# Explicit Formulas for the Weight Enumerators of Some Classes of Deletion Correcting Codes

## TL;DR

This paper introduces a broad class of deletion/insertion correcting codes, providing explicit formulas for their weight enumerators and sizes, which unify and extend many known codes and have potential applications in various fields.

## Contribution

The paper presents a general code framework with explicit weight enumerator formulas, unifying several known codes and deriving new size formulas and solutions to linear congruences.

## Key findings

- Explicit formulas for weight enumerators of the new code class
- Unified formulas for sizes of multiple deletion/insertion codes
- First explicit formula for solutions to arbitrary linear congruences

## Abstract

We introduce a general class of codes which includes several well-known classes of deletion/insertion correcting codes as special cases. For example, the Helberg code, the Levenshtein code, the Varshamov--Tenengolts code, and most variants of these codes including most of those which have been recently used in studying DNA-based data storage systems are all special cases of our code. Then, using a number theoretic method, we give an explicit formula for the weight enumerator of our code which in turn gives explicit formulas for the weight enumerators and so the sizes of all the aforementioned codes. We also obtain the size of the Shifted Varshamov--Tenengolts code. Another application which automatically follows from our result is an explicit formula for the number of binary solutions of an arbitrary linear congruence which, to the best of our knowledge, is the first result of its kind in the literature and might be also of independent interest. Our general result might have more applications/implications in information theory, computer science, and mathematics.