1 Correction to: Quantum Inf Process https://doi.org/10.1007/s11128-019-2234-5

The parameters of the codes in Section 4 of our article [1] are not correctly stated. The inequality in Proposition 5 should be with respect to the dual code \(C^{\perp _s}\). The correct statement of Proposition 5 is

Proposition 5

Assume that a positive integer c satisfies \(2c \le d_H(C^{\perp _s}\setminus \{\mathbf {0}\})-1\), then

$$\begin{aligned} \begin{array}{l} \dim _{{\mathbb {F}}_q} P(C) = \dim _{{\mathbb {F}}_q} C, \;\;\text{ and }\\ \\ \dim _{{\mathbb {F}}_q} P(C)\cap P(C)^{\perp _s}= \dim _{{\mathbb {F}}_q} S(C) = \dim _{{\mathbb {F}}_q} C - 2c. \end{array} \end{aligned}$$

This affects the corresponding inequalities in the statement of Theorem 7, Theorem 8 and Theorem 9. Moreover, the dimension of the codes was erroneously displayed. The correct statements of the above theorems are the following:

Theorem 7

Let \(C \subseteq {\mathbb {F}}_q^{2n}\) be an \({\mathbb {F}}_q\)-linear code with \(\dim _{{\mathbb {F}}_q} C = n-k\) and \(C \subseteq C^{\perp _s}\). Assume that a positive integer c satisfies \(2c \le d_H(C^{\perp _s}\setminus \{\mathbf {0}\})-1\), then the punctured code P(C) provides an

$$\begin{aligned}{}[[n-c, k, \ge d_s(C^{\perp _s}\setminus C); c]]_q \end{aligned}$$

entanglement-assisted code.

Theorem 8

Let \(C \subseteq {\mathbb {F}}_{q^2}^n\) be an \({\mathbb {F}}_{q^2}\)-linear code with \(\dim _{{\mathbb {F}}_{q^2}} C = (n-k)/2\), and suppose that \(C \subseteq C^{\perp _h}\). Let c be a positive integer such that \(c \le d_H(C^{\perp _h}\setminus \{\mathbf {0}\})-1\), then the punctured code \(P_h(C)\) provides an

$$\begin{aligned}{}[[n-c, k, \ge d_H(C^{\perp _h}\setminus C); c]]_q \end{aligned}$$

entanglement-assisted code.

Theorem 9

Let \(C_2 \subseteq C_1 \subseteq {\mathbb {F}}_{q}^n\) be two \({\mathbb {F}}_{q}\)-linear codes such that \(\dim C_i = k_i\), \(1 \le i \le 2\). The standard construction of CSS codes uses \(C_2 \times C_1^\perp \) as the stabilizer. Assume that c is a positive integer such that

$$\begin{aligned} c \le \min \left\{ d_H(C_2^\bot \setminus \{\mathbf {0}\}), d_H(C_1 \setminus \{\mathbf {0}\})\right\} -1, \end{aligned}$$

then the punctured code \(P_h(C_2)\times P_h(C_1^\perp )\) provides an

$$\begin{aligned}{}[[n-c, k_1-k_2, \ge \min \left\{ d_H(C_1\setminus C_2), d_H(C_2^\perp \setminus C_1^\perp ) \right\} ; c]]_q \end{aligned}$$

entanglement-assisted code.

Proofs are the same with the exception that \(c \le d_H (C\setminus \{\mathbf {0}\})-1\) should be replaced by \(c \le d_H (C^{\perp _h}\setminus \{\mathbf {0}\}) -1\), in the proof of Theorem 8 and

$$\begin{aligned} c \le \min \{d_H(C_2 \setminus \{\mathbf {0}\},d_H(C_1^\bot \setminus \{\mathbf {0}\} \} -1 \end{aligned}$$

should be replaced by

$$\begin{aligned} c \le \min \left\{ d_H(C_2^\bot \setminus \{\mathbf {0}\}), d_H(C_1 \setminus \{\mathbf {0}\})\right\} -1, \end{aligned}$$

in the first line of the proof of Theorem 9.