Abstract: We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3,2^{n-m-1},4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the $n$-cube into six cells. An arbitrary binary $(n=2^m-4,2^{n-m},3)$ code $D$, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) with six cells. As a corollary, the codes $C$ and $D$ are completely semiregular; i.e., the weight distribution of such codes depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if $D$ is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable.

Cite: Krotov D.S.
On the binary codes with parameters of triply-shortened 1-perfect codes
Designs, Codes and Cryptography. 2012. V.64. N3. P.275-283. DOI: 10.1007/s10623-011-9574-1 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Mar 16, 2011
Accepted:	Sep 29, 2011
Published online:	Oct 13, 2011

Identifiers:

Web of science:	WOS:000305520100005
Scopus:	2-s2.0-84863782397
Elibrary:	23985142
OpenAlex:	W3100598789

Citing:

DB	Citing
Web of science	8
Scopus	10
Elibrary	5
OpenAlex	15

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