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Article Dans Une Revue Mathematics Année : 2024

Cyclic Codes over a Non-Commutative Non-Unital Ring

Adel Alahmadi
Patrick Solé

Résumé

: In this paper, we investigate cyclic codes over the ring E of order 4 and characteristic 2 defined by generators and relations as E = ⟨a, b | 2a = 2b = 0, a 2 = a, b 2 = b, ab = a, ba = b⟩. This is the first time that cyclic codes over the ring E are studied. Each cyclic code of length n over E is identified uniquely by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, left self-dual, right self-dual, and linear complementary dual (LCD) cyclic codes over E. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over E and quasi-cyclic codes of length 2n over F2 is studied. Motivated by DNA computing, conditions for reversibility and invariance under complementation are derived.
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hal-04629314 , version 1 (29-06-2024)

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Adel Alahmadi, Malak Altaiary, Patrick Solé. Cyclic Codes over a Non-Commutative Non-Unital Ring. Mathematics, 2024, 12 (13), pp.2014. ⟨10.3390/math12132014⟩. ⟨hal-04629314⟩
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