On Undetected Error Probability of Binary Matrix Ensembles

TL;DR

This paper analyzes the undetected error probability of binary matrix ensembles, deriving error exponents and variance formulas, highlighting differences between sparse and dense ensembles, with implications for error detection performance.

Contribution

It introduces the first derivation of the error exponent and closed-form variance expressions for the undetected error probability in Bernoulli matrix ensembles.

Findings

Error exponent behavior differs between sparse and dense ensembles.

Closed-form expressions for variance of undetected error probability.

Covariance formula for weight distribution derived as a byproduct.

Abstract

In this paper, an analysis of the undetected error probability of ensembles of binary matrices is presented. The ensemble called the Bernoulli ensemble whose members are considered as matrices generated from i.i.d. Bernoulli source is mainly considered here. The main contributions of this work are (i) derivation of the error exponent of the average undetected error probability and (ii) closed form expressions for the variance of the undetected error probability. It is shown that the behavior of the exponent for a sparse ensemble is somewhat different from that for a dense ensemble. Furthermore, as a byproduct of the proof of the variance formula, simple covariance formula of the weight distribution is derived.