Problems of Information Transmission, Vol. 40, No. 4, 2004, pp. 337–355. Translated from Problemy Peredachi Informatsii, No. 4, 2004, pp. 48–67.
Original Russian Text Copyright
2004 by V. Zinoviev, D. Zinoviev.
Classiﬁcation of Steiner Quadruple Systems
of Order 16 and Rank at Most 13
V. A. Zinoviev and D. V. Zinoviev
Institute for Information Transmission Problems, RAS, Moscow
Received February 10, 2004; in ﬁnal form, June 17, 2004
Abstract—A Steiner quadruple system SQS(v)oforderv is a 3-design T (v, 4, 3,λ)withλ =1.
In this paper we describe all nonisomorphic systems SQS(16) that can be obtained by the
generalized concatenated construction (GC-construction). These Steiner systems have rank at
most 13 over F
. In particular, there is one system SQS(16) of rank 11 (points and planes of
the aﬃne geometry AG(4, 2)), ﬁfteen systems of rank 12, and 4131 systems of rank 13. All
these Steiner systems are resolvable.
A t-design (tactical conﬁguration) T (v, k, t, λ)isapair(X, B), where X is a v-set and B is a
collection of k-subsets of X such that every t-subset of X is contained in exactly λ members of B.
Such a design is called a Steiner system and is denoted by S(v,k, t)ifλ = 1. Steiner systems
S(v, k, t) were introduced in 1844 by Woolhouse , who posed the following question: For which
integers v, k, t does a system S(v, k, t) exist? This problem remains unsolved until today.
AsystemS(v, 3, 2) is called a Steiner triple system (brieﬂy, STS(v)), and a system S(v, 4, 3) is
called a Steiner quadruple system (brieﬂy, SQS(v)). The necessary condition for the existence of a
system SQS(v)isv ≡ 2 or 4 (mod 6). In 1847, Kirkman  described the construction of a system
) equivalent to points and planes of the m-dimensional aﬃne space over F
. In 1853,
Steiner  asked for the existence of systems S(v, t +1,t). In 1908, Barrau  established the
uniqueness of S(8, 4, 3) and S(10, 4, 3). But it was not until 1960 that Hanani  proved that the
necessary condition v ≡ 2or4(mod6)fortheexistenceofanS(v, 4, 3) is also suﬃcient.
Two systems, SQS(X, B) and SQS(X
), are isomorphic if there is a bijection α: X → X
that maps the quadruples of B to those of B
.Anautomorphism of a system SQS(X, B)isan
isomorphism of (X, B) to itself. To determine the number of nonisomorphic SQS(v), which we
denote by N(v), is the major problem in this area. The result of Barrau  means that N(v)=1
for v ≤ 10, and Mendelson and Hung  derived with the help of a computer that N(14) = 4.
In  it was shown that N(16) ≥ 8. Using computer-assisted computations, Gibbons, Mathon,
and Corneil  proved that N (16) ≥ 282. The knowledge of all nonisomorphic 1-factorizations
(the complete graph on 8 vertices) together with their automorphism groups allowed Lindner
and Rosa , using the classical doubling construction, to obtain the bound N(16) ≥ 31 021. They
slightly improved this bound in : N(16) ≥ 31 301. No progress has been made in this regard
since  appeared (see [11, 12]).
Our purpose is to describe all nonisomorphic systems SQS(16) of rank at most 13 over F
main result can be formulated as follows. Among the nonisomorphic Steiner systems S(16, 4, 3) of
order v =16there are:
Supported in part by the Russian Foundation for Basic Research, project no. 03-01-00098.
2004 MAIK “Nauka/Interperiodica”