ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 2, pp. 114–133.
Pleiades Publishing, Inc., 2016.
Original Russian Text
V.A. Zinoviev, D.V. Zinoviev, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 2, pp. 15–36.
Generalized Preparata Codes and 2-Resolvable
Steiner Quadruple Systems
V. A. Zinoviev
and D. V. Zinoviev
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow, Russia
Received March 17, 2015; in ﬁnal form, November 15, 2015
Abstract—We consider generalized Preparata codes with a noncommutative group opera-
tion. These codes are shown to induce new partitions of Hamming codes into cosets of these
Preparata codes. The constructed partitions induce 2-resolvable Steiner quadruple systems
S(n, 4, 3) (i.e., systems S(n, 4, 3) that can be partitioned into disjoint Steiner systems S(n, 4, 2)).
The obtained partitions of systems S(n, 4, 3) into systems S(n, 4, 2) are not equivalent to such
partitions previously known.
AdesignD =(X, B) with parameters v, k,t, λ, denoted by T(v, k,t, λ), is a collection B of
subsets of size k (called blocks) of a set X containing v elements such that every t-subset of X
belongs to exactly λ blocks. A design with λ = 1 is called a Steiner system and is usually denoted
by S(v, k, t) (see survey papers [1, 2] on Steiner systems).
Deﬁnition 1. AsystemS = S(v, k,t)issaidtobe-resolvable,0≤ ≤ t,ifthesetB of blocks
can be partitioned into diﬀerent disjoint subsets B
where for every i the set B
is a Steiner system S(v, k, ).
It is clear that the number r is uniquely determined by the parameters v, k, t, of the design, i.e.,
The problem of -resolvability of an arbitrary Steiner system S(v, k,t)isanactualopenproblem
(see papers [1, 2] and references therein). In particular, it is known that for any (suitable) v,
i.e., v ≡ 2orv ≡ 4 (mod 6), there exists a 1-resolvable system S(v, 4, 3) , and any system
, 4, 3) of rank r<v− m +2over F
is 1-resolvable .
In this paper, 2-resolvable Steiner systems S(v, 4, 3) are considered, i.e., systems S(v, 4, 3) that
can be partitioned into disjoint systems S(v, 4, 2). Taking into account that a system S(v, 4, 2)
exists for v congruent to 1 or 4 modulo 12, we conclude that 2-resolvable systems S(v,4, 3) can
only exist for values v ≡ 4 (mod 12).
The research was carried out at the Institute for Information Transmission Problems of the Russian
Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50-00150.