# Applications of Gaussian Binomials to Coding Theory for Deletion Error Correction

## TL;DR

This paper explores the use of Gaussian binomials in coding theory, specifically for deletion error correction, by establishing new theorems on code cardinalities and deletion spheres.

## Contribution

It introduces novel applications of q-binomial coefficients to analyze and determine properties of deletion-correcting codes, expanding theoretical understanding.

## Key findings

- Cardinalities of certain error-correcting codes are determined.
- A curious phenomenon related to deletion spheres is proved.
- New theorems connect Gaussian binomials with deletion error correction.

## Abstract

We present new applications on $q$-binomials, also known as Gaussian binomial coefficients. Our main theorems determine cardinalities of certain error-correcting codes based on Varshamov-Tenengolts codes and prove a curious phenomenon relating to deletion sphere for specific cases.