# Squares of Matrix-product Codes

## TL;DR

This paper investigates the properties of squares of matrix-product codes, relating them to their constituent codes, and determines bounds and parameters for their minimum distance, with applications in cryptography.

## Contribution

It provides new insights into the structure and parameters of squares of matrix-product codes, extending understanding of their minimum distance and code construction.

## Key findings

- Relates the square of matrix-product codes to constituent codes.
- Determines parameters for squares of specific matrix-product codes.
- Analyzes the minimum distance in Vandermonde matrix-product codes.

## Abstract

The component-wise or Schur product $C*C'$ of two linear error correcting codes $C$ and $C'$ over certain finite field is the linear code spanned by all component-wise products of a codeword in $C$ with a codeword in $C'$. When $C=C'$, we call the product the square of $C$ and denote it $C^{*2}$. Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrix-product codes, a general construction that allows to obtain new longer codes from several "constituent" codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known $(u,u+v)$-construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the squares of other matrix-product codes, for example when the defining matrix is Vandermonde (where the minimum distance is in a certain maximal with respect to matrix-product codes).