Vector-Circulant Matrices over Finite Fields and Related Codes

TL;DR

This paper introduces vector-circulant matrices over finite fields, providing their algebraic properties and demonstrating their use in constructing additive codes with good parameters, especially over .

Contribution

It formalizes the concept of vector-circulant matrices over finite fields and applies them to construct new additive codes with favorable properties.

Findings

Algebraic characterization of vector-circulant matrices

Construction of additive codes over  using vector-circulant matrices

Examples of good half-rate additive codes

Abstract

A vector-circulant matrix is a natural generalization of the classical circulant matrix and has applications in constructing additive codes. This article formulates the concept of a vector-circulant matrix over finite fields and gives an algebraic characterization for this kind of matrix. Finally, a construction of additive codes with vector-circulant based over $\mathbb{F}_4$ is given together with some examples of good half-rate additive codes.