A Hadamard Product of Linear Codes: Algebraic Properties and Algorithms for Calculating It

Abstract

A study is performed of the algebraic properties of the Hadamard product (Schur product, component-wise product) of linear error-correcting codes. The complexity of constructing a product basis using known multiplier bases is discussed. The concept is introduced of quotient, quasi-quotient, and maximal inclusion quasi-quotient obtained from the Hadamard division of one linear code by another. An explicit form of the maximum Hadamard division quasi-quotient is established. A criterion is proved for the existence of a given code of an inverse code in a semiring formed by linear codes of length \(n\) with the operations of sum and product of Hadamard codes. The explicit form of codes that have a reverse code in this semiring is described.