Fq-linear codes over Fq-algebras

Huffman (2013) [12] studied F-q-linear codes over F(q)m and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative F-q-algebra. An F-q-linear code over S of length n is an F-q-submodule of S-n. In this paper, we study F-q-linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes of MDR codes are introduced. We generalize the ordinary and Hermitian trace products over F-q-algebras and we prove the MacWilliams identity with respect to the generalized form. In particular, we obtain Huffman's results on the MacWilliams identity. Among other results, we give a theory to construct a class of quantum codes and the structure of F-q-linear codes over finite commutative graded F-q-algebras. (C) 2020 Elsevier Inc. All rights reserved.