Perfect codes from the dual point of view. I
type: Item
instance of: scholarly article
title: Perfect codes from the dual point of view. I
zbMATH Open document ID: 1156.94017
author: Olof Heden
published in: Discrete Mathematics
publication date: 2009-01-28T00:00:00Z
review text: In this paper \(1\)-error correcting binary perfect codes are considered. The author generalizes the concepts of dual code and parity check matrix for non-linear cases and shows how to use this generalization for the enumeration and classification of some particular classes of perfect \(1\)-error correcting binary codes. A sample of statements proved: \begin{itemize} \item[(i)] There is exactly one equivalence class of perfect codes of length \(31\), rank \(30\) and with dimension of kernel \(23\). \item[(ii)] There are exactly three equivalence classes of perfect codes of length \(15\), rank \(12\) and with dimension of kernel \(9\). \item[(iii)] There exists a perfect code \(C\) such that there is no Hamming code \(H\) with \[ \ker(C)\subseteq H \subseteq(C)^{\perp}. \] \end{itemize}
reviewed by: Piroska Lakatos
Mathematics Subject Classification ID: 94B25
Mathematics Subject Classification ID: 94B65
zbMATH DE Number: 5499755
zbMATH Keywords: perfect codes
MaRDI profile type: Publication
full work available at URL: j.disc.2007.11.037
OpenAlex ID: W1999220767
cites work: On the ranks and kernels problem for perfect codes
cites work: Q5187201
cites work: On Perfect Codes and Tilings: Problems and Solutions
cites work: Error Detecting and Error Correcting Codes
cites work: A generalized Lloyd theorem and mixed perfect codes.
cites work: On the reconstruction of perfect codes
cites work: A full rank perfect code of length 31
cites work: On the classification of perfect codes: side class structures
cites work: Q5709000
cites work: Q3701331
cites work: Codes of Steiner triple and quadruple systems
cites work: Q3148508
cites work: Q4081126
cites work: A General Product Construction for Error Correcting Codes
cites work: Q4488580
cites work: On perfect codes: Rank and kernel
cites work: On the Nonexistence of Perfect Codes over Finite Fields
DOI: 10.1016/J.DISC.2007.11.037
description: scientific article; zbMATH DE number 5499755