Hyper-bent functions and cyclic codes

Bent functions are those Boolean functions whose Hamming distance to the Reed-Muller code of order 1 equal 2^n-1-2^n/2-1 (where the number n of variables is even). These combinatorial objects, with fascinating properties, are rare. Few constructions are known, and it is difficult to know whether the bent functions they produce are peculiar or not, since no way of generating at random bent functions on 8 variables or more is known.The class of bent functions contains a subclass of functions whose properties are still stronger and whose elements are still rarer. Youssef and Gong have proved the existence of such hyper-bent functions, for every even n. We prove that the hyper-bent functions they exhibit are exactly those elements of the well-known PS_ap class, introduced by Dillon, up to the linear transformations x?δx, . Hyper-bent functions seem still more difficult to generate at random than bent functions; however, by showing that they all can be obtained from some codewords of an extended cyclic code H_n with small dimension, we can enumerate them for up to 10 variables. We study the non-zeroes of H_n and we deduce that the algebraic degree of hyper-bent functions is n/2. We also prove that the functions of class PS_ap are some codewords of weight 2^n-1-2^n/2-1 of a subcode of H_n and we deduce that for some n, depending on the factorization of 2ⁿ-1, the only hyper-bent functions on n variables are the elements of the class , obtained from PS_ap by composing the functions by the transformations x?δx, δ≠0, and by adding constant functions. We prove that non- hyper-bent functions exist for n=4, but it is not clear whether they exist for greater n. We also construct potentially new bent functions for n=12.     

Simple abelian quadruple systems

We generalize a construction of simple cyclic 3-designs due to Köhler (1981) to that of simple abelian 3-designs. We prove that for any abelian group A of order , there exists a simple 3-(v,4,3) design with A?Aut(A) as an automorphism group.     

On the ranks of bent functions

The rank of a bent function is the 2-rank of the associated symmetric 2-design. In this paper, it is shown that it is an invariant under the equivalence relation among bent functions. Some upper and lower bounds of ranks of general bent functions, Maiorana–McFarland bent functions and Desarguesian partial spread bent functions are given. As a consequence, it is proved that almost every Desarguesian partial spread bent function is not equivalent to any Maiorana–McFarland bent function.     

A Note on Certain 2-Groups with Hadamard Difference Sets

Nontrivial difference sets in 2-groups are part of the family of Hadamarddifference sets. An abelian group of order 2^2d+2 has a difference setif and only if the exponent of the group is less than or equal to2 ^d+2. We provide an exponent bound for a more general type of 2-groupwhich has a Hadamard difference set. A recent construction due to Davis and Iiamsshows that we can attain this bound in at least half of the cases.     

Two applications of relative difference sets: Difference triangles and negaperiodic autocorrelation functions

The well-known difference sets have various connections with sequences and their correlation properties. It is the purpose of this note to give two more applications of the (not so well known) relative difference sets: we use them to construct difference triangles (based on an idea of A. Ling) and we show that a certain nonexistence result for semiregular relative difference sets implies the nonexistence of negaperiodic autocorrelation sequences (answering a question of Parker [Even length binary sequence families with low negaperiodic autocorrelation, in: Applied Algebra, Algebraic Algorithms and Error-correcting Codes, Melbourne, 2001, Lecture Notes in Computer Science, vol. 2227, Springer, Berlin, 2001, pp. 200-209.]).     

On the fixing sets of dihedral groups

The fixing number of a graph $\Gamma$ is the minimum number of labeled vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of $\Gamma$. The fixing set of a finite group $G$ is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to $G$. Previously, authors have studied the fixing sets of both abelian groups and symmetric groups. In this article, we determine the fixing set of the dihedral group.     

Regular symmetric groups of boolean functions

We show that with the exception of four known cases: C₃, C₄, C₅, and , all regular permutation groups can be represented as symmetric groups of boolean functions. This solves the problem posed by A. Kisielewicz in the paper [A. Kisielewicz, Symmetry groups of boolean functions and constructions of permutation groups, J. Algebra 199 (1998) 379-403]. A slight extension of our proof yields the same result for semiregular groups.     

^2F4（q2）与2-（v，k，3）对称设计

本文要证明不存在一个非平凡2-（v,k,3）对称设计,它的旗传递自同构群的基柱是^2F4（q2）     

Functions realizing as abelian group automorphisms

Let A be a set and f:A→A a bijective function. Necessary and sufficient conditions on f are determined which makes it possible to endow A with a binary operation ? such that (A,?) is a cyclic group and f∈Aut(A). This result is extended to all abelian groups in case |A| = p², p a prime. Finally, in case A is countably infinite, those f for which it is possible to turn A into a group (A,?) isomorphic to ?ⁿ for some n≥1, and with f∈Aut(A), are completely characterized.     

New 5-designs with automorphism group PSL(2, 23)

Blocks of the unique Steiner system S(5, 8, 24) are called octads. The group PSL(2, 23) acts as an automorphism group of this Steiner system, permuting octads transitively. Inspired by the discovery of a 5-(24, 10, 36) design by Gulliver and Harada, we enumerate all 4- and 5-designs whose set of blocks are union of PSL(2, 23)-orbits on 10-subsets containing an octad. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 147–155, 1999     

Designs from the Group PSL2(q), q Even

Let q be a power of 2 greater than 2 and consider the group G = PSL₂(q). We choose the maximal subgroups of G isomorphic to the dihedral groups D_2(q+1) and D_2(q-1) and present the primitive action of G on the right cosets of these two subgroups. We will find the orbits of the point stabilizer in each case and in the case of D_2(q-1) we will prove there is an orbit Δ of the point stabilizer G_ω, such that Δ ≠ {ω } and whose orbiting under G gives a 1-design with the automorphism group isomorphic to the symmetric group