Stabilizer quantum codes from J-affine variety codes and a new Steane-like enlargement

Abstract

New stabilizer codes with parameters better than the ones available in the literature are provided in this work, in particular quantum codes with parameters $[[127,63, {\ge }12]]_2$ and $[[63,45, {\ge }6]]_4$ that are records. These codes are constructed with a new generalization of the Steane’s enlargement procedure and by considering orthogonal subfield-subcodes—with respect to the Euclidean and Hermitian inner product—of a new family of linear codes, the J-affine variety codes.

Appendix

We devote this appendix to prove Theorem 3 which was stated in the introduction and preliminaries of this paper. To do this, we adapt to our purposes some facts described in [48] and [30]. Consider the vector space ${\mathbb {F}}_q^{2n}$ and the symplectic inner product $(\mathbf {u}|\mathbf {v}) \cdot _s (\mathbf {u}'|\mathbf {v}') = \mathbf {u} \cdot \mathbf {v}' - \mathbf {v} \cdot \mathbf {u}'$. Recall that the weight $\mathrm {w}(\mathbf {u}|\mathbf {v})$ of a word $(\mathbf {u}|\mathbf {v})$ as above is the number of indexes i, $1 \le i \le n$, such that either $u_i$ or $v_i$ (or both) are not zero, where the $u_i$ (respectively, $v_i$) represent the coordinates of the vector $\mathbf {u}$ (respectively, $\mathbf {v}$). Following [2] (see also [9]), to get our stabilizer code, we only need to find a vector subspace S in ${\mathbb {F}}_q^{2n}$ such that $S^{\perp _s} \subseteq S$ with dimension $k_2 + \hat{k}_1$ and minimum distance larger than or equal to that stated in the statement. Let us describe it. Set $G_1$ ($\hat{G}_1$, L, respectively) generator matrices of the codes $C_1$ ($\hat{C}_1$, D, respectively) and let S be the code of ${\mathbb {F}}_q^{2n}$ generated by the matrix

$$\begin{aligned} \left( \begin{array}{cc} L &{} AL \\ G_1 &{} 0 \\ 0 &{} \hat{G}_1 \\ \end{array} \right) , \end{aligned}$$

where A is a fixed point free square matrix (see [30, 48] for its existence). Our hypotheses imply $\hat{k}_1 + k_2 = k_1 + \hat{k}_2$ and that the rows of the previous matrix are linearly independent, therefore, for computing the dimension of S, it suffices to see that the number of rows is $k_2 -k_1 + k_1 + \hat{k}_1 = k_2 + \hat{k}_1$.

Table 16 Defining sets of the codes over ${\mathbb {F}}_7$ in Table 15

Let $H_2$ ($\hat{H}_2$, respectively) be a parity check matrix of the code $C_2$ ($\hat{C}_2$, respectively), one can found a matrix B such that

$$\begin{aligned} \left( \begin{array}{c} H_2 \\ B \\ \end{array} \right) , \left( \left( \begin{array}{c} \hat{H}_2 \\ B \\ \end{array} \right) , \mathrm {respectively} \right) \end{aligned}$$

is a parity check matrix for $C_1$ (respectively, for $\hat{C}_1$). Now defining the matrix $K= BL^t (A^t)^{-1} (B L^t)^{-1}$, it is not difficult to prove that

$$\begin{aligned} \left( \begin{array}{cc} KB &{} B \\ \hat{H}_2 &{} 0 \\ 0 &{} H_2 \\ \end{array} \right) , \end{aligned}$$

is a parity check matrix for the code S and therefore one has that $S^{\perp _s} \subseteq S$.

To end our proof, it only remains to study what happens with the weight $\mathrm {w}(\mathbf {u}|\mathbf {v})$ for elements $(\mathbf {u}|\mathbf {v}) \in S$. First assume $q=2$, a generic element in S has the form $(\mathbf {v}_1 L +\mathbf {v}_2 G_1 | \mathbf {v}_1 AL +\mathbf {v}_3 \hat{G}_1)$, where $\mathbf {v}_1, \mathbf {v}_2, \mathbf {v}_3$ are suitable vectors with coordinates in ${\mathbb {F}}_q$. When $\mathbf {v}_1$ is the zero vector, $\mathbf {u}$ must be in $C_1$ and $\mathbf {v}$ in $\hat{C}_1$, which proves that, in this case, $\mathrm {w}(\mathbf {u}|\mathbf {v})$ must be larger than or equal to the minimum of the values $d_1$ and $\hat{d}_1$. Otherwise, $\mathbf {v}_1 \ne \mathbf {0}$, one can use the property

$$\begin{aligned} w(\mathbf {u}|\mathbf {v}) = \frac{{\mathrm {wt}}(\mathbf {u})+ {\mathrm {wt}}(\mathbf {v}) + {\mathrm {wt}}(\mathbf {u}+\mathbf {v})}{2}, \end{aligned}$$

where ${\mathrm {wt}}$ denotes the standard weight of a word in a code in ${\mathbb {F}}_q^n$, and this concludes the proof since $\mathbf {u} \in C_2$, $\mathbf {v} \in \hat{C}_2$, $\mathbf {u}+\mathbf {v} \in C_3$ and the fact that $(C_1 + \hat{C}_1) \cap D = \{\mathbf {0}\}$ implies that none of the previous vectors are zero.

Let us consider $q \ne 2$, we will only study $w(\mathbf {u}|\mathbf {v})$ for $\mathbf {v}_1 \ne 0$. For convenience, assume that the coordinates $u_{t+1}, u_{t+2}, \ldots , u_n$ of the word $\mathbf {u}$ are zero and that this does not happen with the remaining coordinates. As showed in [30], there exists $\lambda \in {\mathbb {F}}_q$ such that

$$\begin{aligned} w(\mathbf {u}|\mathbf {v}) = t + {\mathrm {wt}} (v_{t+1}, v_{t+2}, \ldots , v_n ) \ge {\mathrm {wt}}(\mathbf {v} - \lambda \mathbf {u}) + \frac{{\mathrm {wt}}(\mathbf {u})}{q} \end{aligned}$$

and, symmetrically, $w(\mathbf {u}|\mathbf {v}) \ge {\mathrm {wt}}(\mathbf {u} - \lambda ' \mathbf {v}) + \frac{{\mathrm {wt}}(\mathbf {v})}{q}$, for some $\lambda ' \in {\mathbb {F}}_q$, holds. This finishes the proof because, as before, our hypotheses imply that $\mathbf {0} \ne \mathbf {v} - \lambda \mathbf {u}$ and $\mathbf {0} \ne \mathbf {u} - \lambda ' \mathbf {v}$ belong to $C_3$, $\mathbf {0} \ne \mathbf {u} \in C_2$ and $\mathbf {0} \ne \mathbf {v} \in \hat{C}_2$.

Remark 3

Notice that the Hamada’s generalization of the Steane’s enlargement, Theorem 2 in this work, is a particular case of Theorem 3 that holds when $C_1 = \hat{C}_1$.