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Demonstration of Theorem 3.2

Demonstration of Theorem 3.2

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Demonstration of Theorem 3.2
Demonstration of Remark 3
A note on QM equivalence of known permutation polynomials
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  • Jan 2025
In (Finite Fields Their Appl. 46, 38–56 2017), Wu et al. defined the notion of quasi-multiplicative (QM) equivalence among permutation polynomials. Other than showing thoroughly, there is no efficient approach to determine whether two given permutation polynomials are QM equivalent or not. This paper provides new results to determine QM equivalence...
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Figure 1. Case g = 2 and q = 1009. The red dots are the values of H ....
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On the L -polynomials of curves over finite fields
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  • Feb 2025
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... We provide a different yet elementary proof of their conjecture, which appears to have been recently resolved by Yadav, Singh, and Gupta [24]. ...
... Remark 4.10. After independently proving Conjecture [23, Conjecture 1] in Theorem 4.9, we discovered that it had already been established as a special case of a recent result by Yadav, Singh, and Gupta [24,Theorem 3.3]. However, our proof differs significantly from theirs, particularly in that we explicitly compute the exponent d rather than assuming it, as was done in their work. ...
New classes of permutation trinomials over finite fields with even characteristic
Preprint
  • Feb 2025
The construction of permutation trinomials of the form Xr(Xα(2m−1)+Xβ(2m−1)+1)X^r(X^{\alpha (2^m-1)}+X^{\beta(2^m-1)} + 1) over \F_{2^{2m}} where α>β\alpha > \beta and r are positive integers, is an active area of research. To date, many classes of permutation trinomials with α≤6\alpha \leq 6 have been introduced in the literature. Here, we present three new classes of permutation trinomials with α>6\alpha>6 and r≥7r \geq 7 over \F_{2^{2m}}. Additionally, we prove the nonexistence of a class of permutation trinomials over \F_{2^{2m}} of the same type for r=9, α=7\alpha=7, and β=3\beta=3 when m>3m > 3. Moreover, we show that the newly obtained classes are quasi-multiplicative inequivalent to both the existing permutation trinomials and to one another. Furthermore, we provide a proof for the recent conjecture on the quasi-multiplicative equivalence of two classes of permutation trinomials, as proposed by Yadav, Gupta, Singh, and Yadav (Finite Fields Appl. 96:102414, 2024).
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