ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 2, pp. 139–147.
Pleiades Publishing, Inc., 2015.
Original Russian Text
I.Yu. Mogilnykh, F.I. Solov’eva, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 2, pp. 57–66.
On Separability of the Classes of Homogeneous
and Transitive Perfect Binary Codes
I. Yu. Mogilnykh and F. I. Solov’eva
Sobolev Institute of Mathematics, Siberian Branch
of the Russian Academy of Sciences, Novosibirsk, Russia
Novosibirsk State University, Novosibirsk, Russia
e-mail: firstname.lastname@example.org, email@example.com
Received December 9, 2014; in ﬁnal form, February 19, 2015
Abstract—By the example of perfect binary codes, we prove the existence of binary homo-
geneous nontransitive codes. Thereby, taking into account previously obtained results, we
establish a hierarchical picture of extents of linearity for binary codes; namely, there is a strict
inclusion of the class of binary linear codes in the class of binary propelinear codes, which are
strictly included in the class of binary transitive codes, which, in turn, are strictly included in
the class of binary homogeneous codes. We derive a transitivity criterion for perfect binary
codes of rank greater by one than the rank of the Hamming code of the same length.
The closest to linear codes in a number of properties (especially in the structure of the auto-
morphism group) are propelinear and transitive codes (for deﬁnitions, see below). The question
of existence of transitive codes that are not propelinear was ﬁrst posed in 2006 by Pujol, Rif`a,
and Solov’eva. Later on, when perfect binary codes of length 15 were classiﬁed (see [1, 2]) and all
transitive and homogeneous perfect codes of length 15 were enumerated, the question naturally
arose on the existence of an inﬁnite series of binary homogeneous codes that are not transitive.
Both questions have been answered positively. The answer to the ﬁrst question was obtained in ,
where it was proved that the well-known binary Best’s code of length 10 with code distance 4,
being transitive, is not propelinear. The existence of an inﬁnite series of transitive nonpropelinear
perfect codes was proved in :
Theorem 1. For any n ≥ 15 there exist perfect binary transitive codes of length n that are not
It should be noted that in  it was proved that only one of the 201 equivalence classes of
transitive perfect codes of length 15 is not propelinear. An answer to the question on the existence
of binary homogeneous nontransitive codes is given in the present paper by an example of perfect
codes. Thus, the inclusion structure of classes of the above-mentioned close-to-linear codes is of
L ⊂ Prl ⊂ Tr ⊂ Hom,
where L is the class of linear binary codes, Prl is the class of propelinear binary codes, Tr is the
class of transitive binary codes, and Hom is the class of homogeneous binary codes.
The research was carried out at the expense of the Russian Science Foundation, project no. 14-11-00555.