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The Optimization of Random Tree Codes for Limited Computational Resources

B. Tan Bacinoglu

TL;DR

This work tackles the problem of reliable communication under limited decoding resources by deriving an achievability bound for irregular random tree codes decoded with a hard computational limit. It models the code as an $(n,k)$-random tree with a branching structure and introduces the SSDGU decoder under an accumulating error cost, decomposing the error bound into computation-limit (CLE) and computation-free (CFE) terms. By optimizing the tree structure with the SBP algorithm, the authors define computationally optimized random tree (CORT) codes that can approach ML performance of pure random codes as the computational budget grows, connecting to classic bounds such as the RCU and Gallager bounds in appropriate limits. The results indicate that practical, SBP-guided code design can yield near-optimal performance under realistic decoding constraints, with considerations for memory and potential extensions to more general decoding measures.

Abstract

In this paper, we introduce an achievability bound on the frame error rate of random tree code ensembles under a sequential decoding algorithm with a hard computational limit and consider the optimization of the random tree code ensembles over their branching structures/profiles and the decoding measure. Through numerical examples, we show that the achievability bound for the optimizated random tree codes can approach the maximum likelihood (ML) decoding performance of pure random codes.

The Optimization of Random Tree Codes for Limited Computational Resources

TL;DR

This work tackles the problem of reliable communication under limited decoding resources by deriving an achievability bound for irregular random tree codes decoded with a hard computational limit. It models the code as an -random tree with a branching structure and introduces the SSDGU decoder under an accumulating error cost, decomposing the error bound into computation-limit (CLE) and computation-free (CFE) terms. By optimizing the tree structure with the SBP algorithm, the authors define computationally optimized random tree (CORT) codes that can approach ML performance of pure random codes as the computational budget grows, connecting to classic bounds such as the RCU and Gallager bounds in appropriate limits. The results indicate that practical, SBP-guided code design can yield near-optimal performance under realistic decoding constraints, with considerations for memory and potential extensions to more general decoding measures.

Abstract

In this paper, we introduce an achievability bound on the frame error rate of random tree code ensembles under a sequential decoding algorithm with a hard computational limit and consider the optimization of the random tree code ensembles over their branching structures/profiles and the decoding measure. Through numerical examples, we show that the achievability bound for the optimizated random tree codes can approach the maximum likelihood (ML) decoding performance of pure random codes.
Paper Structure (7 sections, 6 theorems, 61 equations, 4 tables, 1 algorithm)

This paper contains 7 sections, 6 theorems, 61 equations, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Descriptions
  3. Encoding
  4. Decoding
  5. Main Results
  6. Numerical Simulations
  7. Conclusion

Key Result

Theorem 1

If the SSDGU algorithm with an AEC measure $d(\cdot,\cdot)$ and a computational limit paramater $L$ is used to decode an independent random message $\mathbf{m}$ that is uniformly selected from $\lbrace 0,1\rbrace^{k}$ from the output $\mathbf{y}$ of noisy channel that takes the coded bits $\mathbf{x where $D_{\text{CLE}}[s,d]$ is the computation-limit-error (CLE) bound which is expressed as , whe

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 5 more