ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 2, pp. 130–148.
Pleiades Publishing, Inc., 2011.
Original Russian Text
V.A. Zinoviev, D.V. Zinoviev, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 2, pp. 52–71.
Steiner Systems S(v, k, k − 1): Components and Rank
V. A. Zinoviev and D. V. Zinoviev
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow
Received October 20, 2009; in ﬁnal form, January 18, 2011
Abstract—For an arbitrary Steiner system S(v, k, t), we introduce the concept of a component
as a subset of a system which can be transformed (changed by another subset) without losing
the property for the resulting system to be a Steiner system S(v, k, t). Thus, a component
allows one to build new Steiner systems with the same parameters as an initial system. For
an arbitrary Steiner system S(v, k, k − 1), we provide two recursive construction methods for
inﬁnite families of components (for both a ﬁxed and growing k). Examples of such components
are considered for Steiner triple systems S(v, 3, 2) and Steiner quadruple systems S(v, 4, 3). For
such systems and for a special type of so-called normal components, we ﬁnd a necessary and
suﬃcient condition for the 2-rank of a system (i.e., its rank over F
) to grow under switching of a
component. It is proved that for k ≥ 5 arbitrary Steiner systems S(v, k, k−1) and S(v, k, k−2)
have maximum possible 2-ranks.
A Steiner system S(v, k, t)isapair(X, B), where X is a set of v elements and B is a collection
of k-sets of X (called blocks) such that every t-set of X is contained in exactly one block of B.
AsystemS(v, 3, 2) is called a Steiner triple system (brieﬂy, STS(v)), and a system S(v, 4, 3),
a Steiner quadruple system (brieﬂy, SQS(v)). As was proved in , for a system SQS (v)the
necessary existence condition v ≡ 2orv ≡ 4(mod6)isalsosuﬃcient.
Two Steiner systems S(v, k, t), deﬁned by pairs (X, B)and(X
), are isomorphic if there exists
a one-to-one correspondence α: X → X
which maps blocks of B to blocks of B
of a Steiner system S(v, k, t) deﬁned by a pair (X, B) is a permutation of elements of X which does
not change the set of blocks B. The problem of existence of such Steiner systems S(v, k, t) for an
arbitrary triple of natural numbers v, k, t (2 ≤ t<k≤ v − 2) and the number of nonisomorphic
systems S(v, k, t) (if such systems exist) is a major problem in this area (see surveys on Steiner
The present paper is a natural continuation of our previous paper , where we deﬁned a
transformation of a Steiner quadruple systems S(v, 4, 3). This transformation can be reduced to
a permutation of two coordinate positions in a subset of a system consisting of eight blocks, i.e.,
a typical switching transformation. Generally speaking, this idea is not new. In many areas of
discrete mathematics, including perfect codes with d = 3, Latin squares, and Steiner triple systems
S(15, 3, 2) (we will talk about this later), such switchings have been used for a long time for the
analysis of existence of the corresponding combinatorial conﬁgurations, as well as for constructing
new ones. Without going into details, we only refer to two papers: on components of perfect
codes  and on mobile sets of such codes  (see also references therein).
Supported in part by the Russian Foundation for Basic Research, project no. 09-01-00536.