A Class of Local Non-Chain Rings of Order p5m

This study investigates finite commutative local non-chain rings characterized by the well-established invariants p, n, m, l, and k, where p denotes a prime number. We specifically focus on Frobenius local rings with length l=5 and an index of nilpotency t=4. The significance of Frobenius rings in coding theory arises when specific results of linear codes are applicable to both finite fields and finite Frobenius rings. In light of this, we provide a comprehensive classification and enumeration of Frobenius local rings of order p5m with t=4, highlighting their distinctive properties in relation to varying values of n. This research advances our understanding of the structural features of Frobenius rings and their applications in coding theory.