Construction and Decoding of Convolutional Codes with optimal Column Distances

2026-01-28

AI Summary

This work proposes a novel method for constructing convolutional codes with optimal column distance over arbitrary finite fields, addressing the limitation of traditional maximum distance profile (MDP) convolutional codes that require large field sizes. By establishing a structural connection between the constructed codes and first-order Reed–Muller block codes, the study not only proves for the first time that such codes uniquely achieve the optimal column distance under given parameters but also leverages this insight to design a low-complexity Viterbi decoding algorithm. The approach significantly relaxes the constraint on field size while enabling efficient decoding, thereby enhancing practical applicability without compromising optimal error-correction performance.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The construction of Maximum Distance Profile (MDP) convolutional codes in general requires the use of very large finite fields. In contrast convolutional codes with optimal column distances maximize the column distances for a given arbitrary finite field. In this paper, we present a construction of such convolutional codes. In addition, we prove that for the considered parameters the codes that we constructed are the only ones achieving optimal column distances. The structure of the presented convolutional codes with optimal column distances is strongly related to first order Reed-Muller block codes and we leverage this fact to develop a reduced complexity version of the Viterbi algorithm for these codes.

Problem

• convolutional codes
• optimal column distances
• finite fields
• code construction
• uniqueness

Innovation

• convolutional codes
• optimal column distances
• Maximum Distance Profile
• Reed-Muller codes
• reduced-complexity Viterbi decoding