Decoding convolutional codes over finite rings. A linear dynamical systems approach

TL;DR

This work extends Rosenthal's convolutional-code decoding from finite fields to the finite ring $\mathbb{Z}_{p^r}$ by leveraging minimal input/state/output (I/S/O) representations with the Predictable Degree Property (PDP). It shows how the decoding problem can be reduced to two linear block codes defined over $\mathbb{Z}_{p^r}$ via $\mathrm{Ker}(\Phi_T(\Sigma))$ and $\mathrm{Im}(\Psi_{\Theta}(\Sigma))$, and then constructs a practical decoder by combining Rosenthal's framework with the Greferath-Vellbinger (GV) and a modified Torrecillas-Lobillo-Navarro (TLN) algorithm that operate on $p$-adic expansions. The resulting composite decoder decodes convolutional codes over $\mathbb{Z}_{p^r}$ by exploiting $p$-adic coefficients and block-code decoders, with performance governed by the minimum distance modulo $p$ and the free-distance of the corresponding codes. This advances reliable decoding for ring-based convolutional codes and suggests future extensions to erasure channels in the ring setting.

Abstract

Observable convolutional codes defined over Zpr with the Predictable Degree Property admits minimal input state output representations that behaves well under restriction of scalars. We make use of this fact to present Rosenthal's decoding algorithm for these convolutional codes. When combined with the Greferath-Vellbinger algorithm and a modified version of the Torrecillas-Lobillo-Navarro algorithm, the decoding problem reduces to selecting two decoding algorithms for linear block codes over a field. Finally, we analyze both the theoretical and practical error-correction capabilities of the combined algorithm,

Theorems & Definitions (7)

• Remark 2.4
• Lemma 3.2
• proof
• Theorem 3.8
• proof
• Corollary 3.10
• proof