# Squares of matrix-product codes

Research output: Contribution to journalJournal articleResearchpeer-review

### 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 sense maximal with respect to matrix-product codes).

Original language Danish Finite Fields and Their Applications 62 1071-5797 https://doi.org/10.1016/j.ffa.2019.101606 Published - 2020

### Cite this

@article{bfd725254f2b48d3af29c12e988f1409,
title = "Squares of 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 sense maximal with respect to matrix-product codes).",
author = "Gundersen, {Jaron Skovsted} and Diego Ruano and Ignacio Cascudo",
year = "2020",
doi = "10.1016/j.ffa.2019.101606",
language = "Dansk",
volume = "62",
journal = "Finite Fields and Their Applications",
issn = "1071-5797",

}

In: Finite Fields and Their Applications, Vol. 62, 2020.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Squares of matrix-product codes

AU - Gundersen, Jaron Skovsted

AU - Ruano, Diego

AU - Cascudo, Ignacio

PY - 2020

Y1 - 2020

N2 - 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 sense maximal with respect to matrix-product codes).

AB - 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 sense maximal with respect to matrix-product codes).

U2 - 10.1016/j.ffa.2019.101606

DO - 10.1016/j.ffa.2019.101606

M3 - Tidsskriftartikel

VL - 62

JO - Finite Fields and Their Applications

JF - Finite Fields and Their Applications

SN - 1071-5797

ER -