On Duals of Binary Primitive BCH Codes

Abstract

We treat binary extended cyclic codes of length 2 m over F2. We introduce a class, denoted by {C (t)}2≤t≤m−1, of such codes, whose defining set is characterized by only one cyclotomic coset. We prove that they are duals of extended BCH codes. We study the divisibility of the C (t)’s, and show that it determines the divisibility of all duals of extended BCH codes. Next we obtain a lower bound on their minimum distance, that yields results for several affine-invariant codes. In particular, it gives a bound for all duals of extended BCH codes, which is interesting especially when the Carlitz-Uchiyama bound is negative.