On the Duality of Codes over Non-Unital Commutative Ring of Order p²

This paper establishes an extended theoretical framework centered on the duality of codes constructed over a special class of non-unital, commutative, local rings of order $p^2$, where p is a prime satisfying $p\equiv 1mod4$ or $p\equiv 3mod4$. The work expands the traditional scope of coding theory by developing and adapting a generalized recursive approach to produce quasi-self-dual and self-dual codes within this algebraic setting. While the method for code generation is rooted in the classical build-up technique, the primary focus is on the duality properties of the resulting codes—especially how these properties manifest under different congruence conditions on p. Computational examples are provided to illustrate the effectiveness of the proposed methods.