Bounds on the minimum distance of the duals of extended BCH codes over $$\mathbb{F}_p $$

Abstract

LetC be an extended cyclic code of lengthp m over $$\mathbb{F}_p $$ . The border ofC is the set of minimal elements (according to a partial order on [0,p m −1]) of the complement of the defining-set ofC. We show that an affine-invariant code whose border consists of only one cyclotomic coset is the dual of an extended BCH code if, and only if, this border is the cyclotomic coset, sayF(t, i), ofp t −1−i, with 1 ≦t ≦ m and 0 ≦i < p−1. We then study such privileged codes. We first make precize which duals of extendedBCH codes they are. Next, we show that Weil's bound in this context gives an explicit formula; that is, the couple (t, i) fully determines the value of the Weil bound for the code with borderF(t, i). In the case where this value is negative, we use the Roos method to bound the minimum distance, greatly improving the BCH bound.