Linear codes over \({\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}\)
type: Item
instance of: scholarly article
title: Linear codes over \({\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}\)
zbMATH Open document ID: 1187.94045
author: Bahattin Yildiz
author: Suat Karadeniz
published in: Designs, Codes and Cryptography
publication date: 2010-02-25T00:00:00Z
review text: The authors investigate codes over the ring \(F_2+uF_2 + vF_2 + uv F_2\). These codes are a generalization of codes over \(F_2+uF_2\) and like these codes are equipped with a linear Gray map to binary codes. Lee weights are connected to Hamming weights in the image under this map. Ideals are described in the ring and are used to describe the type of a code and to define independence of generators. MacWilliams relations are given for the complete weight enumerator. Additionally, it is proven that the binary Reed-Muller codes \(RM(r,m)\) are all \(F_2+uF_2 + vF_2 + uv F_2\) linear for \(0 \leq r \leq m.\)
reviewed by: Steven T. Dougherty
Mathematics Subject Classification ID: 94B99
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 13M99
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 5675124
zbMATH Keywords: Lee weights
zbMATH Keywords: Gray maps
zbMATH Keywords: Reed-Muller codes
zbMATH Keywords: complete weight enumerator
zbMATH Keywords: MacWilliams identities
zbMATH Keywords: ideal decompositions
MaRDI profile type: Publication
full work available at URL: s10623-009-9309-8
OpenAlex ID: W2031685689
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DOI: 10.1007/S10623-009-9309-8
description: scientific article; zbMATH DE number 5675124