On rank and kernel of some mixed perfect codes
type: Item
instance of: scholarly article
title: On rank and kernel of some mixed perfect codes
zbMATH Open document ID: 1189.94060
author: Fabio Pasticci
author: Thomas Westerbäck
published in: Discrete Mathematics
publication date: 2009-12-09T00:00:00Z
review text: Let \(Z(n,l)\) denote the direct product \(Z_2\times\cdots\times Z_2\times Z_2^l\), where there are exactly \(n\) factors \(Z_2\) preceding \(Z_2^l\). A word \(\bar{t}\in Z(n,l)\) is a tuple of type \((t_1,\dots,t_n,(t_{n+1},\dots,t_{n+l}))\), where the \(l\)-tuple \((t_{n+1},\dots,t_{n+l})\) is considered as one object instead of consisting of \(l\) bits. The \textit{distance} \(d\) between two words \(\bar{c},\bar{c}'\in Z(n,l)\) is the number of positions in which \(\bar{c}\) and \(\bar{c}'\) differ in the first \(n\) positions, plus one if \(\bar{c}\) and \(\bar{c}'\) are not equal in the \(l\)-tuple at the end of \(\bar{c}\) and \(\bar{c}'\). A \textit{perfect 1-error correcting code} or \textit{mixed perfect code }in \(Z(n,l)\) is a subset \(C\) of \(Z(n,l)\) such that for every \(\bar{x}\in Z(n,l)\), there is a unique \(\bar{c}\in Z(n,l)\) satisfying \(d(\bar{x},\bar{c})\leq 1\). For \(l=0\), this reduces to the classical definition of a binary perfect 1-error correcting code. The \textit{kernel} ker\((C)\) of a subset \(C\subseteq Z(n,l)\) is the set of vectors \(\bar{p}\in Z(n,l)\) for which \(C+\bar{p}=C\). There is a vast literature on binary perfect 1-error correcting codes in \(Z_2^n\). In this article, the authors discuss mixed perfect codes in \(Z(n,l)\). As concrete results, they give a lower bound and an upper bound for the rank \(k\) of the kernel of a mixed perfect code in \(Z(n,l)\) depending on the rank \(r\) of the mixed perfect code, and they give two general constructions for mixed perfect codes in \(Z(n,l)\). The first construction gives a linear mixed perfect code and the second construction gives a non-linear mixed perfect code. As an interesting feature of the linear mixed perfect codes obtained via Construction 1, the sharpness of one of the upper bounds for the rank \(k\) of the kernel of a mixed perfect code is proven.
Mathematics Subject Classification ID: 94B60
Mathematics Subject Classification ID: 94B65
zbMATH DE Number: 5644066
zbMATH Keywords: mixed perfect code
zbMATH Keywords: rank
zbMATH Keywords: Fourier coefficient
reviewed by: Leo Storme
MaRDI profile type: Publication
full work available at URL: j.disc.2008.06.037
OpenAlex ID: W2066472993
cites work: On the ranks and kernels problem for perfect codes
cites work: The classification of some perfect codes
cites work: Q5187201
cites work: On Perfect Codes and Tilings: Problems and Solutions
cites work: A full rank perfect code of length 31
cites work: A new construction of group and nongroup perfect codes
cites work: On the reconstruction of perfect codes
cites work: Group Partition, Factorization and the Vector Covering Problem
cites work: Group Partitions and Mixed Perfect Codes
cites work: Resolving the Existence of Full-Rank Tilings of Binary Hamming Spaces
cites work: On perfect codes: Rank and kernel
cites work: Q4146667
cites work: Full-Rank Tilings of $\mathbbF^8_\!2$ Do Not Exist
DOI: 10.1016/J.DISC.2008.06.037
description: scientific article; zbMATH DE number 5644066