Automorphisms and Equivalence of Bent Functions and of Difference Sets in Elementary Abelian 2-Groups

Geometric and design-theoretic aspects of semibent functions II

“…We note that the spread S 1 -in this case n is odd -defines the quadratic form Q(u, x, y, z) = tr(x y) + uz + z. For a = 1 it can be shown that S a defines a bent function of twisted parabolic type in the sense of [7]. (d) A semibent function with a linear structure in 2n + 1 variables has degree ≤ n. Hence in dimension 5 a function from Theorem 3.5 is quadratic and thus has an extension and by part (a) it has an extension in dimension 7 too.…”

Dimensional doubly dual hyperovals and bent functions

“…) is a bent function, which was called standard parabolic of degree m in [4]. Finally, we recall that two boolean functions f 1 and f 2 on V are called equivalent iff there exist T ∈ GL(V ), v ∈ V , a linear functional λ on V , and a ∈ F 2 such that f 2 (x) = f 1 (xT + v) + λ(x) + a.…”

“…The proof is complete. Here we recall that the nondegenerate, quadratic forms were called bent functions of standard type in [4]. I.e.…”

“…We recall from [4]: Let Q be a quadratic form of (+)-type on V = V (2n, 2), S(V ) the set of singular vectors, and U a totally singular subspace of dimension m. Then the characteristic function of…”

“…In order to describe the bent functions associated with Examples 4.3, 4.6, and 4.9, we introduce a series of bent functions related to the bent functions of parabolic type from [4]. (b) We need to find W 0 such that the anisotropic part has dimension 1 or 2.…”