V. T. Markov

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.

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.

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.

Cyclic Generalized Galois Rings

ABSTRACT The role played by fields in relation to Galois Rings corresponds to semifields if the associativity is dropped, that is, if we consider Generalized Galois Rings instead of (associative) Galois rings. If S is a Galois ring and pS is the set of zero divisors in S, S* = S\ pS is known to be a finite {multiplicative} Abelian group that is cyclic if, and only if, S is a finite field, or S = ℤ/nℤ with n = 4 or n = p r for some odd prime p. Without associativity, S* is not a group, but a loop. The question of when this loop can be generated by a single element is addressed in this article.

Bent and hyper-bent functions over a field of 2 l elements

AbstractWe study the parameters of bent and hyper-bent (HB) functions in n variables over a field

Recursive MDS-codes and recursive differentiable quasigroups

P = \mathbb{F}_q

Application of non-associative groupoids to the realization of an open key distribution procedure

with q = 2ℓ elements, ℓ > 1. Any such function is identified with a function F: Q → P, where