Structure and Enumeration of Constacyclic Codes over Cube-Zero Local Rings of Order q5

We investigate λ-constacyclic codes of length n over finite commutative local rings R of characteristic p and order q5, where q=pm is an odd prime power, whose Jacobson radical N satisfies N3=0≠N2, under the coprimality condition gcd(n,p)=1. In this setting, exactly two radical types occur, namely (3,1) and (2,2), determined by the dimensions of N/N2 and N2. For each type, we provide an explicit classification of the underlying rings and analyze the induced radical filtration of the ambient algebra Aλ=R[X]/⟨Xn−λ⟩. We prove that every λ-constacyclic code is uniquely determined by its residual component in Aλ/J(Aλ) together with two torsion components arising from the radical chain J(Aλ)⊃J(Aλ)2⊃0. This residual–torsion decomposition yields explicit generating sets; in particular, every λ-constacyclic code admits a generating set consisting of at most five elements. Furthermore, we derive exact enumeration formulas for all λ-constacyclic codes. In the type (2,2) case, the enumeration is governed by linear-algebraic constraints over the Chinese Remainder Theorem residue fields and, in the anisotropic class, depends on quadratic character values determined by the extension degrees. In the type (3,1) case, the enumeration is controlled by the dimension of the radical of the induced symmetric bilinear form on the top radical layer, equivalently by the rank class of the associated canonical matrix.