ISSN 0032-9460, Problems of Information Transmission, 2012, Vol. 48, No. 2, pp. 102–126.
Pleiades Publishing, Inc., 2012.
Original Russian Text
V.A. Zinoviev, D.V. Zinoviev, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 2, pp. 21–47.
Steiner Triple Systems S(2
− 1, 3, 2)
of Rank 2
− m +1 over
V. A. Zinoviev
and D. V. Zinoviev
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow
Received December 19, 2011; in ﬁnal form, April 11, 2012
Abstract—Steiner systems S(2
− 1, 3, 2) of rank 2
− m + 1 over the ﬁeld F
A new recursive method for constructing Steiner triple systems of an arbitrary rank is proposed.
The number of all Steiner systems of rank 2
−m+1 is obtained. Moreover, it is shown that all
Steiner triple systems S(2
− 1, 3, 2) of rank r ≤ 2
− m + 1 are derived, i.e., can be completed
to Steiner quadruple systems S(2
, 4, 3). It is also proved that all such Steiner triple systems
are Hamming; i.e., any Steiner triple system S(2
− 1, 3, 2) of rank r ≤ 2
− m +1 overthe
occurs as the set of words of weight 3 of a binary nonlinear perfect code of length 2
A Steiner system S(v, k, t)isapair(X, B)whereX is a set of v elements 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.
AsystemS(v,3, 2) is called a Steiner triple system (brieﬂy STS(v)). A system S(v, 4, 3) is called
a Steiner quadruple system (brieﬂy SQS(v)). In  it is proved that the necessary condition
v ≡ 2 or 4 (mod 6) for the existence of systems SQS(v) is also suﬃcient.
Two Steiner systems S(v, k, t) given by pairs (X, B)and(X
)areisomorphic if there is a
bijection α : X → X
which maps blocks of B into blocks of B
.Anautomorphism of a Steiner
system S(v, k,t)givenbyapair(X, B) is a permutation of elements of X which does not change the
set B of blocks. The existence problem for such Steiner systems S(v, k, t) for any triple of natural
numbers v, k,t (2 ≤ t<k≤ v − 2) and ﬁnding the number of nonisomorphic Steiner systems
S(v, k, t), should they exist, is a major problem in this area; see surveys on Steiner systems [2–4].
A Steiner system S(v, k, t) is resolvable if it splits into trivial Steiner systems S(v,k, 1), where,
of course, k divides v.
In recent works [5,6], all diﬀerent Steiner triple systems STS(v) and quadruple systems SQS(v+1)
or order v =2
− 1andv +1=2
, respectively, with 2-rank (i.e., rank over the ﬁeld F
were enumerated. In , the authors of the present paper enumerated all diﬀerent Steiner quadruple
systems SQS(v)oforderv =2
and 2-rank r ≤ v − m + 1. It turned out that all of such quadruple
systems are resolvable.
The goal of the present work is to enumerate all diﬀerent Steiner triple systems STS(v)oforder
− 1 of the next possible rank, r =2
− m +1, over F
. We propose a new recursive method
for constructing triple systems STS(v). In particular, all systems of order v =2
− 1 and of rank
not greater than 2
− m + 1 can be constructed by this method. We also obtain the number of
Supported in part by the Russian Foundation for Basic Research, project no. 12-01-00905.