Binomial moments of the distance distribution and the probability of undetected error

In [1] K. A. S. Abdel-Ghaffar derives a lower bound on the probability of undetected error for unrestricted codes. The proof relies implicitly on the binomial moments of the distance distribution of the code. We use the fact that these moments count the size of subcodes of the code to give a very simple proof of the bound in [1] by showing that it is essentially equivalent to the Singleton bound. This proof reveals connections of the probability of undetected error to the rank generating function of the code and to related polynomials (Whitney function, Tutte polynomial, and higher weight enumerators). We also discuss some improvements of this bound. Finally, we analyze asymptotics. We show that an upper bound on the undetected error exponent that corresponds to the bound of [1] improves known bounds on this function.