Cyclic Codes over a Non-Local Non-Unital Ring
Résumé
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.
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