A Class of Narrow-Sense BCH Codes

Abstract

BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers. 1) The dimension of narrow-sense BCH codes of length ((q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> -1)/λ) with designed distance 2 ≤ δ ≤ ((q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Γ(m+1)/2⌉</sup> - 1)/(λ) + 1) is settled, where λ is any factor of (q - 1). 2) The weight distributions of two classes of narrow-sense BCH codes of length ((q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> - 1)/2) with designed distance δ = (((q - 1)q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sup> - q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⌊(m-1)12⌋</sup> - 1)/2) and δ = (((q - 1)q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sup> - q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⌊(m+1)/2⌋</sup> - 1)/2) are determined. 3) The weight distribution of a class of BCH codes of length ((q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> - 1)/(q - 1)) is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized.