Exact and Approximate Expressions for the Probability of Undetected Errors of Varshamov–Tenengol´ts Codes

Computation of the undetected error probability for error detecting codes over the Z-channel is an important issue, explored only in part in previous literature. In this paper, Varshamov-Tenengol´ts (VT) codes are considered. First, an exact formula for the probability of undetected errors is given. It can be explicitly computed for small code lengths (up to approximately 25). Next, some lower bounds that can be explicitly computed up to almost twice this length are studied. A comparison to the Hamming codes is given. It is further shown that heuristic arguments give a very good approximation that can easily be computed even for large lengths. Finally, Monte Carlo methods are used to estimate performance for long code lengths.