Abstract: We show that any binary (n=2^k-3, 2^{n-k}, 3) code C1 is a cell of an equitable partition (perfect coloring) (C1,C2,C3,C4) of the n-cube with the quotient matrix ((0, 1, n-1, 0) (1, 0, n-1, 0) (1, 1, n-4, 2) (0, 0, n-1, 1)). Now the possibility to lengthen the code C1 to a 1-perfect code of length n + 2 is equivalent to the possibility to split the cell C 4 into two distance-3 codes or, equivalently, to the biparticity of the graph of distances 1 and 2 of C4. In any case, C1 is uniquely embedable in a twofold 1-perfect code of length n+2 with some structural restrictions, where by a twofold 1-perfect code we mean that any vertex of the space is within radius 1 from exactly two codewords. By one example, we briefly discuss 2-(n,3,2) multidesigns with similar restrictions. We also show a connection of the problem with the problem of completing latin hypercuboids of order 4 to latin hypercubes.

Cite: Krotov D.S.
On the binary codes with parameters of doubly-shortened 1-perfect codes
Designs, Codes and Cryptography. 2010. V.57. N2. P.181-194. DOI: 10.1007/s10623-009-9360-5 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Jul 25, 2009
Accepted:	Dec 19, 2009
Published online:	Jan 20, 2010

Identifiers:

Web of science:	WOS:000280072200006
Scopus:	2-s2.0-77955665454
Elibrary:	15314482
OpenAlex:	W4234318926

Citing:

DB	Citing
Web of science	11
Scopus	14
Elibrary	11
OpenAlex	13

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