Several families of negacyclic BCH codes and their duals

Negacyclic BCH codes are a special subclasses of negacyclic codes, and have the best parameters known in many cases. A family of good negacyclic BCH codes are the q-ary narrow-sense negacyclic BCH codes of length \(n=(q^m-1)/2\), where q is an odd prime power. Little is known about the true minimum distance of this family of negacyclic BCH codes and the dimension of this family of negacyclic BCH codes with large designed distance. The main objective of this paper is to study three subfamilies of this family of negacyclic BCH codes. The dimension and true minimum distance of a subfamily of the q-ary narrow-sense negacyclic BCH codes of length n are determined. The dimension and good lower bounds on the minimum distance of two subfamilies of the q-ary narrow-sense negacyclic BCH codes of length n are presented. The minimum distances of the duals of the q-ary narrow-sense negacyclic BCH codes of length n are also investigated. As will be seen, the three subfamilies of negacyclic BCH codes are sometimes distance-optimal and sometimes have the same parameters as the best linear codes known.