# On squares of cyclic codes

## TL;DR

This paper investigates the properties of the square of cyclic codes, focusing on their minimum distance and dimension, with implications for cryptography and code construction.

## Contribution

It provides general results on the minimum distance of squared cyclic codes and introduces constructions for codes with optimal parameters.

## Key findings

- Derived bounds on the minimum distance of code squares
- Constructed cyclic codes with large dimension and high minimum distance of the square
- Identified conditions for codes where both code and square have optimal minimum distances

## Abstract

The square $C^{*2}$ of a linear error correcting code $C$ is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in $C$. Squares of codes have gained attention for several applications mainly in the area of cryptography, and typically in those applications one is concerned about some of the parameters (dimension, minimum distance) of both $C^{*2}$ and $C$. In this paper, motivated mostly by the study of this problem in the case of linear codes defined over the binary field, squares of cyclic codes are considered. General results on the minimum distance of the squares of cyclic codes are obtained and constructions of cyclic codes $C$ with relatively large dimension of $C$ and minimum distance of the square $C^{*2}$ are discussed. In some cases, the constructions lead to codes $C$ such that both $C$ and $C^{*2}$ simultaneously have the largest possible minimum distances for their length and dimensions.