The Unit Group of the Ring of Nega-Circulant Matrices over Finite Fields

Abstract

Circulant and nega-circulant matrices have been of interest due to their nice algebraic structures and wide applications. This thesis focuses on the algebraic characterization and enumeration of the unit group of the ring of n x n nega-circulant matrices over a finite field Fq. Based on the well-known fact that the ring of n x n nega-circulant matrices over Fq is isomorphic to the quotient ring Fq[x]/, the study is given through the factorization of xn+1 in Fq[x]. The results are presented into two cases where gcd(n, q) = 1 and where n is any arbitrary positive integer. This study provides not only the algebraic structures and enumeration of such unit groups but also efficient methods for computation. In some cases, simplified and efficient formulas for the number of units in these matrix rings are presented.

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