ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 3, pp. 265–275.
Pleiades Publishing, Inc., 2016.
Original Russian Text
I.Yu. Mogilnykh, F.I. Solov’eva, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 3, pp. 73–83.
On the Symmetry Group of the Mollard Code
I. Yu. Mogilnykh
and F. I. Solov’eva
Sobolev Institute of Mathematics, Siberian Branch
of the Russian Academy of Sciences, Novosibirsk, Russia
Received August 4, 2015; in ﬁnal form, February 1, 2016
Abstract—We study the symmetry group of a binary perfect Mollard code M(C, D)oflength
tm + t + m containing as its subcodes the codes C
formed from perfect codes C
and D of lengths t and m, respectively, by adding an appropriate number of zeros. For the
Mollard codes, we generalize the result obtained in  for the symmetry group of Vasil’ev codes;
namely, we describe the stabilizer Stab
Sym(M(C, D)) of the subcode D
in the symmetry
group of the code M (C, D) (with the trivial function). Thus we obtain a new lower bound on
the order of the symmetry group of the Mollard code. A similar result is established for the
automorphism group of Steiner triple systems obtained by the Mollard construction but not
necessarily associated with perfect codes. To obtain this result, we essentially use the notions
of “linearity” of coordinate positions (points) of a nonlinear perfect code and a nonprojective
Steiner triple system.
In the literature there are few results devoted to the analysis of symmetry groups of perfect
codes, even of binary ones. Studying such groups and automorphism groups is important, since
these groups reﬂect symmetries of codes and elucidate their ﬁne structure. In the present paper we
essentially use two characteristics of “linearity” of coordinate positions (points) in binary nonlinear
perfect codes and nonprojective Steiner triple system, not necessarily associated with perfect codes.
One of these characteristics, related to Pasch conﬁgurations, was rather widely used in the literature,
and the second, μ-linearity, was ﬁrst proposed in  (see below for deﬁnitions and comments). Both
invariants are important in studying profound structural properties of perfect codes. It is known
that the symmetry group of a perfect codes containing the zero vector is closely related with the
automorphism group of its Steiner triple system.
According to the result of , every ﬁnite group is isomorphic to the symmetry group of some
binary perfect single-error-correcting code. In  it was proved that the symmetry group of any
perfect code of length n is isomorphic to the symmetry group of the subcode consisting of all its
codewords of weight (n−1)/2. However, these profound results do not provide complete information
on the structure of symmetry groups and automorphism groups of perfect codes.
It is well known  that the symmetry group Sym(H) of a Hamming code H of length n is iso-
morphic to the general linear group GL(log(n+1), 2) and is maximal in the class of perfect codes .
On the other hand, existence of classes of perfect codes (both nonsystematic and systematic) with
trivial automorphism groups was considered in [7–9]. Finally we note that automorphism groups
of arbitrary perfect codes were also studied in [6, 10–13]. For basic deﬁnitions used in this paper,
Supported by the Russian Foundation for Basic Research, project nos. 15-01-05867 and 13-01-00463.
Supported by the Russian Foundation for Basic Research, project no. 16-01-00499.