Some self-dual codes and isodual codes constructed by matrix product codes.

In 2020, Cao et al. proved that any repeated-root constacyclic code is monomially equivalent to a matrix product code of simple-root constacyclic codes. In this paper, we study a family of matrix product codes with wonderful properties, which is a generalization of linear codes obtained from the [ u + v | u - v ] -construction and [ u + v | λ - 1 u - λ - 1 v ] -construction. Then we show that any λ -constacyclic code (not necessary repeated-root λ -constacyclic code) of length N over the finite field F q with gcd (q - 1 ord (λ) , N) ≥ 2 , where ord (λ) is the order of λ in the cyclic group F q ∗ = F q \ { 0 } , is a matrix product code of some constacyclic codes. It is a highly interesting question that the existence of sequences { C 1 , C 2 , C 3 ,... } of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances, i.e., C i is an [ n (C i) , k (C i) , d (C i) ] q -linear code such that lim i → + ∞ n (C i) = + ∞ and lim i → + ∞ d (C i) n (C i) > 0. Based on the [ u + v | λ - 1 u - λ - 1 v ] -construction, we construct several families of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances by using Reed-Muller codes, projective Reed-Muller codes. And we construct some new Euclidean isodual λ -constacyclic codes with square-root-like minimum Hamming distances from Euclidean self-dual cyclic codes and Euclidean self-dual negacyclic codes by monomial equivalences. [ABSTRACT FROM AUTHOR]