This paper investigates the algebraic structure of free convolutional codes over the finite local ring Z_{p^r}. We introduce a new structural invariant, the Residual Structural Polynomial, denoted by Delta_p(C) in F_p[D]. We construct this invariant via encoders which are reduced internal degree matrices (RIDM). We formally demonstrate that Delta_p(C) is an intrinsic characteristic of the code, invariant under equivalent RIDMs. A central result of this work is the establishment that Delta_p(C) serves as an algebraic criterion for intrinsic catastrophicity: we prove that a free code C admits a non-catastrophic realization if and only if Delta_p(C) is a monomial of the form D^s. Furthermore, we establish a fundamental duality theorem, proving that Delta_p(C) = Delta_p(C^perp). This result reveals a deep structural symmetry, showing that the "catastrophicity" of a free code is preserved under orthogonality.

翻译：本文研究了有限局部环 Z_{p^r} 上自由卷积码的代数结构。我们引入了一种新的结构不变量——残差结构多项式，记作 F_p[D] 中的 Delta_p(C)。我们通过约化内度矩阵（RIDM）编码器来构造该不变量。我们正式证明了 Delta_p(C) 是码的一个内在特征，在等价 RIDM 下保持不变。本工作的一个核心结果是确立了 Delta_p(C) 可作为内在灾难性的代数判据：我们证明了一个自由码 C 存在非灾难性实现当且仅当 Delta_p(C) 是形如 D^s 的单式。此外，我们建立了一个基本对偶定理，证明了 Delta_p(C) = Delta_p(C^perp)。这一结果揭示了一种深刻的结构对称性，表明自由码的“灾难性”在正交性下得以保持。