New quantum codes from matrix-product codes over small fields

In this paper, we provide methods for constructing Hermitian dual-containing (HDC) matrix-product codes over $$\mathbb {F}_{q^2}$$ from some non-singular matrices and a special sequence of HDC codes and determine parameters of obtained matrix-product codes when the input matrix and sequence of HDC codes satisfy some conditions. Then, using some nested HDC BCH codes with lengths $$n=\frac{q^4-1}{a} (a=1 ~$$ or $$~ a=q\pm 1)$$ , we construct some HDC matrix-product codes with lengths $$N=$$ 2n or 3n and derive nonbinary quantum codes with length N from these matrix-product codes via Hermitian construction. Four classes of quantum codes over $$\mathbb {F}_{q}$$ ( $$3\le q\le 5$$ ) are presented, whose parameters are better than those in the literature. Besides, some of our new quantum codes can exceed the quantum Gilbert-Varshamov (GV) bound.