Permutation polynomials over ${\mathbb{F}}_{{q}^{2}}$ constructed from self-conjugate reciprocal polynomials

Xiang Kang; Xianping Liu; Xiaofang Xu

doi:10.3934/amc.2025041

Article Contents

2026, Volume 21: 42-56. Doi: 10.3934/amc.2025041

This volume Previous Article A family of cyclic codes with increasing dimensions and fixed minimum distances Next Article Additive skew polycyclic codes over $ \mathbb{F}_{p^2} $

Permutation polynomials over ${\mathbb{F}}_{{q}^{2}}$ constructed from self-conjugate reciprocal polynomials

1.
School of Mathematics and Statistics, Hubei Minzu University, Enshi, China
2.
School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, China

^*Corresponding author: Xiaofang Xu
^*Corresponding author: Xiaofang Xu

Received: January 2025

Revised: June 2025

Early access: August 26, 2025

Published online: August 26, 2025

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Abstract

Motivated by the fact that a polynomial permutes $ \mathbb{F}_{q^{2}} $ if and only if the corresponding self-conjugate reciprocal (SCR) polynomial has no roots in the unit circle, we propose four classes of permutation polynomials over $ \mathbb{F}_{q^{2}} $ by investigating the SCR polynomials with the forms $ f(0)^{-1}f(x)f^{*}(x) $ and $ x^{nq}f(x^{q}+x^{-q}) $, where $ f(x)\in \mathbb{F}_{q^{2}}[x] $ and $ f^{*}(x) = x^{\operatorname{deg}(f)}f(\frac{1}{x}) $. Moreover, the two forms of SCR polynomials can provide effective tools to construct new permutation polynomials.

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Mathematics Subject Classification: 11T06, 11T71, 12E10.

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Table 1. Known PPs derived from SCR polynomials over $ \mathbb{F}_{q^{2}} $

$ f(x) $	Conditions	Reference
$ x^{r+d(q-1)}+\beta^{-1}x^{r} $	see Corollary 5.3 in [46]	[46]
$ x^{s}B(x^{q-1})(ax^{2(q-1)}+bx^{q-1}+a^{q}) $	see Theorem 3.5, 3.7 in [30]	[30]
$ x^{s}B(x^{q-1})(a^{q}x^{2q(q-1)}+bx^{q(q-1)}+a) $	see Theorem 3.8, 3.9 in [30]	[30]
$ x^{s}B(x^{q-1})(ax^{3(q-1)}+a^{q}) $	see Theorem 3.10 in [30]	[30]
$ x^{s}B(x^{q-1})(ax^{3(q-1)}+bx^{2(q-1)}+bx^{q-1}+a^{q}) $	see Theorem 3.11 in [30]	[30]
$ x^{s}B(x^{q-1})(bx^{q(q-1)}+bx^{q-1}+2a) $	see Theorem 3.12 in [30]	[30]
$ x^{s}B(x^{q-1})(bx^{q(q-1)}+bx^{q-1}+a^{q}+a) $	see Theorem 3.13 in [30]	[30]
$x^{s}B(x^{q-1})((\delta x^{q-1}-\beta \delta^{q})^{m}+$ $\\ b_{1}(\delta x^{q-1}-\beta \delta^{q})^{m-1}(x^{q-1}-\beta)+\cdots +$ $b_{m-1}(\delta x^{q-1}-\beta \delta^{q})(x^{q-1}-\beta)^{m-1}\\ $ $+b_{m}(x^{q-1}-\beta)^{m})$	see Theorem 3.15 in [30]	[30]
$ x^{2n(q-1)+r}+(a+a^{-1})x^{n(q-1)+r}+x^{r} $	Theorem 3.1	This paper
$x^{2n(q-1)+r} + a^{-1}b\, x^{(2n-1)(q-1)+r} + \\ $ $bx^{(n+1)(q-1)+r} +\\ $ $(a^{-1} + a^{-1}b^{2} + a)x^{n(q-1)+r} \\ $ $ + a^{-1}b\, x^{(q-1)+r} + bx^{(n-1)(q-1)+r} + x^{r} $	Theorem 3.3	This paper
$ x^{r}\left( (x^{2(q^{2}-q)}+1)^{n}+ax^{n(q^{2}-q)}\right) $	Theorem 3.5	This paper
$ x^{4(q^{2} - q) + r} + b\, x^{3(q^{2} - q) + r} + a\, x^{2(q^{2} - q) + r} + b\, x^{(q^{2} - q) + r} + x^{r} $	Theorem 3.7	This paper

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Table 2. Known PPs of the form $ a_{1}x^{r}+a_{2}x^{s(q-1)+r}+a_{3}x^{t(q-1)+r} $ over $ \mathbb{F}_{q^2} $

$ f(x) $	Conditions	Reference
$ a_{1}x+a_{2}x^{q}+x^{2q-1} $	see Theorem A, B in [14]	[14]
$ a_{1}^{q}x^{kq(q-1)+1}+a_{1}x^{k(q-1)+1}+a_{2}x $	see Proposition 1 in [20]	[20]
$ a_{1}x^{-(q-1)+1}+a_{2}x^{(q-1)+1}+a_{3}x $	see Theorem 4 in [20]	[20]
$ x^{3(q-1)+3}+a_{1}x^{(q-1)+3}+a_{2}x^{3} $	see Theorem 4, 8 in [26]	[26]
$ x^{3}+a_{1}x^{(q-1)+3}+a_{2}x^{2(q-1)+3} $	see Theorem 3, 4 in [27]	[27]
$ x^{3}+a_{1}x^{2(q-1)+3}+a_{2}x^{3(q-1)+3} $	see Theorem 7, 8 in [27]	[27]

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Table 3. Known PPs of the form $ x^r + a_1x^{s_1(q-1)+r} + a_2x^{s_2(q-1)+r} + a_3x^{s_3(q-1)+r} + a_4x^{s_4(q-1)+r} $ over $ \mathbb{F}_{q^2} $

$ f(x) $	Conditions	Reference
$ x^{5}+x^{q+4}+x^{3q+2}+x^{4q+1}+x^{5q} $	$ q=2^{m}, m\not \equiv0(\operatorname{mod}4) $	[41]
$ x^{3}+x^{2q+1}+x^{3q}+x^{4q-1}+x^{-q+4} $	$ q=2^{m} $, $ m $ is odd	[41]
$ x^{5}+x^{q+4}+x^{2q+3}+x^{4q+1}+x^{5q} $	$ q=2^{m}, m\not \equiv0(\operatorname{mod}4) $	[41]
$ x^{3}+x^{q+2}+x^{3q}+x^{4q-1}+x^{-q+4} $	$ q=2^{m} $, $ m $ is odd	[41]
$ x^{7}+x^{2q+5}+x^{3q+4}+x^{5q+2}+x^{6q+1} $	$ q=2^{m}, \operatorname{gcd}(m, 3)=1 $	[41]
$ x^{5}+x^{q+4}+x^{3q+2}+x^{4q+1}+x^{6q-1} $	$ q=2^{m} $, $ m $ is odd with $ \operatorname{gcd}(m, 3)=1 $ or $ m \equiv2(\operatorname{mod}4) $	[41]
$ x^{5}+x^{3q+2}+x^{4q+1}+x^{5q}+x^{6q-1} $	$ q=2^{m}, m \equiv2(\operatorname{mod}4) $	[41]
$ x^{7}+x^{3q+4}+x^{4q+3}+x^{5q+2}+x^{6q+1} $	$ q=2^{m}, m \equiv0(\operatorname{mod}4) $	[41]
$ x^{9}+x^{3q+6}+x^{6q+3}+x^{7q+2}+x^{9q} $	$ q=2^{m}, m $ is odd	[41]
$ x^{9}+x^{2q+7}+x^{3q+6}+x^{6q+3}+x^{9q} $	$ q=2^{m}, m $ is odd	[41]
$ x^{q+6}+x^{2q+5}+x^{5q+2}+x^{8q-1}+x^{-q+8} $	$ q=2^{m}, \operatorname{gcd}(m, 3)=1 $, $ m \not\equiv2(\operatorname{mod}4) $	[41]
$ x^{2q+5}+x^{5q+2}+x^{6q+1}+x^{8q-1}+x^{-q+8} $	$ q=2^{m}, \operatorname{gcd}(m, 3)=1 $, $ m \not\equiv2(\operatorname{mod}4) $	[41]
$ x+a_{1}x^{\frac{1}{4}(q-1)+1}+a_{2}x^{\frac{1}{2}(q-1)+1}+a_{3}x^{\frac{3}{4}(q-1)+1}+a_{4}x^{q} $	see Theorem 3.1 in [45]	[45]
$ x+x^{\frac{1}{17}(q-1)+1}+x^{\frac{8}{17}(q-1)+1}+x^{\frac{16}{17}(q-1)+1}+x^{\frac{15}{17}(q-1)+1} $	$ q=2^{m} $, $ \operatorname{gcd}(17, q+1)=1 $	[45]
$ x+x^{\frac{11}{13}(q-1)+1}+x^{\frac{9}{13}(q-1)+1}+x^{\frac{2}{13}(q-1)+1}+x^{q} $	$ q=2^{m} $, $ \operatorname{gcd}(13, q+1)=\operatorname{gcd}(5, q+1)=1 $	[45]
$ x+x^{3q-2}+x^{2q-1}+x^{q^{2}-q+1}+x^{q^{2}-2q+2} $	$ q=2^{m} $	[19]
$ x^{7q-5}+x^{3q-1}+x^{q^{2}-q+2}+x^{q^{2}-5q+6}+x^{q^{2}-7q+8} $	$ q=2^{m}, m\not\equiv0(\operatorname{mod}7) $	[19]
$ a^{2}x^{i(6q^{2}-6q)+1}+a^{2}x^{i(6q-6)+1}+(a^{2}+b^{2})x^{i(2q^{2}-2q)+1}+(a^{2}+b^{2})x^{i(2q-2)+1}+c^{2}x $	see Theorem 2 in [32]	[32]

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