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Construction and Decoding of Convolutional Codes with optimal Column Distances

Julia Lieb, Michael Schaller

TL;DR

The paper addresses constructing convolutional codes with optimal column distances over a fixed finite field, a task challenging for general MDP codes that require very large fields. It develops Construction 1 based on MacDonald and first-order Reed-Muller codes to achieve optimal column distances for selected parameter sets and proves a converse: such optimality characterizes the construction up to a right-monomial transform. Leveraging the first-order RM/MacDonald structure, it introduces an efficient, reduced-complexity Viterbi decoding scheme, and extends the approach to two alternative constructions (Constructions 2 and 3) for different rate regimes. The resulting framework yields significant decoding-efficiency gains (complexity $O(N q^2 n \log_q(n))$) while preserving strong distance properties, with practical impact for streaming data transmission over moderate finite fields.

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.

Construction and Decoding of Convolutional Codes with optimal Column Distances

TL;DR

The paper addresses constructing convolutional codes with optimal column distances over a fixed finite field, a task challenging for general MDP codes that require very large fields. It develops Construction 1 based on MacDonald and first-order Reed-Muller codes to achieve optimal column distances for selected parameter sets and proves a converse: such optimality characterizes the construction up to a right-monomial transform. Leveraging the first-order RM/MacDonald structure, it introduces an efficient, reduced-complexity Viterbi decoding scheme, and extends the approach to two alternative constructions (Constructions 2 and 3) for different rate regimes. The resulting framework yields significant decoding-efficiency gains (complexity ) while preserving strong distance properties, with practical impact for streaming data transmission over moderate finite fields.

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.
Paper Structure (13 sections, 21 theorems, 140 equations, 1 figure, 1 algorithm)

This paper contains 13 sections, 21 theorems, 140 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear block codes
  4. Convolutional codes
  5. The Viterbi algorithm for convolutional codes
  6. Construction of convolutional codes with optimal column distances
  7. Efficient decoding of MacDonald codes
  8. Efficient decoding of first order Reed Muller Codes
  9. Comparing a received word to all codewords of a MacDonald code
  10. Comparing a received word to all codewords of certain subcodes of MacDonald codes
  11. Improved Viterbi algorithm for a class of optimal convolutional codes
  12. Alternative Constructions with good column distances and efficient decoding algorithm
  13. Conclusion

Key Result

Theorem 2.3

van1998introduction Let $C\subset\mathbb F_q^n$ be a linear block code with $d:=d_{min}(C)<n(1-\frac{1}{q})$. Then,

Figures (1)

  • Figure 1: Trellis for the code of Example \ref{['example:viterbi']} with the sent codeword highlighted

Theorems & Definitions (56)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Definition 2.8
  • ...and 46 more