A class of narrow-sense BCH codes over F_q of length q~m-1/2

摘要

BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorial design. This paper studies a class of linear codes of length qm-1 2 over Fq with special trace representation, where q is an odd prime power. With the help of the inner distributions of some subsets of association schemes of quadratic forms, we determine the weight enumerators of these codes. Determining some cyclotomic coset leaders di of cyclotomic cosets modulo qm- 1 2, we prove that narrow-sense BCH codes of length qm-1 2 with designed distance di = qm-qm-1 2 - 1 - q <= m-3 2 <=+i- 1 2 have the corresponding trace representation, and have the minimal distance d = di and the Bose distance dB = di, where 1 <= i <= [m+11/6].