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Algorithms for Computing the Free Distance of Convolutional Codes

Zita Abreu, Joachim Rosenthal, Michael Schaller

TL;DR

This work tackles the challenging problem of computing the free distance $d_{free}$ of convolutional codes, a key performance metric for error correction. It systematically analyzes existing approaches, demonstrates the incorrectness of the Heapmod approach, and proposes both a generalized, optimized FAST algorithm and a new bidirectional method that combines FAST with Larsen’s ideas to handle codes of all rates, degrees, and over any finite field. The contributions include rigorous comparisons, practical algorithmic improvements, and an open-source implementation, expanding the toolkit for reliable distance computation in convolutional codes. The results have practical impact for designing codes with robust error protection and for understanding the trade-offs in distance computation across different code configurations.

Abstract

The free distance of a convolutional code is a reliable indicator of its performance. However its computation is not an easy task. In this paper, we present some algorithms to compute the free distance with good efficiency that work for convolutional codes of all rates and over any field. Furthermore we discuss why an algorithm which is claimed to be very efficient is incorrect.

Algorithms for Computing the Free Distance of Convolutional Codes

TL;DR

This work tackles the challenging problem of computing the free distance of convolutional codes, a key performance metric for error correction. It systematically analyzes existing approaches, demonstrates the incorrectness of the Heapmod approach, and proposes both a generalized, optimized FAST algorithm and a new bidirectional method that combines FAST with Larsen’s ideas to handle codes of all rates, degrees, and over any finite field. The contributions include rigorous comparisons, practical algorithmic improvements, and an open-source implementation, expanding the toolkit for reliable distance computation in convolutional codes. The results have practical impact for designing codes with robust error protection and for understanding the trade-offs in distance computation across different code configurations.

Abstract

The free distance of a convolutional code is a reliable indicator of its performance. However its computation is not an easy task. In this paper, we present some algorithms to compute the free distance with good efficiency that work for convolutional codes of all rates and over any field. Furthermore we discuss why an algorithm which is claimed to be very efficient is incorrect.
Paper Structure (8 sections, 1 theorem, 6 equations, 8 figures, 3 algorithms)

This paper contains 8 sections, 1 theorem, 6 equations, 8 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Naive computation of the Free Distance
  4. The Heapmod algorithm is wrong
  5. The optimized FAST algorithm
  6. A new algorithm
  7. Conclusion and Future Work
  8. Comments on Larsen's Algorithm

Key Result

Theorem 1

Let $\mathcal{C}$ be an $(n,k,\delta)$ convolutional code. Then, $d_{free}(\mathcal{C})\leq(n-k)\left(\left\lfloor \frac{\delta}{k}\right\rfloor+1\right)+\delta+1.$ This bound is called the generalized Singleton bound.

Figures (8)

  • Figure 1: Naive computation of the free distance for a $(1,2,2)$ convolutional code with $G(z)=1 + z^2z^2 \in \mathbb{F}_{2}[z]^{1 \times 2}$.
  • Figure 2: Illustration of the Heapmod algorithm.
  • Figure 3: Idea behind the (optimized) Fast algorithm for a convolutional code of rate $1/n$ over $\mathbb{F}_2$.
  • Figure 4: Efficiencies of the FAST algorithm and the optimized FAST algorithm.
  • Figure 5: Another perspective of Figure \ref{['graphic1']}.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 1: $\space$ rosenthal1999maximum
  • Example 1
  • Example 2