Two families of BCH codes and new quantum codes

Abstract

In this paper, two families of non-narrow-sense (NNS) BCH codes of lengths $$n=\frac{q^{2m}-1}{q^2-1}$$ and $$n=\frac{q^{2m}-1}{q+1}$$ ( $$m\ge 3)$$ over the finite field $$\mathbf {F}_{q^2}$$ are studied. The maximum designed distances $$\delta ^\mathrm{new}_\mathrm{max}$$ of these dual-containing BCH codes are determined by a careful analysis of properties of the cyclotomic cosets. NNS BCH codes which achieve these maximum designed distances are presented, and a sequence of nested NNS BCH codes that contain these BCH codes with maximum designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these NNS BCH codes. The new quantum codes presented here include many classes of good quantum codes, which have parameters better than those constructed from narrow-sense BCH codes, negacyclic and constacyclic BCH codes in the literature.