On Two Classes of Primitive BCH Codes and Some Related Codes

Abstract

BCH codes are an interesting type of cyclic codes and have wide applications in communication and storage systems. Generally, it is very hard to determine the minimum distances of BCH codes. In this paper, we determine the weight distributions of two classes of primitive BCH codes C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q,m,δ(2))</sub> and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q,m,δ(3))</sub> and their extended codes, which solve two problems proposed by Ding et al. It is shown that the extended codes C̅ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q,m,δ(2))</sub> have four nonzero weights. We also employ the Hartmann-Tzeng bound to present the minimum distance of the dual code C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q,m,δ(2))</sub> for q ≥ 5. Inspired by the idea, we then determine the dimensions of a class of cyclic codes and give lower bounds on their minimum distances, which is greatly improved comparing with the BCH bound. Some optimal codes are obtained.