Constacyclic codes over finite local Frobenius non-chain rings of length 5 and nilpotency index 4

The family of finite local Frobenius non-chain rings of length 5 and nilpotency index 4 is determined, as a by-product all finite local Frobenius non-chain rings with p⁵ elements (p a prime) and nilpotency index 4 are given. And the number and structure of γ-constacyclic codes over those rings, of length relatively prime to the characteristic of the residue field of the ring, are determined.