2020/07/28 by Somphong Jitman, Jitman, Somphong · 1 citation
Computer Science · Engineering · Mathematics · #11C20 #13F10 #15B33 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2007.14123
openalex publication_date 2020/07/28 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Circulant matrices over finite fields and over commutative finite chain rings\nhave been of interest due to their nice algebraic structures and wide\napplications. In many cases, such matrices over rings have a closed connection\nwith diagonal matrices over their extension rings. In this paper, the\ndeterminants of diagonal and circulant matrices over commutative finite chain\nrings R with residue field mathbbFq are studied. The number of n\×\nn diagonal matrices over R of determinant a is determined for all\nelements a in R and for all positive integers n. Subsequently, the\nenumeration of nonsingular n\× n circulant matrices over R of\ndeterminant a is given for all units a in R and all positive integers\nn such that \gcd(n,q)=1. In some cases, the number of singular n\× n\ncirculant matrices over R with a fixed determinant is determined through\nthe link between the rings of circulant matrices and diagonal matrices. As\napplications, a brief discussion on the determinant of diagonal and circulant\nmatrices over commutative finite principal ideal rings is given. Finally, some\nopen problems and conjectures are posted\n