On the Minimum Distances of Some Families of BCH Codes

BCH codes form an important class of cyclic codes, which have applications in communication and data storage systems. Although the BCH bound provides a lower bound on the minimum distance of BCH codes, determining the true minimum distances of BCH codes is a very challenging problem. In this paper, we settle the minimum distances of a number of infinite families of narrow-sense BCH codes. By explicitly constructing the locator polynomials for minimum weight codewords, we obtain many families of primitive and non-primitive BCH codes withd= δ, wheredis the minimum distance of aq-ary BCH code of lengthn, designed distance δ, and offsetb, denoted by C(q,n,δ,b). For primitive BCH codes, we obtain infinite families of BCH codes over F3and F4satisfyingd= δ, where δ 2 {5, 6, 7, 8}. Moreover, we construct several infinite families ofq-ary BCH codes withd= δ, where 2 ≤ δ ≤q−1. For δ =qt+1, we prove that the BCH code C(q,qm−1,qt+1,1)hasd= δ for allmsatisfyingm≡ 0 (modpt), wherepdenotes the characteristic of Fq. In the paper by Ding et al., IEEE Trans. Inf. Theory 61(5): 2351-2356, it was conjectured that the minimum distance of C(q,qm−1,qt+1,1)is always equal to its Bose distancedB. Our result confirms this conjecture for the casem≡ 0 (modpt). For non-primitive BCH codes, we construct a family of BCH codes C(q, qp−1/λ ,p+1,1)withd= δ =p+ 1, wherepis an odd prime,q = pewithp∤eand λ |q− 1.