On catastrophicity of convolutional codes and their encoders over pʳ

This paper investigates the existence of minimal p-encoders for convolutional codes over Zpʳ, where p is a prime. This addresses a conjecture from {k}, which posits that every such code admits a minimal p-encoder, implying that all convolutional codes over Zpʳ are noncatastrophic when input sequences are restricted to coefficients in \0, , 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 Zpʳ to be noncatastrophic in the usual sense (where input coefficients are from Zpʳ). Based on these findings, we affirm the conjecture by providing a constructive method for obtaining a minimal p-encoder for any convolutional code over Zpʳ.