On the b-Symbol Distances of Matrix Product Codes, Constacyclic Codes, and Reed-Muller Codes

Matrix product codes are generalizations of some well-known classes of codes, including generalized Reed-Muller codes and repeated-root constacyclic codes. Recently, a bound for the minimum symbol-pair distance of a matrix product code was given by (Luo et al., 2023), leading to the creation of new families of MDS symbol-pair codes. In this paper, we provide lower and upper bounds for the minimum b-symbol distance of matrix product codes, which naturally extends some of the results by (Luo et al., 2023). Examples meeting the bounds are included to illustrate our results. As an initial application of these new bounds, we establish that constacyclic codes (repeated-root or otherwise) meeting specific criteria can indeed be classified as matrix product codes, and present bounds on the minimum b-symbol distance of such constacyclic codes. Additionally, we determine all the minimum b-symbol distances of Reed-Muller codes as a secondary application of the new bounds.