Permutation polynomials of degree 8 over finite fields of characteristic 2

Fan, Xiang Ning

doi:10.1016/j.ffa.2020.101662

Cited by 11 publications

(13 citation statements)

References 11 publications

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“…Further, when a 1 = 0 (and thus a 3 0), we can assume that a 3 = 1, as q − 1 = 26 is coprime to 5. Note that a 2 = −a 3 and (a 4 , a 3 , a 2 , a 1 ) listed as follows: (11,1,12,6), (11,1,21,22), (15,8,16,12), (15,16,13,7), (16,8,7,1), (19,5,10,20), (20, 12, 2, 6). 12, 12, 4, 1), (17,3,9,17), (18,12,4,10), (18,17,13,16) 3, 18, 1), (11,3,5,1), (18,11,0,6) The output is Ideal(1), which indicates that polynomials x 1 and x 3 cannot both vanish at (a 1 , a 2 , .…”

Section: Explicit Resultsmentioning

confidence: 99%

“…, a 1 ), among which three tuples (4, 2, 5, 0, 4, 0), (10, 2, 2, 0, 4, 0) and (12,2,6,0,4,0) give three linearly related X. Fan [13] Algorithm 11 To list PPs of degree 8 in normalised form over F 13 , (4,11,5,8), (7,11,8,1), (10,5,2,1), (10,9,2,2), (12,11,6, 1) (1, 1)…”

Section: Explicit Resultsmentioning

confidence: 99%

“…(1, 9, 6, 4), (4,2,5,10), (4,3,5,7), (4,4,5,12), (7,5,8,5), (9,5,5,10), (9,8,5,7), (12,10,6, 3) (2, 1) (0, 3, 0, 0), (1,8,6,5), (1,9,6,3), (2,9,11,1), (5,8,7,6), (6,12,8,4), (7,4,8,12), (8,0,7,…”

Section: Explicit Resultsmentioning

confidence: 99%

“…When a 1 = 0, we can assume that a 5 ∈ {0, 1}, as q − 1 = 10 is coprime to 3. When a 1 = a 5 = 0, we a 4 , a 3 , a 2 ) (0, 0) (0, 0, 2, 0), (0, 0, 4, 0), (2,6,2,10), (2,7,3,8) (1, 0) (0, 2, 0, 5), (1, 1, 0, 4), (1, 4, 0, 1), (1,4,5,1), (1,9,1,7), (1,10,5,6), (2,1,4,3), (2,4,0,8), (2,5,0,6), (3,2,3,10), (4, 1, 0, 1), (4,6,4,3), (4,7,0,10), (4,8,3,6), (4,9,9,…”

Section: Explicit Resultsmentioning

confidence: 99%

“…, 0),(8,7,6, 0),(9, 0,4,1),(9,3,10,7),(9,7,2,4), (9, 7, 3, 4), (9, 10, 3, 10), (10, 2, 6, 3), (10, 8, 3, 9) (4, 1) (0, 3, 0, 3), (1, 1, 10, 2), (1, 4, 10, 10), (2, 3, 8, 8), (2, 7, 9, 0), (3, 0, 5, 3), (3, 1, 7, 0), (3, 1, 9, 0), (3, 3, 9, 5), (3, 5, 0, 10), (3, 5, 7, 10), (4, 1, 2, 10), (5, 5, 10, 0), (5, 10, 1, 8), (5, 10, 7, 8), (6, 3, 9, 7), (6, 7, 2, 5), (7, 2, 8, 0), (7, 5, 10, 1), (7, 9, 0, 6), (8, 3, 10, 1),…”

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confidence: 99%

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Permutation Polynomials of Degree 8 Over Finite Fields of Odd Characteristic

Fan¹

2019

Bull. Aust. Math. Soc.

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We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form.

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Section: Explicit Resultsmentioning

confidence: 99%

Section: Explicit Resultsmentioning

confidence: 99%

Section: Explicit Resultsmentioning

confidence: 99%

Section: Explicit Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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Permutation Polynomials of Degree 8 Over Finite Fields of Odd Characteristic

Fan¹

2019

Bull. Aust. Math. Soc.

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Improved Lower Bounds for Permutation Arrays Using Permutation Rational Functions

Bereg

Malouf

Morales

et al. 2021

Arithmetic of Finite Fields

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New lower bounds for permutation arrays using contraction

Bereg

Miller

Mojica

et al. 2019

Des. Codes Cryptogr.

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We consider rational functions of the form V (x)/U (x), where both V (x) and U (x) are polynomials over the finite field F q . Polynomials that permute the elements of a field, called permutation polynomials (P P s), have been the subject of research for decades. Let P 1 (F q ) denote Z q ∪ {∞}. If the rational function, V (x)/U (x), permutes the elements of P 1 (F q ), it is called a permutation rational function (PRF). Let N d (q) denote the number of PPs of degree d over F q , and let N v,u (q) denote the number of PRFs with a numerator of degree v and a denominator of degree u. It follows that N d,0 (q) = N d (q), so PRFs are a generalization of PPs. The number of monic degree 3 PRFs is known [11]. We develop efficient computational techniques for N v,u (q), and use them to show N 4,3 (q) = (q + 1)q 2 (q − 1) 2 /3, for all prime powers q ≤ 307, N 5,4 (q) > (q + 1)q 3 (q − 1) 2 /2, for all prime powers q ≤ 97, and N 4,4 (p) = (p + 1)p 2 (p − 1) 3 /3, for all primes p ≤ 47. We conjecture that these formulas are, in fact, true for all prime powers q. Let M (n, D) denote the maximum number of permutations on n symbols with pairwise Hamming distance D. Computing improved lower bounds for M (n, D) is the subject of much current research with applications in error correcting codes. Using PRFs, we obtain significantly improved lower bounds on M (q, q − d) and M (q + 1, q − d), for d ∈ {5, 7, 9}.

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Permutation polynomials of degree 8 over finite fields of characteristic 2

Cited by 11 publications

References 11 publications

Permutation Polynomials of Degree 8 Over Finite Fields of Odd Characteristic

Permutation Polynomials of Degree 8 Over Finite Fields of Odd Characteristic

Improved Lower Bounds for Permutation Arrays Using Permutation Rational Functions

New lower bounds for permutation arrays using contraction

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