New Techniques for Upper-Bounding the ML Decoding Performance of Binary Linear Codes

TL;DR

This paper introduces new techniques to tighten upper bounds on the ML decoding error probability of binary linear codes over AWGN channels, improving existing bounds with minimal added complexity.

Contribution

The paper presents novel methods based on pair-wise and triplet-wise error probabilities to refine union bounds on decoding performance, utilizing geometric properties of bipolar vectors.

Findings

Improved upper bounds on ML decoding error probability.

Techniques applicable to bit-error probability analysis.

Bounds involve only the Q-function, maintaining low complexity.

Abstract

In this paper, new techniques are presented to either simplify or improve most existing upper bounds on the maximum-likelihood (ML) decoding performance of the binary linear codes over additive white Gaussian noise (AWGN) channels. Firstly, the recently proposed union bound using truncated weight spectrums by Ma {\em et al} is re-derived in a detailed way based on Gallager's first bounding technique (GFBT), where the "good region" is specified by a sub-optimal list decoding algorithm. The error probability caused by the bad region can be upper-bounded by the tail-probability of a binomial distribution, while the error probability caused by the good region can be upper-bounded by most existing techniques. Secondly, we propose two techniques to tighten the union bound on the error probability caused by the good region. The first technique is based on pair-wise error probabilities, which can be further tightened by employing the independence between the error events and certain components of the received random vectors. The second technique is based on triplet-wise error probabilities, which can be upper-bounded by proving that any three bipolar vectors form a non-obtuse triangle. The proposed bounds improve the conventional union bounds but have a similar complexity since they involve only the $Q$-function. The proposed bounds can also be adapted to bit-error probabilities.