Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles

Abstract

Additive Hadamard cocycles are a natural generalization of presemifields. In this paper, we study divisible designs and semiregular relative difference sets obtained from additive Hadamard cocycles. We show that the designs obtained from additive Hadamard cocycles are flag transitive. We introduce a new product construction of Hadamard cocycles. We also study additive Hadamard cocycles whose divisible designs admit a polarity in which all points are absolute. Our main results include generalizations of a theorem of Albert and a theorem of Hiramine from presemifields to additive Hadamard cocycles. At the end, we generalize Maiorana-McFarland's construction of bent functions to additive Hadamard cocycles.

Partial difference sets and amorphic group schemes from pseudo-quadratic bent functions

“…Proposition 2.2 is a direct consequence of Lemma 4.5 in [10]. We will call a nondegenerate quadratic form over F q a quadratic bent function over F q .…”

“…We define H 1 (α) = {x ∈ F q | tr q/q (αx) = 0 and x = 0} and 10 and C 20 be the subgroups of index q + 1 and q 2 n + 1, respectively, in F * q 2 m+n . Then C 10 , C 20 = S q 2 m+n and S q 2 m+n is the union of (q + 1)/2 cosets of C 10 and N q 2 m+n is the union of (q 2 n + 1)/2 cosets of C 20 .…”

“…Bent functions were studied in detail by Dillon in [15,16] and generalized by Kumar, Scholtz and Welch in [26]. Our definition of bent functions is taken from [10] based upon their connection with splitting semi-regular relative difference sets (see [4] for details). By the Fourier inversion formula, one has…”

Paley type group schemes from cyclotomic classes and Arasu-Dillon-Player difference sets

“…Feng in [25] constructed a family of Paley type group schemes in extra-special p-groups of order p 3 and of exponent p for p > 3, and his construction was generalized by Chen and Polhill in [17] using the flag group of finite fields (see [12]). In [36], Muzychuk obtained a large number of Paley type group schemes in F q 3 and showed that the number of inequivalent such schemes grows exponentially.…”