Cyclic Codes over a Non-Local Non-Unital Ring

Adel Alahmadi (1) , Malak Altaiary (1) , Patrick Solé (2)

1 KAU - King Abdul Aziz University
2 I2M - Institut de Mathématiques de Marseille

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.

Mots clés: non-unitary rings; cyclic codes; self-orthogonal codes

Domaines: Théorie de l'information et codage [math.IT]

Adel Alahmadi, Malak Altaiary, Patrick Solé. Cyclic Codes over a Non-Local Non-Unital Ring. Mathematics, 2024, ⟨10.3390/math12060866⟩. ⟨hal-04506243⟩