On catastrophicity of convolutional codes and their encoders over ${\mathbb{Z}}_{p^r}$

This paper investigates the existence of minimal $p$-encoders for convolutional codes over ${\mathbb{Z}}_{p^r}$, where $p$ is a prime. This addresses a conjecture from \cite{k}, which posits that every such code admits a minimal $p$-encoder, implying that all convolutional codes over ${\mathbb{Z}}_{p^r}$ are noncatastrophic when input sequences are restricted to coefficients in $\{0,\dots ,p-1\}$. Our contributions include the introduction of a new polynomial invariant that characterizes free codes, which enables us to establish a necessary and sufficient condition for a free code over ${\mathbb{Z}}_{p^r}$ to be noncatastrophic in the usual sense (where input coefficients are from ${\mathbb{Z}}_{p^r}$). Based on these findings, we affirm the conjecture by providing a constructive method for obtaining a minimal $p$-encoder for any convolutional code over ${\mathbb{Z}}_{p^r}$.