Some Nonprimitive BCH Codes and Related Quantum Codes

Abstract

Let$q$be a prime power and$m\geq 3$be odd. Suppose that$n=\frac {q^{2m}-1}{a}$with$a|(q^{m}+1)$and$3\leq a \leq 2(q^{2}-q+1)$. This paper mainly determines the actual maximum designed distance of Hermitian dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes over$\mathbb {F}_{q^{2}}$of length$n$. Firstly, we give the maximum designed distance$\delta _{m,a}^{R}$of narrow-sense Hermitian dual-containing BCH codes. Secondly, we show that there are also non-narrow-sense ones of designed distance up to$\delta _{m,a}^{R}$. It is worth mentioning that our maximum designed distance$\delta _{m,a}^{R}>\lceil \frac {a}{2}\rceil \delta _{m}^{A}$, where$\delta _{m}^{A}$is given by Alyet al.(IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1183-1188, 2007). Thus, many families of Hermitian dual-containing BCH codes with relatively large designed distance are obtained. Using the Hermitian construction to them, we can subsequently construct different classes of nonprimitive quantum codes, which are new in the sense that their parameters are not covered in the literature.