A class of constacyclic BCH codes and new quantum codes

Abstract

Constacyclic BCH codes have been widely studied in the literature and have been used to construct quantum codes in latest years. However, for the class of quantum codes of length $$n=q^{2m}+1$$n=q2m+1 over $$F_{q^2}$$Fq2 with q an odd prime power, there are only the ones of distance $$\delta \le 2q^2$$?≤2q2 are obtained in the literature. In this paper, by a detailed analysis of properties of $$q^{2}$$q2-ary cyclotomic cosets, maximum designed distance $$\delta _\mathrm{{max}}$$?max of a class of Hermitian dual-containing constacyclic BCH codes with length $$n=q^{2m}+1$$n=q2m+1 are determined, this class of constacyclic codes has some characteristic analog to that of primitive BCH codes over $$F_{q^2}$$Fq2. Then we can obtain a sequence of dual-containing constacyclic codes of designed distances $$2q^2 2q^2$$d>2q2 can be constructed from these dual-containing codes via Hermitian Construction. These newly obtained quantum codes have better code rate compared with those constructed from primitive BCH codes.