V. L. Kurakin

Linear recurring sequences over rings and modules

Here we present some fundamental concept and results of the theory of linear recurring sequences over rings and modules and their applications. Of course, the authors give in more detail those results that are close to their mathematical interests. In particular, an attempt has been made to construct a general algebraic theory of k-LRS over modules, paying explicit attention to periodic k-sequences, to properties of linear recurrences over finite rings and especially over Galois rings, and also to methods of constructing codes baed on such recurrences.

Linear Codes and Polylinear Recurrences over Finite Rings and Modules (A Survey)

We give a short survey of the results obtained in the last several decades that develop the theory of linear codes and polylinear recurrences over finite rings and modules following the well-known results on codes and polylinear recurrences over finite fields. The first direction contains the general results of theory of linear codes, including: the concepts of a reciprocal code and the MacWilliams identity; comparison of linear code properties over fields and over modules; study of weight functions on finite modules, that generalize in some natural way the Hamming weight on a finite field; the ways of representation of codes over fields by linear codes over modules. The second one develops the general theory of polylinear recurrences; describes the algebraic relations between the families of linear recurrent sequences and their periodic properties; studies the ways of obtaining “good” pseudorandom sequences from them. The interaction of these two directions leads to the results on the representation of linear codes by polylinear recurrences and to the constructions of recursive MDS-codes. The common algebraic foundation for the effective development of both directions is the Morita duality theory based on the concept of a quasi-Frobenius module.

Linear and Polylinear Recurring Sequences over Abelian Groups and Modules

In this paper, the well-known results in the theory of polylinear recurring sequences over fields and their recent generalizations for sequences over modules with commutative rings of coefficients are extended to the class of polylinear sequences over modules with noncommutative rings of coefficients. Possible noncommutativity of the main ring of coefficients requires naturally a study of polylinear recurring sequences over bimodules. To estimate the linear complexity of the considered sequences, we introduce and study polylinear (k-linear) shift registers. A criterion for the theory of polylinear recurring sequences over fields to be adequately generalized in our case is the property of the main bimodule to be quasi-Frobenius with the Artinian (respectively, from the left and the right sides) rings of coefficients (i.e., so-called Artinian duality context).

Polylinear Recurring Sequences over a Bimodule

The main goal of the paper is to extend some of the known results of the theory of polylinear recurring sequences over fields and their generalizations for sequences over modules with commutative rings of coefficients to the case of noncommutative rings of coefficients. Possible noncommutativity of the main ring causes to consider polylinear sequences over a bimodule. To estimate linear complexity of the sequences in question we consider the notion of polylinear (k-linear) shift register. In fact, the theory of polylinear recurring sequences over fields admits a rather complete extension in this generality if the main bimodule is an Artinian duality context.