Cyclic Codes over a Non-Commutative Non-Unital Ring
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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