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Applicable Algebra in Engineering, Communication and Computing
Published in: Applicable Algebra in Engineering, Communication and Computing 2/2020

27-07-2019 | Original Paper

Linear codes over \(\mathbb {F}_2 \times (\mathbb {F}_2+v\mathbb {F}_2)\) and the MacWilliams identities

Authors: Fatma Çalışkan, Refia Aksoy

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 2/2020

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Abstract

In this work, we study linear codes over the ring \(\mathbb {F}_2 \times (\mathbb {F}_2+v\mathbb {F}_2)\) and their weight enumerators, where \(v^2=v\). We first give the structure of the ring and investigate linear codes over this ring. We also define two weights called Lee weight and Gray weight for these codes. Then we introduce two Gray maps from \(\mathbb {F}_2 \times (\mathbb {F}_2+v\mathbb {F}_2)\) to \(\mathbb {F}_2^3\) and study the Gray images of linear codes over the ring. Moreover, we prove MacWilliams identities for the complete, the symmetrized and the Lee weight enumerators.
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Metadata
Title
Linear codes over and the MacWilliams identities
Authors
Fatma Çalışkan
Refia Aksoy
Publication date
27-07-2019
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 2/2020
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-019-00397-9

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