This article studies the cardinality of perfect multi deletion binary codes. The lower bound for any perfect deletion code with fixed code length and number of deletions, and the asymptotic achievability of Levenshtein's upper bound are shown.

This paper presents a formula for the cardinality of a class of BAD correcting codes proposed at ISIT2017. Furthermore, we show that the cardinality is approximately optimal. In other words, the ratio of the cardinality of the code and that of maximal cardinality BAD correcting code converges to 1 for sufficiently large length.