A lower bound on the undetected error probability and strictly optimal codes

Error detection is a simple technique used in various communication and memory systems to enhance reliability. We study the probability that a q-ary (linear or nonlinear) block code of length n and size M fails to detect an error. A lower bound on this undetected error probability is derived in terms of q, n, and M. The new bound improves upon other bounds mentioned in the literature, even those that hold only for linear codes. Block codes whose undetected error probability equals the new lower bound are investigated. We call these codes strictly optimal codes and give a combinatorial characterization of them. We also present necessary and sufficient conditions for their existence. In particular, we find all values of n and M for which strictly optimal binary codes exist, and determine the structure of all of them. For example, we construct strictly optimal binary-coded decimal codes of length four and five, and we show that these are the only possible lengths of such codes