Deletion correcting codes meet the Littlewood–Offord problem

Abstract

In this paper, we make a novel connection between information theory and additive combinatorics; more specifically, between deletion/insertion correcting codes and the celebrated Littlewood–Offord problem. We see how results from one area can have an impact on the other area and vice versa. In particular, a result on the Littlewood–Offord problem gives a nice upper bound for the size of the Levenshtein code and the Helberg code (and possibly other variants of these codes). Also, a recent result on the deletion correcting codes gives a modular analogue of the Littlewood–Offord problem which generalizes the results of Vaughan and Wooley (Q J Math Oxf Ser (2) 42(2):379–386, 1991) (obtained using tools from analytic number theory and properties of exponential sums) and of Griggs (Bull Am Math Soc (N.S.) 28:329–333, 1993) (obtained using a combinatorial argument). This novel connection might opens up new doors to research in these or other related areas.