Quadratic permutations, complete mappings and mutually orthogonal Latin squares.

*(English)*Zbl 1442.11165Summary: 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.

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) |