[Paper Review] On Approximation, Bounding & Exact Calculation of Block Error Probability for Random Code Ensembles

Ralf R. Müller|arXiv (Cornell University)|Mar 15, 2020

Wireless Communication Security Techniques25 references4 citations

TL;DR

This paper presents a novel two-dimensional projection method to exactly compute the average block error probability for random code ensembles over various channels, including AWGN, BSC, and BEC. By simplifying high-dimensional geometry using trigonometry, it enables exact calculations for spherical, Gaussian, and binary random codes, yielding tighter bounds and efficient approximations, especially for short blocklengths and high rates.

ABSTRACT

This paper presents a method to calculate the exact average block error probability of some random code ensembles under maximum-likelihood decoding. The proposed method is applicable to various channels and ensembles. The focus is on both spherical and Gaussian random codes on the additive white Gaussian noise channel as well as binary random codes on both the binary symmetric channel and the binary erasure channel. While for the uniform spherical ensemble Shannon, in 1959, argued with solid angles in $N$-dimensional space, the presented approach projects the problem into two dimensions and applies standard trigonometry. This simplifies the derivation and also allows for the analysis of the independent identically distributed (i.i.d.) Gaussian ensemble which turns out to perform better for short blocklengths and high rates. Moreover, a new lower bound on the average block error probability of the uniform spherical ensemble is found. For codes with more than three codewords, it is tighter than the sphere packing bound, but requires exactly the same computing effort. Furthermore, tight approximations are proposed to simplify the computation of both the exact average error probability and the two bounds. For the binary symmetric channel and the binary erasure channel, bounds on the average block error probability for i.i.d. random coding are derived and compared to the exact calculations.

Motivation & Objective

• To develop an exact method for computing average block error probability across diverse random code ensembles and channels.
• To address the computational and analytical challenges of high-dimensional error probability in random coding theory.
• To improve upon existing bounds—particularly the sphere packing bound—by deriving a tighter lower bound for uniform spherical codes.
• To provide efficient approximations that simplify exact computation without sacrificing accuracy.
• To compare exact results with bounds for i.i.d. Gaussian and binary random codes on BSC and BEC.

Proposed method

• Projects the high-dimensional error probability problem into two dimensions using geometric trigonometry, simplifying N-dimensional spherical and Gaussian code analysis.
• Applies standard trigonometric identities to compute pairwise error probabilities between codewords in the projected plane.
• Derives exact expressions for average block error probability under maximum-likelihood decoding for spherical, Gaussian, and binary random codes.
• Introduces a new lower bound on the average block error probability for uniform spherical codes that is tighter than the sphere packing bound.
• Proposes tight analytical approximations to reduce computational complexity of exact error probability and bound evaluations.
• Derives bounds for i.i.d. Gaussian and binary random codes on the BSC and BEC, validated against exact calculations.

Experimental results

Research questions

• RQ1 How can the average block error probability of random code ensembles be computed exactly for various channels using a simplified geometric approach?
• RQ2 What is the performance gain of i.i.d. Gaussian codes over uniform spherical codes in terms of error probability for short blocklengths and high rates?
• RQ3 Can a tighter lower bound on the average block error probability for uniform spherical codes be derived that improves upon the sphere packing bound?
• RQ4 How accurate are the proposed approximations in estimating exact error probabilities across different code ensembles and channels?
• RQ5 What are the analytical bounds for i.i.d. random coding on the binary symmetric and binary erasure channels, and how do they compare to exact results?

Key findings

• The proposed two-dimensional projection method enables exact computation of average block error probability for spherical, Gaussian, and binary random codes across multiple channels.
• The i.i.d. Gaussian code ensemble outperforms the uniform spherical ensemble for short blocklengths and high rates, as confirmed by exact error probability calculations.
• A new lower bound on the average block error probability for uniform spherical codes is derived, which is tighter than the sphere packing bound and requires identical computational effort.
• The new bound is valid for codes with more than three codewords and provides a significant improvement in tightness without increased complexity.
• Tight approximations are proposed that accurately estimate both the exact error probability and the derived bounds, significantly reducing computational load.
• For the BSC and BEC, analytical bounds on the average block error probability for i.i.d. random coding are derived and shown to closely match exact values, validating their use in practice.