The undetected error probability threshold of m-out-of-n codes

The well-known m-out-of-n code Ω_n^m consists of all binary vectors of length n and weight m. It is known that it is good for error detection (in the technical sense, that is, the probability of undetected error P_ud(Ω_n ^m,p)⩽P_ud(Ω_n^m,1/2) for all p, 0⩽p⩽1/2) only for a few small values of m and n. It is therefore of interest to determine (bounds for) the threshold in general, that is, find the range of bit-error probabilities p for which P_ud (Ω_n^m,p)⩽P_ud (Ω_n^m,1/2). In this article such bounds are given