Several new classes of MDS symbol-pair codes derived from matrix-product codes

In order to correct the pair-errors generated during the transmission of modern high-density data storage that the outputs of the channels consist of overlapping pairs of symbols, a new coding scheme named symbol-pair code is proposed. The error-correcting capability of the symbol-pair code is determined by its minimum symbol-pair distance. For such codes, the larger the minimum symbol-pair distance, the better. It is a challenging task to construct symbol-pair codes with optimal parameters, especially, maximum-distance-separable (MDS) symbol-pair codes. In this paper, the permutation equivalence codes of matrix-product codes with underlying matrices of orders $3$ and $4$ are used to extend the minimum symbol-pair distance, four new classes of MDS symbol-pair codes and a new class of AMDS symbol-pair codes are derived.