Construction of new quantum codes via Hermitian dual-containing matrix-product codes

Abstract

In 2001, Blackmore and Norton introduced an important tool called matrix-product codes, which turn out to be very useful to construct new quantum codes of large lengths. To obtain new and good quantum codes, we first give a general approach to construct matrix-product codes being Hermitian dual-containing and then provide the constructions of such codes in the case $$s{\mid }(q^{2}-1)$$ , where s is the number of the constituent codes in a matrix-product code. For $$s{\mid } (q+1)$$ , we construct such codes with lengths more flexible than the known ones in the literature. For $$s{\mid } (q^{2}-1)$$ and $$s{\not \mid } (q+1)$$ , such codes are constructed in an unusual manner; some of the constituent codes therein are not required to be Hermitian dual-containing. Accordingly, by Hermitian construction, we present two procedures for acquiring quantum codes. Finally, we list some good quantum codes, many of which improve those available in the literature or add new parameters.