Linear Codes Over a General Infinite Family of Rings and Macwilliams-Type Relations

Abstract

We study structural properties of linear codes over the ring R k which is defined by R [ v 1 , v 2 , . . . , v k ] with conditions v 2 i = v i for i = 1 , 2 , . . . , k , where R is any finite commutative Frobenius ring. We describe these linear codes in terms of necessary and sufficient conditions involving Gray maps, and we use these characterizations to construct Hermitian and Euclidean self-dual linear codes of this ring of arbitrary given length. We also derive MacWilliams-type relations for these codes with respect to Hamming weight enumerator as well as complete and symmetrized weight enumerators. As an application of the obtained results, we construct several optimal linear codes over Z 4 .