On the Straight-Line Bound for the Undetected Error Exponent

We derive a one-parametric family of lower bounds for the undetected error probability of a code in a binary symmetric channel. With an optimally chosen parameter, these bounds lead to the so-called straight-line bound for the undetected error exponent. The straight-line bound is asymptotically exact if the Varshamov-Gilbert bound for the distance of binary codes is asymptotically exact.