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
$4\leq q\leq 9$
.