Cyclic Codes over a Commutative Non-Unital Ring

This paper investigates cyclic codes over the commutative non-unital ring I=⟨a,b∣2a=2b=0,a2=b,ab=0⟩, a ring of order four and characteristic two. We develop a structural description of linear and cyclic codes over I, analyze their residue and torsion codes, and characterize self-dual codes. A Gray map from In to F22n is introduced and shown to transform cyclic codes into binary quasi-cyclic codes of index two. Numerical examples for lengths 2≤n≤7 illustrate the diversity of cyclic codes over I and highlight their potential applications.