Maximally nonlinear functions and bent functions.
The topic to which the present paper belongs has earlier been studied in several works of Dobbertin. Let \(GF(2^n)\) be the finite field of size \(2^n\). The mappings (denoted by \(F\)) from \(GF(2^n)\) to itself are studied. A quantity \(L(F)\) is introduced, it serves as a measure of the linearity of \(F\). (If \(L(F)< L(G)\), then \(F\) is farther from being linear then \(G\) is.) It is known that the bent functions (i.e., functions from \(GF(2^n)\) to \(GF(2)\) which are at maximum Hamming distance from the first-order Reed-Muller code) are (essentially) coinciding with the elementary Hadamard difference sets. Consider a fixed odd \(n\), denote \(\lfloor n/2\rfloor\) by \(m\). It is known that the minimum of \(L(F)\) is \(2^{m+1}\). The author studies the properties of the mappings \(F\) yielding minimal \(L(F)\); among others, a new construction is given, by use of these \(F\)'s, which produces elementary Hadamard difference sets. (For even values of \(n\), the equality of the minimum of \(L(F)\) and \(2^{m+1}\) is a conjecture.)
scientific article; zbMATH DE number 1392716