The hull of a linear code $C$ is the intersection of $C$ with its dual. To the best of our knowledge, there are very few constructions of binary linear codes with the hull dimension $\ge 2$ except for self-orthogonal codes. We propose a building-up construction to obtain a plenty of binary $[n+2, k+1]$ codes with hull dimension $\ell, \ell +1$, or $\ell +2$ from a given binary $[n,k]$ code with hull dimension $\ell$. In particular, with respect to hull dimensions 1 and 2, we construct all binary optimal $[n, k]$ codes of lengths up to 13. With respect to hull dimensions 3, 4, and 5, we construct all binary optimal $[n,k]$ codes of lengths up to 12 and the best possible minimum distances of $[13,k]$ codes for $3 \le k \le 10$. As an application, we apply our binary optimal codes with a given hull dimension to construct several entanglement-assisted quantum error-correcting codes(EAQECC) with the best known parameters.
翻译:线性代码 $C $C 的船体是 $C 与 $C 的交叉点。 据我们所知,除了自体代码外,只有很少几处建造带有船体维度的二元线性代码 $G 2美元。我们建议建造一个建筑结构,以获得大量包含船体维度的二元 $n+2, k+1美元代码 $ell +1美元,或者从给定的二进制 $C $(n,k) $+2美元代码与船体维度的二元+2美元。特别是,在船体维度1和2方面,我们建造了所有最优的二进制[n, k] 长度代码,直到 13 4 和 5。关于船体维度,我们建造了所有最优的双进最佳的 $[n,k] $12 和最短可能最短的最低距离 $[13,k] 美元代码。作为应用我们的双元最佳标准,用一个特定船体维度的二元标准来构建几个缠绕辅助量子错误校准代码(EAQEC) 。