New bounds for b-Symbol Distances of Matrix Product Codes

09/16/2023

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by   Pan Xu, et al.

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Matrix product codes are generalizations of some well-known constructions of codes, such as Reed-Muller codes, [u+v,u-v]-construction, etc. Recently, a bound for the symbol-pair distance of a matrix product code was given in <cit.>, and new families of MDS symbol-pair codes were constructed by using this bound. In this paper, we generalize this bound to the b-symbol distance of a matrix product code and determine all minimum b-symbol distances of Reed-Muller codes. We also give a bound for the minimum b-symbol distance of codes obtained from the [u+v,u-v]-construction, and use this bound to construct some [2n,2n-2]_q-linear b-symbol almost MDS codes with arbitrary length. All the minimum b-symbol distances of [n,n-1]_q-linear codes and [n,n-2]_q-linear codes for 1≤ b≤ n are determined. Some examples are presented to illustrate these results.