Hadamard product of linear codes: algebraic properties and computation algorithms

Recieved: 07/03/2023

Accepted: 07/20/2023

Published: 11/28/2023

Keywords: Hadamard product, Shur product, Component-wise product, McEliece public-key cryptosystem, algorithm, Hadamard quotient, Hadamard quasi quotient, Hadamard maximum quasi quotient

To cite this article

Chizhov I.V. Hadamard product of linear codes: algebraic properties and computation algorithms. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2023. N 4, p.61-73 https://doi.org/ 10.55959/MSU/0137–0782–15–2023–47–4–61–73.

This work is licensed under a Сreative Commons Atribiution - NonCommercial 4.0 International (CC BY-NC 4.0)

Chizhov I.V. (Lomonosov Moscow State University (MSU))

Abstract

The paper considers the algebraic properties of the Hadamard product (Shur product, componentwise product) of error-correcting linear codes. We discuss the question of the complexity of constructing the basis of Hadamard’s product by known bases of factors. Also, we introduce the concept of quotients, quasi quotients, and maximal (concerning inclusion) quasi quotient from the Hadamard division of one linear code to another. We establish an explicit form of the maximal quasi quotients of the Hadamard division. It proved the criterion of existence for a given code of an inverse code in a semiring formed by linear codes of length n with operations of sum and Hadamard product of codes. We also describe the explicit form of codes with an inverse code in this semiring.