On the binary codes with parameters of triply-shortened 1-perfect codes |
| |
Authors: | Denis S. Krotov |
| |
Affiliation: | 1. Sobolev Institute of Mathematics, pr. Akademika Koptyuga 4, Novosibirsk, 630090, Russia 2. Mechanics and Mathematics Department, Novosibirsk State University, Pirogova 2, Novosibirsk, 630090, Russia
|
| |
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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|