On generating invertible circulant binary matrices with a prescribed number of ones

Fabšič, Tomáš; Grošek, Otokar; Nemoga, Karol; Zajac, Pavol

doi:10.1007/s12095-017-0239-4

Cited by 4 publications

(3 citation statements)

References 5 publications

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“…By setting s = , the following corollary can be obtained immediately. [7]). Let q be a prime power and let n be a positive integer such that gcd(n, q) = .…”

Section: Circulant Matrices Over Cfcrsmentioning

confidence: 99%

“…Then cn(R, a) = cn(R, ) for all a ∈ U(R).. Determinants of Nonsingular Circulant Matrices over CFCRsNonsingular n × n circulant matrices over a CFCR are studied and the number cn(R, a) is determined for all a ∈ U(R) and for all positive integers n such that gcd(n, q) = . By Corollary 4.3, it is enough to derive only the number cn(R, ).Let NSCn(R) = {A ∈ Cn(R) | det(A) ∈ U(R)} denote the set of nonsingular n × n circulant matrices over a CFCR R. The number |NSCn(Fq)| which is key to determine cn(R, ) is given in[16, Proposition 1] and[7] via the ring isomorphism T : Cn(R) → R[X]/ X n − de ned by cir(a , a , . .…”

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confidence: 99%

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Determinants of some special matrices over commutative finite chain rings

Jitman

2020

Special Matrices

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Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field 𝔽q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinants of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted

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“…By setting s = , the following corollary can be obtained immediately. [7]). Let q be a prime power and let n be a positive integer such that gcd(n, q) = .…”

Section: Circulant Matrices Over Cfcrsmentioning

confidence: 99%

mentioning

confidence: 99%

Determinants of some special matrices over commutative finite chain rings

Jitman

2020

Special Matrices

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“…Lemma 4.4 ([16, Proposition 1] and [6]). Let q be a prime power and let n be a positive integer such that gcd(n, q) = 1.…”

Section: Determinants Of Nonsingular Circulant Matrices Over Cfcrsmentioning

confidence: 99%

Determinants of some Special Matrices over Commutative Finite Chain Rings

Jitman¹

2020

Preprint

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Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field F q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinant of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems

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