### 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 |
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Journal | Finite Fields and Their Applications |

Volume | 62 |

ISSN | 1071-5797 |

DOIs | |

Publication status | Published - 2020 |

### Cite this

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**Squares of matrix-product codes.** / Gundersen, Jaron Skovsted; Ruano, Diego; Cascudo, Ignacio.

Research output: Contribution to journal › Journal article › Research › peer-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 -