Construction of New Matrix-Product Codes and Their Applications

In this paper, from special code chains $\mathcal {C}_{1} \supseteq \mathcal {C}_{2}~\supseteq \mathcal {C}_{3}$ such that $\mathcal {C}^{\bot _{h}}_{1}\subseteq \mathcal {C}_{3}$ and $\mathcal {C}^{\bot _{h}}_{2}\subseteq \mathcal {C}_{2}$ , some Hermitian dual-containing (HDC) matrix-product (MP) codes are presented, where $\mathcal {C}_{3}$ is not HDC. By studying some constacyclic codes of lengths $n=q^{2}\pm 1$ and $n=\frac {q^{2}-1}{2}$ , we construct many HDC MP codes of length $3n$ . Consequently, many $q$ -ary quantum codes with larger designed distance $d\geq q+1$ are obtained from these MP codes, where $4\leq q\leq 9$ .