Quadratic permutations, complete mappings and mutually orthogonal Latin squares. (English) Zbl 1442.11165

Samardjiska, Simona; Gligoroski, Danilo

Math. Slovaca 67, No. 5, 1129-1146 (2017).

Summary: We investigate the permutation behavior of a special class of Dembowski-Ostrom polynomials over a finite field of characteristic 2 of the form \(P(X) = L_1(X)(L_2(X)+L_1(X)L_3(X))\) where \(L_1, L_2, L_3\) are linearized polynomials. To our knowledge, the given class has not been studied previously in the literature. We identify several new types of permutation polynomials of this class. While most of the newly identified polynomials are linearly equivalent to permutation monomials, we show that there exist subclasses that are not affine equivalent to monomials, and we describe their forms.
One of the newly identified classes contains a subclass of complete mappings. We use these complete mappings to define new sets of mutually orthogonal Latin squares, as well as new vectorial bent functions from the Maiorana-McFarland class. Moreover, the quasigroup polynomials obtained in the process are different and inequivalent to the previously known ones.

MSC:

11T06	Polynomials over finite fields
94B25	Combinatorial codes
05B15	Orthogonal arrays, Latin squares, Room squares
11T71	Algebraic coding theory; cryptography (number-theoretic aspects)

Keywords:

permutation polynomials; Dembowski-Ostrom polynomials; complete mappings; quasigroup polynomials; mutually orthogonal Latin squares (MOLS)