Linear codes over F2 + μF2 + νF2 + μνF2

Bahattin Yildiz; Suat Karadeniz

doi:10.1007/s10623-009-9309-8

Linear codes over F₂ + μF₂ + νF₂ + μνF₂

Bahattin Yildiz, Suat Karadeniz

Research output: Contribution to journal › Article › peer-review

87 Scopus citations

Abstract

In this work, we investigate linear codes over the ring F₂ + μF₂ + νF₂ + μνF₂. We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43-65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32-45, 1999). We then characterize the F₂ + μF ₂ + νF₂ + μνF₂ -linearity of binary codes under the Gray map and give a main class of binary codes as an example of F₂ + μF₂ + νF₂ + μνF₂ -linear codes. The duals and the complete weight enumerators for Fdbl;₂ + μF₂ + νF₂ + μνF ₂ -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over F₂ + μF₂ + νF₂ + μνF₂ are obtained.

Original language	English (US)
Pages (from-to)	61-81
Number of pages	21
Journal	Designs, Codes, and Cryptography
Volume	54
Issue number	1
DOIs	https://doi.org/10.1007/s10623-009-9309-8
State	Published - Jan 2010
Externally published	Yes

Keywords

Complete weight enumerator
Gray maps
Ideal decompositions
Lee weights
Mac Williams identities
Reed-Muller codes

ASJC Scopus subject areas

Computer Science Applications
Applied Mathematics

Access to Document

10.1007/s10623-009-9309-8

Cite this

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title = "Linear codes over F2 + μF2 + νF2 + μνF2",

abstract = "In this work, we investigate linear codes over the ring F2 + μF2 + νF2 + μνF2. We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43-65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32-45, 1999). We then characterize the F2 + μF 2 + νF2 + μνF2 -linearity of binary codes under the Gray map and give a main class of binary codes as an example of F2 + μF2 + νF2 + μνF2 -linear codes. The duals and the complete weight enumerators for Fdbl;2 + μF2 + νF2 + μνF 2 -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over F2 + μF2 + νF2 + μνF2 are obtained.",

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AB - In this work, we investigate linear codes over the ring F2 + μF2 + νF2 + μνF2. We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43-65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32-45, 1999). We then characterize the F2 + μF 2 + νF2 + μνF2 -linearity of binary codes under the Gray map and give a main class of binary codes as an example of F2 + μF2 + νF2 + μνF2 -linear codes. The duals and the complete weight enumerators for Fdbl;2 + μF2 + νF2 + μνF 2 -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over F2 + μF2 + νF2 + μνF2 are obtained.

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KW - Reed-Muller codes

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