Generalizations of the BCH bound

Cyclic codes generated by polynomials having multiple sets of do — 1 roots in consecutive powers of a nonzero field element are considered and some generalizations of the BCH bound are presented. In particular, it is shown, among other results, that if g(x) GF(q)[x[ is the generator polynomial of a cyclic code Vn of length n such that g(βl+i1c1+i2c2 = 0 for i1 = 0, 1,…, d0 — 2 and i2 = 0,1,…,s, where β GF(qm) is a nonzero element of order n and c1 , c2 are relatively prime to n, then the minimum distance of Vn is at least d0 + s