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In this paper, by the Hasse-Weil bound, we determine the necessary and sufficient condition on coefficients $a_1,a_2,a_3inmathbb{F}_{2^n}$ with $n=2m$ such that $f(x) = {x}^{3cdot2^m} + a_1x^{2^{m+1}+1} + a_2 x^{2^m+2} + a_3x^3$ is an APN function over $mathbb{F}_{2^n}$. Our result resolves the first half of an open problem by Carlet in International Workshop on the Arithmetic of Finite Fields, 83-107, 2014.
For the finite field $mathbb{F}_{2^{3m}}$, permutation polynomials of the form $(x^{2^m}+x+delta)^{s}+cx$ are studied. Necessary and sufficient conditions are given for the polynomials to be permutation polynomials. For this, the structures and properties of the field elements are analyzed.
How information is encoded in bio-molecular sequences is difficult to quantify since such an analysis usually requires sampling an exponentially large genetic space. Here we show how information theory reveals both robust and compressed encodings in
Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding in cite{D} constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions ov
We study a class of general quadrinomials over the field of size $2^{2m}$ with odd $m$ and characterize conditions under which they are permutations with the best boomerang uniformity, a new and important parameter related to boomerang-style attacks.
Cognitive radios have been studied recently as a means to utilize spectrum in a more efficient manner. This paper focuses on the fundamental limits of operation of a MIMO cognitive radio network with a single licensed user and a single cognitive user