On the Undetected Error Probability of Binary Matrix Ensembles

In this paper, an ensemble analysis of the undetected error probability of a standard error detection scheme based on a sparse binary parity check matrix is presented. The
Bernoulli ensemble
, the members of which are considered to be matrices generated from an i.i.d. Bernoulli source, is primarily considered herein. The main contributions of the present study are (i) the derivation of the error exponent of the average undetected error probability and (ii) closed form expressions for the variance of the undetected error probability. The behavior of the exponent for an ensemble of sparse matrices is shown to be somewhat different from that for an ensemble of dense matrices. Furthermore, as a byproduct of the proof of the variance formula, a simple covariance formula of the weight distribution is derived.