Finite Quasi-Frobenius Modules and Linear Codes

Abstract

The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting.

Code equivalence characterizes finite Frobenius rings

“…Theorem 2.2 has been generalized to the context of linear codes defined over modules (to be defined in Section 4). Greferath, Nechaev, and Wisbauer show in [5] that every finite ring R has a Frobenius bimodule R (its character bimodule) and that codes over R have the extension property for Hamming weight. When R is a Frobenius ring, R is isomorphic to R as one-sided modules, so that Theorem 2.2 is a special case of the result in [5].…”

“…Kurakin et al [8] first introduced such codes for alphabets that are modules over a finite commutative ring. Later, Greferath et al [5] developed the theory for modules over an arbitrary finite ring. Codes over modules provide exactly the right setting for the strategy of Dinh and López-Permouth described in Remark 2.5.…”

“…The author acknowledges with pleasure his intellectual debt to the work of Dinh and López-Permouth [4] and the work of Greferath et al [5]. The author thanks several anonymous referees for their helpful suggestions.…”

On semiperfect rings of injective dimension one

“…The finite Frobenius rings have many important applications in coding theory (see, for example, [5], [6], [8]). …”

Isometry groups of combinatorial codes

“…As it was shown in numerous papers [4,6,13], for linear codes over module alphabets an analogue of the MacWilliams Extension Theorem does not hold in general. This means that there may exist codes with automorphisms that do not extend to monomial maps.…”