Cyclic Codes over a Non-Commutative Non-Unital Ring

Adel Alahmadi (), Malak Altaiary and Patrick Solé
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Adel Alahmadi: Research Group of Algebraic Structures and Applications, Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Malak Altaiary: Research Group of Algebraic Structures and Applications, Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Patrick Solé: I2M Lab, CNRS, Centrale Marseille, Aix Marseille University, 13009 Marseilles, France

Mathematics, 2024, vol. 12, issue 13, 1-17

Abstract: 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 ∣ 2 a = 2 b = 0 , a 2 = a , b 2 = b , a b = a , b a = 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 2 n over F 2 is studied. Motivated by DNA computing, conditions for reversibility and invariance under complementation are derived.

Keywords: non-unitary rings; cyclic codes; self-orthogonal codes; Gray map (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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