Dual distance of BCH codes (Q1090660)

The authors consider the embedding of binary BCH codes B(d,m) of length \(2^ m-1\) with designed distance \(d=2t+1\), into arbitrary, shortened binary Reed-Muller codes R(r,m) of order r for which the dual distance is known. This yields an upper bound on the dual distance of the codes B(d,m). This problem has not been solved for most codes. Using some results from number theory they prove that the code B(d,m) with the designed distance \(d\geq \delta (s,m)+2\) is a subcode of the code R(m-s- 1,m); where the parameter \(\delta\) (s,m) is the generator largest in magnitude of the cyclotomic class of the code R(m-s-1,m). The results presented in the paper enable to improve the lower bound on rational trigonometric sums in fields of characteristic 2.