The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group.

Dickson, L. E.

doi:10.2307/1967217

Cited by 210 publications

(149 citation statements)

References 0 publications

Supporting

Mentioning

147

Contrasting

Unclassified

Order By: Relevance

“…(1, 0, 5, 0, 2), (1, 0, 5, 0, 7), (1,0,5,8,6), (1,1,5,2,5), (1,2,5,9,8), (1,4,5,8,0), (1,8,5,8,0), (1,5,5,1,5), (2,0,9,0,8), (2,0,9,0,9),…”

Section: Non-exceptional Pps Of Degreementioning

confidence: 99%

“…By a permutation polynomial (PP) over F q , we mean a polynomial f ∈ F q [x] with the associated mapping a → f (a) permuting F q . Arose in Hermite [6] and Dickson [2], the theory of PPs over have been a hot topic of study for more than one hundred years. However, the basic problem of classification or characterization of PPs of prescribed forms are still challenging.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Chahal and Ghorpade [1,Remark 3.4] replaced n 4 in Lemma 1 to (n − 2)(n − 3) + (n − 2) 2 (n − 3) 2 + 2(n 2 − 1) 2 2 which is less than n 2 (n − 2) 2 . Moreover, our preprint [3] further refined this bound as follows.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A classification of permutation polynomials of degree 7 over finite fields

Fan

2019

Finite Fields and Their Applications

View full text Add to dashboard Cite

show abstract

“…(1, 0, 5, 0, 2), (1, 0, 5, 0, 7), (1,0,5,8,6), (1,1,5,2,5), (1,2,5,9,8), (1,4,5,8,0), (1,8,5,8,0), (1,5,5,1,5), (2,0,9,0,8), (2,0,9,0,9),…”

Section: Non-exceptional Pps Of Degreementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A classification of permutation polynomials of degree 7 over finite fields

Fan

2019

Finite Fields and Their Applications

View full text Add to dashboard Cite

show abstract

“…We will require elements of Dickson's classification of onevariable permutation polynomials over finite fields of low degree [27]. By Lemma 3.1, a hidden variable λ prescribes to W (p 1 , q 1 , p 2 , q 2 ) the outcome λ 1 p 1 + λ 2 q 1 + λ 3 p 2 + λ 4 q 2 .…”

Section: Typementioning

confidence: 99%

Logical paradoxes in quantum computation

Silva

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

View full text Add to dashboard Cite

The precise features of quantum theory enabling quantum computational power are unclear. Contextuality-the denial of a notion of classical physical reality-has emerged as a promising hypothesis: e.g. Howard et al. showed that the magic states needed to practically achieve quantum computation are contextual.Strong contextuality, as defined by Abramsky-Brandenburger, is an extremal form of contextuality describing systems that exhibit logically paradoxical behaviour.After introducing number-theoretic techniques for constructing exotic quantum paradoxes, we present large families of strongly contextual magic states that are computationally optimal in the sense that they enable universal quantum computation via deterministic injection of gates of the Clifford hierarchy. We thereby bolster a refinement of the resource theory of contextuality that emphasises the computational power of logical paradoxes. IntroductionIssues. Identification of the precise features of quantum theory accounting for quantum advantages over classical devices, and of the mechanisms by which they do so, remains an open problem in quantum computation. Physicists and quantum information theorists seek high-level principles behind quantum advantages in order to optimise resources and serve as guideposts towards novel protocols and algorithms.A promising, emerging hypothesis is contextuality: a concept from the foundations of quantum mechanics first articulated by Bell-Kochen-Specker [13,47] that is commonly understood as the denial of a classical notion of reality. Essentially, contextuality is the failure of correlated probabilistic data to be reproducible by a classical probability model. It subsumes nonlocality-the failure of separated systems to respect a strong classical assumption about causality [14]-as a special case. With these concepts, tensions between classical and quantum physics, e.g. the Einstein-Podolsky-Rosen paradox [28], are formally articulated to a degree that experiments (e.g. [10,39]) can be performed to refute classical ontological assumptions.An important lesson of recent studies of contextuality is that it is a logical phenomenon that is most clearly understood independently of quantum physics, the setting in which it first appeared

show abstract

“…So, finding new permutation polynomials is of great interest in both theoretical and applied aspects. The study of permutation polynomials has a long history [6,12] and many recent results are surveyed in [14].…”

Section: Introductionmentioning

confidence: 99%

Two types of permutation polynomials with special forms

Zheng

Yuan

2019

Finite Fields and Their Applications

View full text Add to dashboard Cite

Let q be a power of a prime and F q be a finite field with q elements. In this paper, we propose four families of infinite classes of permutation trinomials having the form cx − x s + x qs over F q 2 , and investigate the relationship between this type of permutation polynomials with that of the form (x q − x + δ) s + cx. Based on this relation, many classes of permutation trinomials having the form (x q − x + δ) s + cx without restriction on δ over F q 2 are derived from known permutation trinomials having the form cx − x s + x qs .

show abstract

The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group.

Cited by 210 publications

References 0 publications

A classification of permutation polynomials of degree 7 over finite fields

A classification of permutation polynomials of degree 7 over finite fields

Logical paradoxes in quantum computation

Two types of permutation polynomials with special forms

Contact Info

Product

Resources

About