_1,\ldots ,j_m\}}g_{j}(x)\prod _{j\in \{1,\ldots ,t\}\backslash \{j_1,\ldots ,j_m\}} h_{j}(x)h_{j}^{*}(x)\rangle .$

• (3) $\textrm{Sum}(C)=\langle \prod _{j\in \{j_1,\ldots ,j_m\}}g_{j}(x)\rangle .$

• (4) there exists a QEC code Q with parameters $[[n+e,n-2k+e,d(Q)]]_q$, where $k=n-(\sum _{s=1}^l \textrm{deg}(f_{i_s}(x))+\sum _{t=1}^m \textrm{deg}(g_{j_t}(x))$, $e=\sum _{s=1}^l \textrm{deg}(f_{i_s}(x))+\sum _{t=1}^m \textrm{deg}(g_{j_t}(x)+\sum _{i\in \{1,\ldots ,s\}}\textrm{deg}(f_{i}(x))+\sum _{j\in \{j_1,\ldots ,j_m\}}{deg}(g_{j}(x))+\sum _{j\in \{1,\ldots ,t\}\backslash \{j_1,\ldots ,j_m\}} ({deg}(h_{j}(x))+{deg}(h_{j}^{*}(x)))-n$, and $d(Q)\ge \textrm{min}\{d(C^{\perp }),d(C+C^{\perp })+1\}$.

FormalPara Theorem 3.7

Let $n=q-1$, $\delta \ge 1$, and $2\delta -1\le k\le \frac{n-1}{2}$. Then there exists a QEC code with parameters $[[2n+2\delta -1,2n-2k+2\delta -1,d(Q)]]_q$, where $d(Q)\ge \textrm{min} \{k+1,k-2\delta +3\}$.

FormalPara Theorem 3.9

Let $C_1$ and $C_2$ be $[n, k_1, d_1]_q$ and $[n, k_2, d_2]_q$ linear codes over $\mathbb {F}_q$, respectively. Write $C = C_1\curlyvee C_2$. Then there exists a QEC code with parameters $[[4n-(k_1+k_2+l_1+l_2),4n-3(k_1+k_2)-l_1-l_2,d(Q)\ge \textrm{min}\{d(C^{\perp }), d(Sum(C))+1\}]]_q$, where

$$\begin{aligned} & l_1=\textrm{dim}_{\mathbb {F}_q}(C_1\cap C_1^{\perp }), l_2=\textrm{dim}_{\mathbb {F}_q}(C_2\cap C_2^{\perp }),\\ & d(C^{\perp }))=\textrm{min} \{2d(C_1^{\perp }),2d(C_2^{\perp }), \textrm{max}\{d(C_1^{\perp }),d(C_2^{\perp })\}\}, \end{aligned}$$

and

$$\begin{aligned} d(Sum(C))=\textrm{min} \{2d(Sum(C_1)),2d(Sum(C_2)), \textrm{max}\{d(Sum(C_1),d(Sum(C_2)\}\}. \end{aligned}$$

FormalPara Lemma 3.14

Let q be a prime power and $n\le q + 1$ be a positive integer. Let $\{\alpha _1, \ldots , \alpha _{n-1}\}$ be $n-1$ distinct elements of $\mathbb {F}_q$. Further, let $P(x)=f(x)g(x)$ and $Q(x)=g(x)h(x)$ where f(x), g(x) and h(x) are three distinct monic irreducible polynomials in $\mathbb {F}_q[x]$ such that

• (1) $\textrm{deg}(P(x)) + \textrm{deg}(Q(x)) = n$.

• (2) $\textrm{gcd}(P(x)Q(x), ~\prod _{i=1}^{n}(x-\alpha _i))=1$.

• (3) $\frac{P(\alpha _i)Q(\alpha _i)}{\prod _{1\le j\le n-1,j\ne i}(\alpha _i-\alpha _j))}=v_i^2$, for every $1 \le i \le n-1$. Suppose that $\textrm{deg}(f(x))=t$ and $\textrm{deg}(g(x))=k$ where $0\le t \le n-2k$ and $k\le \lfloor \frac{n}{2}\rfloor$. Then, there exists a QEC code with parameters $[[2n-2k-t,2n-4k-3t, d]]_q$ where $d\ge \textrm{min}\{k+t+1,k+2\}$.

Table 1 Code comparisons

Table 2 A comparison of new QEC codes

Table 3 New QEC codes from Theorem 3.9

Table 4 A comparison of new QEC codes

Table 5 New QEC codes from two cyclic codes over $\mathbb {F}_{4}$

FormalPara Theorem 3.15

Let $q=l^2$ and $l>7$ is an odd prime power. Further, let n, k, t be integers such that $2\le k\le \lfloor \frac{n-2}{2}\rfloor , 2\le t\le n-2k$ and $8\le n\le l+1$.

• (1) There exists a QEC code with parameters $[[2n-2k-t,2n-4k-3t, d]]_q$ where $d\ge \textrm{min}\{k+t+1,k+2\}$.

• (2) There exists a QEC WMDS code with parameters $[[2n-2k-2,2n-4k-6, d]]_q$, where $d\ge k+2$.

FormalPara Theorem 3.16

Let $q=2^s$ and $s\ge 3, 8\le n\le q+1$ and $2\le k\le \lfloor \frac{n-2}{2}\rfloor$. Then there exists a QEC WMDS code with parameters $[[2n-2k-2,2n-4k-6, d]]_q$, where $d\ge k+2$.

FormalPara Theorem 3.20

Let $\mathbb {F}_4=\{0,1,w,w^2: 1+w+w^2=0,w^3=1\}$. For $i=1,2$, suppose that $C_i\subset \mathbb {F}_{4}^{n}$ is an $[n,k_i,d_i]_{4}$ code. Let $C=[C_1,C_2]A$ where $A=\begin{pmatrix}w& 1\\ 1& w\end{pmatrix}$. Write $d_{C_i^{\perp }}=d(C_i^{\perp })$ and $d_{\textrm{Sum}(C_i)}=d(\textrm{Sum}(C_i))$ for $i=1,2$. Then there exists a QEC code with parameters $[[4n-(k_1+k_2+k),4n-3(k_1+k_2)-k, d]]_{4}$, where

$$\begin{aligned} k=\textrm{dim}_{\mathbb {F}_4}(\textrm{Hull}(C_1))+\textrm{dim}_{\mathbb {F}_4}(\textrm{Hull}(C_2)), \end{aligned}$$

and

$$\begin{aligned} d\ge \textrm{min}\{ \textrm{min} \{ 2d_{C_1^{\perp }},d_{C_2^{\perp }} \}, \textrm{min}\{ 2d_{\textrm{Sum}(C_1)},d_{\textrm{Sum}(C_2)} \}+1 \}. \end{aligned}$$

This affects the corresponding parameters of the QEC codes in the statement of Tables 1, 2, 3, 4 and 5. Moreover, the length and dimension of the QEC codes were erroneously displayed. The correct statements are in the Tables 1, 2, 3, 4 and 5 of this paper.