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Steiner systems S(v, k, k - 1): components and rank
Zinoviev V., Zinoviev D. Problems of Information Transmission47 (2):130-148,2011.Type:Article
Date Reviewed: Nov 13 2012

A Steiner system is a t-(v,k,&lgr;) design, for which &lgr;=1; in other words, in a pair (P,B), with P a set of points, |P|=v, and B a set of blocks, every element of B is incident with exactly k points, and all t points are contained in exactly one block. Let S(v,k,t) be a Steiner system. The authors’ objective is to construct new Steiner systems from a given one.

The authors first describe the connection of a Steiner triple system S(v,k,t) with a binary (v,k,2(k-t+1),N) code, that is, a code with words of length v, minimum distance 2(k-t+1), and N codewords of fixed weight k, where N is the number of blocks of S(v,k,t) through one given point. Clearly, this code is a subset of the space En.

Given a set of vectors XEn, of weight ln-1, let D(X) be the set of vectors of weight l-1 that are covered by vectors of X. In short, a set KEn is a component of order n of S(v,k,t), if there exists a set LEn such that D(K) = D(L). More conditions are needed to describe the concept exactly.

The authors give examples of components of Steiner triple and quadruple systems, that is, respectively, (k,t)=(3,2), (4,3), and describe a construction method for components of arbitrary Steiner systems S(v,k,k-1).

This paper contributes to several aspects of the theory of designs, and especially to the theory of Steiner systems. The potential construction of new Steiner systems from existing ones is a particularly interesting result. Through the connection between designs and linear codes, some results might also have an impact in coding theory.

Reviewer:  Jan De Beule Review #: CR140667 (1302-0124)
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