On catastrophicity of convolutional codes and their encoders over \Z_p^r

2025/09/08 by Mohammed El Oued, Oued, Mohammed El
Computer Science · #Coding theory and cryptography #Cellular Automata and Applications #Error Correcting Code Techniques

paper · pdf · doi:10.48550/arxiv.2509.06670

Abstract

This paper investigates the existence of minimal p-encoders for convolutional codes over ℤ_p^r, where p is a prime. This addresses a conjecture from \citek, which posits that every such code admits a minimal p-encoder, implying that all convolutional codes over ℤ_p^r 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 ℤ_p^r to be noncatastrophic in the usual sense (where input coefficients are from 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 ℤ_p^r.

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