More classes of permutation pentanomials over finite fields with characteristic two. (English) Zbl 07905715
Summary: Let \(q = 2^m\). In this paper, we investigate permutation pentanomials over \(\mathbb{F}_{q^2}\) of the form \(f(x) = x^t + x^{r_1(q - 1) + t} + x^{r_2(q - 1) + t} + x^{r_3(q - 1) + t} + x^{r_4(q - 1) + t}\) with \(\gcd(x^{r_4} + x^{r_3} + x^{r_2} + x^{r_1} + 1, x^t + x^{t - r_1} + x^{t - r_2} + x^{t - r_3} + x^{t - r_4}) = 1\). We transform the problem concerning permutation property of \(f(x)\) into demonstrating that the corresponding fractional polynomial permutes the unit circle \(U\) of \(\mathbb{F}_{q^2}\) with order \(q + 1\) via a well-known lemma, and then into showing that there are no certain solution in \(\mathbb{F}_q\) for some high-degree equations over \(\mathbb{F}_q\) associated with the fractional polynomial. According to numerical data, we have found all such permutations with \(4 \leq t < 100\), \(1 \leq r_i \leq t\), \(i\in[1, 4]\). Several permutation polynomials are also investigated from the fractional polynomials permuting the unit circle \(U\) found in this paper.
MSC:
| 05A05 | Permutations, words, matrices |
| 11T06 | Polynomials over finite fields |
| 11T55 | Arithmetic theory of polynomial rings over finite fields |
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