ISSN 0032-9460, Problems of Information Transmission, 2007, Vol. 43, No. 1, pp. 33–47.
Pleiades Publishing, Inc., 2007.
Original Russian Text
V.A. Zinoviev, D.V. Zinoviev, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 1, pp. 39–55.
On Resolvability of Steiner Systems S(v =2
, 4, 3)
of Rank r ≤ v − m +1over F
V. A. Zinoviev
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received February 14, 2006; in ﬁnal form, September 12, 2006
Abstract—Two new constructions of Steiner quadruple systems S(v,4, 3) are given. Both
preserve resolvability of the original Steiner system and make it possible to control the rank of
the resulting system. It is proved that any Steiner system S(v =2
, 4, 3) of rank r ≤ v − m +1
is resolvable and that all systems of this rank can be constructed in this way. Thus, we
ﬁnd the number of all diﬀerent Steiner systems of rank r = v − m +1.
A Steiner system S(v, k, t)isapair(X, B), where X is a v-set and B is a collection of k-subsets
(blocks) of X such that every t-subset of X is contained in exactly one block of B.Inparticular,
asystemS(v,3, 2) is called a Steiner triple system (denoted by STS(v)), and a system S(v,4, 3)
is called a Steiner quadruple system (denoted by SQS(v)). Hanani  proved that the necessary
condition v ≡ 2orv ≡ 4 (mod 6) for the existence of an S(v, 4, 3) is also suﬃcient.
A Steiner system S(v,4, 3) is called resolvable if it can be split into disjoint sets so that every
set is a Steiner system S(v, 4, 1). After a series of works where authors constructed resolvable
systems S(v,4, 3) for particular values of v, in 1987 Hartman  proved that resolvable Steiner
systems S(v, 4, 3) exist for all suitable v =4u,wherev ≡ 2orv ≡ 4 (mod 6), except for 23 cases.
Those cases where ﬁnally studied in 2005 (see ). Resolvable Steiner systems S(v, 4
, 3) exist for
all suitable values of v, but resolvability of an arbitrary quadruple Steiner system S(v, 4, 3) where v
is a multiple of 4 is still an open question.
Two systems, SQS(X,B) and SQS(X
), are isomorphic if there is a bijection α: X → X
that maps quadruples of B to those of B
.Anautomorphism of SQS(X, B) is an isomorphism
of (X, B) to itself. Finding the number of nonisomorphic SQS(v),whichwedenotebyN(v), is
the major problem in this area. Barrau  proved that N(v)=1forv ≤ 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 eight vertices) together with their automorphism groups allowed 
to obtain, using the classical doubling construction, the bound N(16) ≥ 31 021. The authors of 
slightly improved this bound in : N(16) ≥ 31 301. Since then, there was no progress in this area
(see [10, 11]).
The authors of the present paper enumerated in [12, 13] all nonisomorphic Steiner systems
S(16, 4, 3) of ranks 12, 13, and 14. Let N
(16) be the number of nonisomorphic systems S(16, 4, 3)
Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00226.