Analysis of Sequential Decoding Complexity Using the Berry-Esseen Inequality

TL;DR

This paper introduces a new method using the Berry-Esseen inequality to accurately estimate the computational complexity of sequential decoding algorithms across all blocklengths, improving upon traditional bounds.

Contribution

The study applies the Berry-Esseen theorem to derive bounds for sequential decoding complexity valid for any blocklength, with empirical validation on specific algorithms and SNR conditions.

Findings

The new bounds closely match simulation results at high SNR for simplified GDA.

For MLSDA, the theoretical bounds are near the simulation results across various SNR levels.

The bounds are less accurate at low SNR for simplified GDA, but remain close for MLSDA.

Abstract

his study presents a novel technique to estimate the computational complexity of sequential decoding using the Berry-Esseen theorem. Unlike the theoretical bounds determined by the conventional central limit theorem argument, which often holds only for sufficiently large codeword length, the new bound obtained from the Berry-Esseen theorem is valid for any blocklength. The accuracy of the new bound is then examined for two sequential decoding algorithms, an ordering-free variant of the generalized Dijkstra's algorithm (GDA)(or simplified GDA) and the maximum-likelihood sequential decoding algorithm (MLSDA). Empirically investigating codes of small blocklength reveals that the theoretical upper bound for the simplified GDA almost matches the simulation results as the signal-to-noise ratio (SNR) per information bit ($\gamma_b$) is greater than or equal to 8 dB. However, the theoretical bound may become markedly higher than the simulated average complexity when $\gamma_b$ is small. For the MLSDA, the theoretical upper bound is quite close to the simulation results for both high SNR ($\gamma_b\geq 6$ dB) and low SNR ($\gamma_b\leq 2$ dB). Even for moderate SNR, the simulation results and the theoretical bound differ by at most \makeblue{0.8} on a $\log_{10}$ scale.