Restricted Composition Deletion Correcting Codes

Abstract

We investigate deletion correcting codes and restricted composition codes in particular. Restricted composition codes generalize codes on permutations and multipermutations. We use graph theoretic methods to characterize codes, establish bounds on code size, and describe constructions. The substring partial order has a well-known property. For any string, the number of superstrings of a particular length depends only on the length of the original string. We generalize this property to take compositions into account. For any string, the number of superstrings of a particular composition depends only on the composition of the original string. We present a bijective proof of this fact. We apply this result to obtain a lower bound on the size of restricted composition codes. We obtain an upper bound by analyzing deletion errors when the composition of deleted symbols is restricted to a particular worst case composition. We construct binary restricted composition single-deletion correcting codes and show that they are asymptotically optimal and form an optimal coloring. There is a natural distance on compositions that provides a lower bound on deletion distance. Unrestricted deletion correcting codes can be constructed from the union of restricted composition codes as long as the set of compositions used themselves form a code. The nonbinary single-deletion correcting codes constructed by Tenengolts are a special case of our method.