ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 1, pp. 23–31.
Pleiades Publishing, Inc., 2009.
Original Russian Text
F.I. Solov’eva, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 1, pp. 27–35.
On Transitive Partitions of an n-Cube into Codes
F. I. Solov’eva
Sobolev Institute of Mathematics, Siberian Branch of the RAS, Novosibirsk
Novosibirsk State University
Received January 18, 2008; in ﬁnal form, September 26, 2008
Abstract—We present methods to construct transitive partitions of the set E
of all binary
vectors of length n into codes. In particular, we show that for all n =2
− 1, k ≥ 3, there exist
transitive partitions of E
into perfect transitive codes of length n.
We continue the construction and analysis of transitive objects in an n-cube E
started in [1,2].
In  there were proposed construction methods for transitive codes; in particular, there were
constructed at least k/2
nonequivalent perfect transitive codes of length n =2
− 1, k>4,
with code distance 3. A similar result is true for extended perfect transitive codes. The obtained
transitive codes have various ranks (the rank of a code is the dimension of its linear span): up to
n − log
(n +1)/4, for n =16
− 1, ≥ 1; and from n − log
(n +1), whichis greater by1than
the rank of the linear perfect binary code (the Hamming code), to n −log
(n +1)/2, for every
− 1, k>4.
In , for n →∞, there are constructed exponentially many pairwise nonequivalent transitive
extended perfect binary codes of rank 4n − log
4n (which is greater by 1 than the rank of the
Hamming code) of length 4n. It is easily seen that extending a transitive code with a parity check
always results in a transitive code. The converse is not always true; for instance, in  there is
presented an extended perfect transitive code of length 16 such that neither of its sixteen punctured
codes of length 15 is transitive. Thus, it is reasonable to construct and analyze extended transitive
codes independently of constructing transitive codes that are not extended. Many of known classes
of codes are transitive; for instance, such are all additive codes and all Z
-linear codes (see [1–11]).
To the construction of partitions of the metric n-space E
of all binary vectors of length n with
the Hamming metric into perfect codes, few papers are devoted (see [12–22]). Two constructions
(switching and concatenated) were proposed in ; the ﬁrst of them yields the following nontrivial
lower bound on the number of partitions P
into perfect codes of length n, n>15 (see [20,21]):
In  it is shown that for each n =2
− 1, k ≥ 5, there exists a partition of the set of all
binary vectors of length n into pairwise nonequivalent perfect binary codes of length n with code
distance 3. The paper  is devoted to constructing partitions into pairwise nonparallel Hamming
codes. In , all nonequivalent partitions of E
into Hamming codes were classiﬁed; their number
proved to be 11.
In the present paper we apply the switching partition construction method of  (using the
Vasil’ev codes ), as well as a generalization of this construction using the Mollard codes ,