On a type of permutation rational functions over finite fields

Hou, Xiang‐dong; Sze, Christopher

doi:10.1016/j.ffa.2020.101758

Cited by 9 publications

(5 citation statements)

References 29 publications

(23 reference statements)

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“…Similarly, a polynomial f (x) ∈ F q [x] is called permutation polynomial if f induces a permutation on F q . In [13] the polynomial F (x, y) = [f (x+y)g(x)−f (x)g(x+y)]/y is defined for a rational function ϕ(x) = f (x)/g(y). Now ∆ ϕ (x, y) = F (x, y − x).…”

Section: A Partition Of Morphisms On P 1 Of Degreementioning

confidence: 99%

The extended coset leader weight enumerator of a twisted cubic code

Blokhuis¹,

Pellikaan²,

Szőnyi³

2021

Preprint

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The extended coset leader weight enumerator of the generalized Reed-Solomon [q + 1, q − 3, 5] q code is computed. The computation is considered as a question in finite geometry. For this we need the classification of the points, lines and planes in the projective three space under projectivities that leave the twisted cubic invariant. A line in three space determines a rational function of degree at most three and vice versa. Furthermore the double point scheme of a rational function is studied. The pencil of a true passant of the twisted cubic, not in an osculation plane gives a curve of genus one as double point scheme. With the Hasse-Weil bound on F q -rational points we show that there is a 3-plane containing the passant.

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Section: A Partition Of Morphisms On P 1 Of Degreementioning

confidence: 99%

The extended coset leader weight enumerator of a twisted cubic code

Blokhuis¹,

Pellikaan²,

Szőnyi³

2021

Preprint

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“…◻ Theorem 2. 5 The number N aff (X) of affine rational points of X satisfies N aff (X) = N(F) − (p + 1) . In particular, we have Proof It is a well-known fact that each nonsingular rational point of a curve corresponds to a unique rational place of its function field, see [8,Section 3.1].…”

Section: Corollary 23 As the Full Constant Field Of F Ismentioning

confidence: 99%

“…It is shown in [10] that f b (X) is a permutation of p n for p = 2, 3 and any integer n ≥ 1 . Then Hou and Sze [5] have studied the polynomials f b (X) for p > 3 and arrived at the following result.…”

Section: Introductionmentioning

confidence: 99%

“…Particularly, the case P(X) = X + (X p − X + b) k , i.e., L(X) = X and G(X) = (X p − X + b) with Tr(b) ≠ 0 , where Tr(z) = z + z p + ⋯ + z p n−1 the absolute trace from p n to p . There is a series of papers devoted to the classification of these permutation polynomials, see [5] and references therein. In the case k = p n − 2 the degree of the polynomial is large compared to the size of the finite field, hence one can not directly apply the above method.…”

Section: Introductionmentioning

confidence: 99%

“…We also refer to [5] and references therein for the recent work on f b (X) given in Equation (1.2). It is shown in [10] that f b (X) is a permutation of p n for p = 2, 3 and any integer n ≥ 1 .…”

Section: Introductionmentioning

confidence: 99%

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On a special type of permutation rational functions

Anbar

2022

AAECC

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Let p be a prime and n be a positive integer. We consider rational functions f b (X) = X + 1∕(X p − X + b) over p n with Tr(b) ≠ 0 . In Hou and Sze (Finite Fields Appl 68, Paper No. 10175, 2020), it is shown that f b (X) is not a permutation for p > 3 and n ≥ 5 , while it is for p = 2, 3 and n ≥ 1 . It is conjectured that f b (X) is also not a permutation for p > 3 and n = 3, 4 , which was recently proved sufficiently large primes in Bartoli and Hou (Finite Fields Appl 76, Paper No. 101904, 2021). In this note, we give a new proof for the fact that f b (X) is not a permutation for p > 3 and n ≥ 5 . With this proof, we also show the existence of many elements b ∈ p n for which f b (X) is not a permutation for n = 3, 4.

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