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Low-degree planar monomials in characteristic two. (English) Zbl 1343.51005
Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Y. Zhou [J. Combin. Des. 21, No. 12, 563–584 (2013; Zbl 1290.05038)] introduced an even characteristic analogue which has similar applications. In this paper, the authors focus on monomial functions on \(\mathbb F_q\) of the form \(f(x) = cx^t\). In particular, they show that, if \(q\) is a power of \(2\), \(t\) is an integer with \(0 < t \leq q^{1/4}\) and \(c \in \mathbb F_q^*\), then a monomial function is planar on \(\mathbb F_q\) if and only if \(t\) is a power of \(2\). This extends a result in [K.-U. Schmidt and Y. Zhou, J. Algebr. Comb. 40, No. 2, 503–526 (2014; Zbl 1319.51008)], where the authors proved that \(t=1\) is the only odd value such that \(f(x)\) is planar for some \(c \neq 0\) over infinitely many fields.

MSC:
51E20 Combinatorial structures in finite projective spaces
11T06 Polynomials over finite fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
05B05 Combinatorial aspects of block designs
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