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An Improved Viterbi Algorithm for a Class of Optimal Binary Convolutional Codes

Zita Abreu, Julia Lieb, Michael Schaller

TL;DR

This work tackles decoding of binary convolutional codes with optimal column distances by introducing a reduced-complexity decoder for the class of $k$-partial simplex convolutional codes, built from partial simplex ( Reed–Muller related) codes. The authors integrate fast Hadamard-based decoding of Reed-Muller codes with the Viterbi framework, leveraging permutations and structured trellis navigation to compute distances efficiently. They define and analyze $k$-partial simplex codes and their associated convolutional codes, showing that per-timestep complexity can be reduced to $O(n\log n)$, leading to a total complexity of $O(N\cdot n\log n)$, substantially better than the classical Viterbi scaling. The approach offers practical decoding gains for codes with optimal column distances and sets the stage for extending Hadamard-based acceleration to other code families and for improving sequential decoding methods.

Abstract

The most famous error-decoding algorithm for convolutional codes is the Viterbi algorithm. In this paper, we present a new reduced complexity version of this algorithm which can be applied to a class of binary convolutional codes with optimum column distances called k-partial simplex convolutional codes.

An Improved Viterbi Algorithm for a Class of Optimal Binary Convolutional Codes

TL;DR

This work tackles decoding of binary convolutional codes with optimal column distances by introducing a reduced-complexity decoder for the class of -partial simplex convolutional codes, built from partial simplex ( Reed–Muller related) codes. The authors integrate fast Hadamard-based decoding of Reed-Muller codes with the Viterbi framework, leveraging permutations and structured trellis navigation to compute distances efficiently. They define and analyze -partial simplex codes and their associated convolutional codes, showing that per-timestep complexity can be reduced to , leading to a total complexity of , substantially better than the classical Viterbi scaling. The approach offers practical decoding gains for codes with optimal column distances and sets the stage for extending Hadamard-based acceleration to other code families and for improving sequential decoding methods.

Abstract

The most famous error-decoding algorithm for convolutional codes is the Viterbi algorithm. In this paper, we present a new reduced complexity version of this algorithm which can be applied to a class of binary convolutional codes with optimum column distances called k-partial simplex convolutional codes.
Paper Structure (8 sections, 6 theorems, 23 equations, 1 figure, 2 algorithms)

This paper contains 8 sections, 6 theorems, 23 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Convolutional Codes
  3. Viterbi decoding for Convolutional Codes
  4. Decoding of Reed-Muller Codes
  5. Decoding of Partial simplex Codes
  6. Improved Viterbi algorithm for a class of optimal convolutional codes
  7. Complexity Analysis
  8. Conclusion

Key Result

Theorem 1

Let $\mathcal{C}$ be a $(2^{\delta}(2^k-1),k,\delta)$ convolutional code with generator matrix $G(z)=\sum_{i=0}^{\lceil\frac{\delta}{k}\rceil}G_iz^i\in\mathbb F_2[z]^{k\times 2^{\delta}(2^k-1)}$ where $(G_0^\top\ \cdots\ \ G_{\mu-1}^\top \ \tilde{G}_{\mu}^\top)^\top=S(\delta+k)_k$. $\mathcal{C}$ is and these column distances are optimal in the sense of Definition defopt. Moreover,

Figures (1)

  • Figure 1: Trellis for Algorithm \ref{['Viterbi']} applied to Example \ref{['exviterbi']}.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6: isit23
  • Theorem 1: isit23
  • Example 1
  • Theorem 2
  • proof
  • ...and 7 more