ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 3, pp. 284–288.
Pleiades Publishing, Inc., 2016.
Original Russian Text
D.S. Krotov, A.Yu. Vasil’eva, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 3, pp. 92–96.
On Perfect Codes That Do Not Contain
D. S. Krotov
and A. Yu. Vasil’eva
Sobolev Institute of Mathematics, Siberian Branch
of the Russian Academy of Sciences, Novosibirsk, Russia
Received December 30, 2015; in ﬁnal form, March 10, 2016
Abstract—We show that for every length of the form 4
− 1 there exists a binary 1-perfect
code that does not contain any Preparata-like code.
Binary 1-error-correcting perfect codes (below, perfect codes) and codes with the parameters
of Preparata codes (below, Preparata codes) are two remarkable inﬁnite families of binary codes
having many common properties. Many of these properties are related to the fact that any code
with the parameters (length, size, code distance) of codes from a considered classes induces a regular
partition of the space. Every Preparata code is contained in some perfect code .
A perfect code can contain several nonequivalent Preparata codes. In the present paper, we
consider the question of the existence of perfect codes that do not contain any Preparata code (of
course, we consider only lengths of the form 4
− 1, for which Preparata codes exist ).
As is known since the paper , the number of nonequivalent perfect codes grows doubly ex-
ponentially with respect to the length (a survey of some constructions can be found in [4, 5]).
Before , only a linear perfect code, the Hamming code, was known; moreover, because of some
strict metric-invariant properties of perfect codes [6, 7], it was conjectured that there are no other
1-error-correcting perfect binary codes.
Presently, only two classes of Preparata codes are known. The ﬁrst class contains the original
Preparata codes  and their generalizations [8–10] (the paper  contains the most general known
representation for codes of this class). These codes are nonlinear, but the containing perfect code,
the Hamming code, is linear. The second class is the class of Z
-linear Preparata codes. Strictly
speaking, the corresponding distance-6 extended Preparata codes are Z
-linear. The ﬁrst known
series of such codes was published in . As was shown in , the number of nonequivalent
-linear extended Preparata codes of length n =4
grows super-polynomially in n for almost all
values of m. All Preparata codes from the second class are contained in the same, up to equivalence,
perfect code, whose extension is Z
-linear. This perfect code is nonlinear if n>15 .
Thus, for every length n =4
− 1 > 15, only two nonequivalent perfect codes containing
Preparata codes are known. The fraction of these codes among the set of all perfect codes is
unknown. It is natural to conjecture that these fraction is negligibly small; however, before the
current work, there was no perfect code known to have length 4
− 1 ≥ 63 and guaranteedly not
contain any Preparata code. In the present paper, we give a short proof of the existence of such
The research was carried out at the expense of the Russian Science Foundation, project no. 14-11-00555.