Cyclic Codes over a Non-Local Non-Unital Ring

Author:

Alahmadi Adel1ORCID,Altaiary Malak12,Solé Patrick3ORCID

Affiliation:

1. Research Group of Algebraic Structures and Applications, Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

2. Math Department, Shaqra University, Shaqra 11961, Saudi Arabia

3. I2M Lab (CNRS, Aix Marseille University, Centrale Marseille), 13009 Marseilles, France

Abstract

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H=⟨a,b∣2a=2b=0,a2=0,b2=b,ab=ba=0⟩. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2n over F2 is studied.

Funder

The Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia

Publisher

MDPI AG

Reference25 articles.

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5. Cyclic Codes over Z4 of Even Length;Dougherty;Des. Codes Cryptogr.,2006

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