Frobenius Local Rings of Length 5 and Index of Nilpotency 3

This paper investigates finite local non-chain rings associated by the well-known invariants p,n,m,l, and k, where p is a prime number. In particular, we provide a comprehensive characterization of Frobenius local rings of length l=5 and index of nilpotency t=3, where t the index of nilpotency of the maximal ideal. The relevance of Frobenius rings is notable in coding theory, as it has been demonstrated that two classical results by MacWilliams—the Extension Theorem and the MacWilliams identities—are applicable not only to finite fields but also to finite Frobenius rings. We, therefore, classify and count Frobenius local rings of order p5m with t=3, outlining their properties in connection with various values of n.