ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 4, pp. 317–332.
Pleiades Publishing, Inc., 2009.
Original Russian Text
V.A. Zinoviev, D.V. Zinoviev, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 4, pp. 26–42.
On One Transformation of Steiner
Quadruple Systems S(v, 4, 3)
V. A. Zinoviev and D. V. Zinoviev
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received June 16, 2008; in ﬁnal form, August 12, 2009
Abstract—A transformation of Steiner quadruple systems S(v, 4, 3) is introduced. For a given
system, it allows to construct new systems of the same order, which can be nonisomorphic to
the given one. The structure of Steiner systems S(v, 4, 3) is considered. There are two diﬀerent
types of such systems, namely, induced and singular systems. Induced systems of 2-rank r can
be constructed by the introduced transformation of Steiner systems of 2-rank r − 1 or less.
A suﬃcient condition for a Steiner system S(v, 4, 3) to be induced is obtained.
A Steiner system S(v, k, t)isapair(X, B), where X is a v-set and B is a collection of k-subsets
of X (called blocks) such that every t-subset of X is contained in exactly one element of B.
In particular, a system S(v, 3, 2) is called a Steiner triple system (and is denoted by STS(v)), and
asystemS(v, 4, 3) is called a Steiner quadruple system (and is denoted by SQS(v)). In  it is
proved that a necessary condition v ≡ 2 or 4 (mod 6) for the existence of S(v, 4, 3) is also suﬃcient.
A Steiner system S(v, 4, 3) is said to be resolvable if the set of blocks can be partitioned into
distinct pairwise disjoint subsets such that each subset is a (trivial) Steiner system S(v, 4, 1).
By eﬀorts of a series of authors (see [2, 3] and references therein) it was proved that resolvable
systems S(v, 4, 3) exist for all suitable values of v (which are multiples of 4).
Concerning Steiner systems, see surveys [4–6].
Two systems SQS(X, B) and SQS(X
)areisomorphic if there is a bijection α: X → X
maps quadruples of B to those of B
.Anautomorphism of SQS(X, B) is a permutation of elements
of X which does not change the set of blocks B. Finding the number of nonisomorphic systems
SQS(v) (as well as of nonisomorphic resolvable systems SQS(v)) is the main problem in this area.
Denote by N(v) the number of nonisomorphic systems S(v, 4, 3). After eﬀorts of many authors,
itwasprovedinthatN(16) ≥ 31 301. There was no progress in this direction (see [5, 6]) until
2003, when all nonisomorphic systems S(16, 4, 3) of 2-rank 12 were found in . Then, in , all
systems SQS(16) with rank less than or equal to 13, and then, in [9, 10], those with rank 14 were
classiﬁed. Finally, in  all nonisomorphic Steiner systems S(16, 4, 3) were found.
These results can be formulated as follows. Among nonisomorphic Steiner systems S(16, 4, 3)
of order v =16there are:
– one system S(16, 4, 3) of rank 11 (points and planes of the aﬃne geometry AG(4, 2));
– 15 systems S(16, 4, 3) of rank 12 [7, 8];
– 4131 systems S(16, 4, 3) of rank 13 ;
Supported in part by the Russian Foundation for Basic Research, project no. 09-01-00536.