Cycles of linear permutations over a finite field

Gary L. Mullen; Theresa P. Vaughan

doi:10.1016/0024-3795(88)90179-6

Cycles of linear permutations over a finite field

Gary L. Mullen, Theresa P. Vaughan

Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We study the cycle structure of those permutations of the finite field F_qⁿ of the form L(x) = ∑ i=0 n-1a_ix^qⁱ where each a_i ε{lunate} F_q. For such a permutation, the problem of finding its cycle decomposition of F_qⁿ can be reduced to finding its cycle decomposition on certain T-invariant subspaces of F_qⁿ, where T is the operator defined by T : x → x^q. If L₁(x) and L₂(x)M are in the above form, we say that L₁(x) and L₂(x) are equivalent if L₁(x) and L₂(x) induce the same cycle decomposition of F_qⁿ, and we say they are strongly equivalent if they induce the same cycle decomposition in every T-invariant subspace of F_qⁿ. We show that these notions are not the same, and we give characterizing theorems for each.

Original language	English (US)
Pages (from-to)	63-82
Number of pages	20
Journal	Linear Algebra and Its Applications
Volume	108
Issue number	C
DOIs	https://doi.org/10.1016/0024-3795(88)90179-6
State	Published - Sep 1988

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/0024-3795(88)90179-6

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title = "Cycles of linear permutations over a finite field",

abstract = "We study the cycle structure of those permutations of the finite field Fqn of the form L(x) = ∑ i=0 n-1aixqi where each ai ε{lunate} Fq. For such a permutation, the problem of finding its cycle decomposition of Fqn can be reduced to finding its cycle decomposition on certain T-invariant subspaces of Fqn, where T is the operator defined by T : x → xq. If L1(x) and L2(x)M are in the above form, we say that L1(x) and L2(x) are equivalent if L1(x) and L2(x) induce the same cycle decomposition of Fqn, and we say they are strongly equivalent if they induce the same cycle decomposition in every T-invariant subspace of Fqn. We show that these notions are not the same, and we give characterizing theorems for each.",

author = "Mullen, {Gary L.} and Vaughan, {Theresa P.}",

note = "Funding Information: *This author was sponsored in part by the National Security Agency under grant agreement MDA 904-85H-0013.",

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T1 - Cycles of linear permutations over a finite field

AU - Mullen, Gary L.

AU - Vaughan, Theresa P.

N1 - Funding Information: *This author was sponsored in part by the National Security Agency under grant agreement MDA 904-85H-0013.

PY - 1988/9

Y1 - 1988/9

N2 - We study the cycle structure of those permutations of the finite field Fqn of the form L(x) = ∑ i=0 n-1aixqi where each ai ε{lunate} Fq. For such a permutation, the problem of finding its cycle decomposition of Fqn can be reduced to finding its cycle decomposition on certain T-invariant subspaces of Fqn, where T is the operator defined by T : x → xq. If L1(x) and L2(x)M are in the above form, we say that L1(x) and L2(x) are equivalent if L1(x) and L2(x) induce the same cycle decomposition of Fqn, and we say they are strongly equivalent if they induce the same cycle decomposition in every T-invariant subspace of Fqn. We show that these notions are not the same, and we give characterizing theorems for each.

AB - We study the cycle structure of those permutations of the finite field Fqn of the form L(x) = ∑ i=0 n-1aixqi where each ai ε{lunate} Fq. For such a permutation, the problem of finding its cycle decomposition of Fqn can be reduced to finding its cycle decomposition on certain T-invariant subspaces of Fqn, where T is the operator defined by T : x → xq. If L1(x) and L2(x)M are in the above form, we say that L1(x) and L2(x) are equivalent if L1(x) and L2(x) induce the same cycle decomposition of Fqn, and we say they are strongly equivalent if they induce the same cycle decomposition in every T-invariant subspace of Fqn. We show that these notions are not the same, and we give characterizing theorems for each.

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

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ER -

Cycles of linear permutations over a finite field

Abstract

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