Spectrum of Sizes for Perfect Deletion-Correcting Codes

One peculiarity with deletion-correcting codes is that perfect t-deletion-correcting codes of the same length over the same alphabet can have different numbers of codewords, because the balls of radius t with respect to the Levenshtĕın distance may be of different sizes. There is interest, therefore, in determining all possible sizes of a perfect t-deletion-correcting code, given the length n and the alphabet size q. In this paper, we determine completely the spectrum of possible sizes for perfect q-ary 1-deletion-correcting codes of length three for all q, and perfect q-ary 2-deletion-correcting codes of length four for almost all q, leaving only a small finite number of cases in doubt.