Two generalizations of ideal matrices and their applications

TL;DR

This paper introduces two new generalizations of ideal matrices, explores their properties, and applies them to construct generalized quasi-cyclic codes, expanding the theoretical framework and coding applications.

Contribution

It proposes generalized ideal matrices and double ideal matrices, analyzes their properties, and applies them to construct a new class of quasi-cyclic codes.

Findings

Defined generalized ideal matrices and double ideal matrices.

Verified ranks and linear independence of these matrices.

Constructed generalized quasi-cyclic codes using double ideal matrices.

Abstract

In this paper, two kinds of generalizations of ideal matrices, generalized ideal matrices and double ideal matrices. are obtained and studied, The concepts of generalized ideal matrices and double ideal matrices are proposed, and their ranks and maxima.linearly independent groups are verified.The initial motivation to study double cyclic matrices is to study the quasi cyclic codes of the fractional index. In this paper, the generalized form of the quasi cyclic codes, i.e. the {\phi}-quasi cyclic codes. and the construction of the generated matrix are given by the double ideal matrix.