ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 3, pp. 289–298.
Pleiades Publishing, Inc., 2016.
Original Russian Text
I.Yu. Mogil’nykh, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 3, pp. 97–107.
On Extending Propelinear Structures of the
Nordstrom–Robinson Code to the Hamming Code
I. Yu. Mogil’nykh
Sobolev Institute of Mathematics, Siberian Branch
of the Russian Academy of Sciences, Novosibirsk, Russia
Received September 8, 2015; in ﬁnal form, May 24, 2016
Abstract—A code is said to be propelinear if its automorphism group contains a subgroup
which acts on the codewords regularly. Such a subgroup is called a propelinear structure on
the code. With the aid of computer, we enumerate all propelinear structures on the Nord-
strom–Robinson code and analyze the problem of extending them to propelinear structures
on the extended Hamming code of length 16. The latter result is based on the description of
partitions of the Hamming code of length 16 into Nordstrom–Robinson codes via Fano planes,
presented in the paper. As a result, we obtain a record-breaking number of propelinear struc-
tures in the class of extended perfect codes of length 16.
Consider the vector space F
of dimension n over the ﬁeld of two elements. The Hamming
distance between two elements of F
is deﬁned to be the number of coordinates in which they
diﬀer. A set C, C ⊂ F
, is called a binary code with parameters (n, M, d), where n is the code
length, |C| = M, and the minimum distance between distinct codewords in C is d.Acodeof
length n containing the zero vector 0
is said to be reduced.
By the automorphism group Aut(F
equipped with the Hamming metric, we
mean the group of its isometries with respect to the composition operation. It is well known that
) is conﬁned to transformations of the form (x, π), with π ∈ S
and x ∈ F
(x, π)(y)=x + π(y)=
The composition (x, π) · (y, π
) of automorphisms (x, π)and(y, π
) is the automorphism
(x, π) · (y, π
)=(x + π(y),π· π
In the group Aut(F
) we select a subgroup Sym(F
), called the symmetry group:
,π): π ∈ S
The automorphism (symmetry) group of a code C, denoted by Aut(C) (respectively, Sym(C)),
is the stabilizer of the set of its codewords in the automorphism (respectively, symmetry) group
The Nordstrom–Robinson code  has 256 codewords, length 16, and is an optimal code with
code distance 6 (it was independently constructed in ). It is known  that a code with these
The results of Section 3 of the paper are obtained under the support of the Russian Foundation for Basic
Research, project no. 13-01-00463; results of Section 4 are funded by the Russian Science Foundation,
project no. 14-11-00555.