The Modular Subset-Sum Problem and the size of deletion correcting codes

In this paper, using some results on the deletion correcting codes, we give an equivalent form of the Modular Subset-Sum Problem which is of significant importance in computer science and (quantum) cryptography. We also, using Ramanujan sums and their properties, give an explicit formula for the size of the Levenshtein code which has found many interesting applications, for examples, in studying DNA-based data storage and distributed message synchronization.