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.(i) The dimension of narrow-sense BCH codes of length q m´1 λ with designed distance 2 ď δ ď q rpm`1q{2s´1 λ`1 is settled, where λ is any factor of q´1.(ii) The weight distributions of two classes of narrow-sense BCH codes of length q m´1 2 with designed distance δ " pq´1qq m´1´qtpm´1q{2u´1 2 and δ " pq´1qq m´1´qtpm`1q{2u´1 2 are determined. (iii) The weight distribution of a class of BCH codes of length q m´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.

New Explicit Good Linear Sum-Rank-Metric Codes

“…From the calculation in [28] we have the following result on F q -linear sum-rank-metric codes of the matrix size n × n. Theorem 3.3. Let q be a prime power, λ and n be two positive integers satisfying λ|q n − 1, u be an odd positive integer, u 0 , .…”

“…From construction 2 Theorem 2.1, it is natural to use several Hamming BCH codes over F q n of length q un − 1 to construct sum-rank-metric codes of block length q un − 1. From some previous calculations of dimensions of Hamming BCH codes in [13,28], we get the lower bound of the dimension of these sumrank-metric codes. It is interesting to notice that the parameters of these linear sum-rank-metric codes from Hamming BCH codes can be compared with the parameters of these sum-rank BCH codes developed by algebraic method in [17].…”

Two classes of narrow-sense BCH codes and their duals

“…Let m ≥ q, b ≡ m − 1 (mod q − 1). When b = 0, b = 1 or b = q − 2, δ 1 and |C δ 1 | have been given in [35,Lemma 16].…”

Five families of the narrow-sense primitive BCH codes over finite fields

“…There are some papers on the narrow-sense primitive BCH codes [1,2,4,9,12,13,17,18,20]. Recently, some excellent survey on parameters of the narrow-sense primitive BCH codes has been given in [6][7][8][9]13,15,19,21]. Next, we list the main known results on the dimension, the Bose distance and the minimum distance of the narrow-sense primitive BCH codes in Table 1.…”