Contemporary Mathematics Volume 93 ( 1989) The S. D. Berman Work on Coding Theory and on Theory of Threshold Functions B. N. GARTSHTEIN AND 1.1. GRUSHKO S. D. Berman's works form an essential contribution to discrete math- ematics, especially to algebraic coding theory and the theory of threshold functions. The first part of the survey written by I. I. Grushko contains the main results on the topic of algebraic coding. The second half written by B. N. Gartshtein contains a small part of the results in the theory of thresh- old functions that S. D. Berman had an opportunity to publish just before his sudden death. The most of Berman's papers on the topic of coding is concerned with the creation of the theory of group codes. In the framework of coding theory one usually concentrates on the calculation of code parameters. We will only consider a linear code V, i.e. a linear subspace in the n-dimensional linear space over a field F. In this case the code parameters are defined as follows: n is block length: the number k( V) of information symbols is the dimension of V over F the Hamming weight wt(x) of an element x E V is the number of nonzero components in the decomposition x = EgEG agg, ag E F the code distance is d(V) = mino ixEV wt(x) the rate of the code V is the ratio k(V)In. The first paper contains two parts I 1-21 and two significant results. At first, we will give some introduced in I 1 I definitions. Let G be a finite group, F an arbitrary field and FG a group algebra of the group G over F. Any ideal V of F G is called an F G-eode. If G is abelian, then F G-eode is called abelian. If the characteristic ofF does not divide the order of G, then FG- code is called semisimple otherwise it is called modular. If G is a p-group and the characteristic ofF is p, then V is called a p-code. All these definitions are the natural generalizations of some classes of codes well-known in those days: cyclic codes, i.e. ideals in the group algebra FG of cyclic group G over the field F, and Reed-Muller codes (RM-codes), i.e. ideals in the group algebra FG of the group G of type (2, 2, ... , 2) over the field F = {0, 1 }. © 1989 American Mathematical Society 0271-4132/89$1.00 + $.25 per page http://dx.doi.org/10.1090/conm/093/1003337
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