A. A. Nechaev

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.

Kerdock code in a cyclic form

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Trace-Function on a Galois Ring in Coding Theory

Patterns of the distribution of elements in words of linear codes over a Galois ring and in their representations over a Galois field are investigated. Often they may be evaluated using numbers of some special solutions of the equation defined by the trace-function on a Galois ring. Here the solutions of such an equation over a Galois ring R=GR(q2,4) of characteristic 4 are enumerated. It allows us in particular to describe the complete weight enumerators of the base linear code K R (m) and the appropriate Kerdock code K q (m+1) over a Galois field of q=21 elements.Results based on properties of special quadrics over GF(21) arise by description of the 2-adic decomposition of the trace-function.

Linear recurring sequences over modules

The aim of this paper is to extend some fundamental and applied results of the theory of linear recurring sequences over fields to the case of polylinear recurring sequences over rings and modules. Quasi-Frobenius modules and Galois rings play a very special role in this project.

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 recurring sequences over Galois rings. (Russian, English)

Linear recurrences of maximal period over a Galois ring and over a residue class ring modulo p are studied. For any such recurrence, the coordinate sequences (in p-adic and some other expansions) are considered as linear recurring sequences over a finite field. Upper and lower bounds for the ranks (linear complexities) of these coordinate sequences are obtained. The results are based on using the properties of Galois rings and the trace-function on such rings.

Formal Duality of Linearly Presentable Codes over a Galois Field

In [4] it was shown, that the weight enumerators of two binary ℤ4-linearly dual codes satisfy the McWilliams identity (i.e. these codes are formally dual). If we consider an arbitrary Galois ring R=GR(q2, p2) of characteristic p2 and a pair of R-linearly dual codes over a Galois field GF(q) this result is not preserved. We propose the approach to correcting this disadvantage. The titled codes are presented as codes inthe alphabet ℜ=RS q (q,2), being a Reed-Solomon code. The appropriate exact weight enumerators of these codes are reduced to some projective weight enumerators (obtained by identifying of variables) which satisfy the McWilliams identity for linear codes over GF(q). We discuss ways of “optimal” identifying of variables such that the corresponding projective weight enumerators allow us to construct complete weight enumerators of the initial codes over GF(q).

COORDINATE SETS OF GENERALIZED GALOIS RINGS

A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic pn, for some prime number p, such that its top-factor is a semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to , the so-called Teichmuller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop is right (or left) primitive.

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).

Approximation of Boolean functions by monomial ones

Every Boolean function of n variables is identified with a function F : Q → P, where Q = GF(2 n ), P = GF(2). A. Youssef and G. Gong showed that for n = 2λ there exist functions F which have equally bad approximations not only by linear functions (that is, by functions tr (μx), where μ ∈ Q* and tr: Q → P is the trace function), but also by proper monomial functions (functions tr(μxδ ), where (δ, 2 n − 1) = 1). Such functions F were called hyper-bent functions (HB functions, HBF), and for any n = 2λ a non-empty class of HBF having the property F(0) = 0 was constructed. This class consists of the functions F(x) = such that the equation F(x) = 1 has exactly (2 λ − 1)2 λ−1 solutions in Q. In the present paper, we give some essential restrictions on the parameters of an arbitrary HBF showing that the class of HBF is far less than that of bent functions. In particular, we show that any HBF is a bent function having the degree of nonlinearity λ, and for some n (for instance, if λ > 2 and 2 λ − 1 is prime, or λ ∈ {4,9,25,27}) the class of HBF is exhausted by the functions F(x) = described by A. Youssef and G. Gong. For n = 4, in addition to 10 HBF listed above there exist 18 more HBF with property F(0) = 0. The question of whether there exist other hyper-bent functions for n > 4 remains open.