On some d-dimensional dual hyperovals in

In [H. Taniguchi, On d-dimensional dual hyperovals in PG(2d,2), Innov. Incidence Geom., in press], we construct d-dimensional dual hyperovals in PG(2d,2) from quasifields of characteristic 2. In this note, we show that, if d-dimensional dual hyperovals in PG(2d,2) constructed from nearfields are isomorphic, then those nearfields are isomorphic. Some results on dual hyperovals constructed from quasifields are also proved.     

Distance regular graphs arising from dimensional dual hyperovals

In [12], A. Pasini and S. Yoshiara studied the distance regular graphs constructed from the Yoshiara dual hyperovals. In this note, we prove that the incidence graphs of the semibiplanes constructed from dimensional dual hyperovals are distance regular graphs if the dual hyperovals are doubly dual hyperovals (DDHOs). This generalizes the result in [12].     

On d-Dimensional Dual Hyperovals

d-dimensional dual hyperovals in a projective space of dimension n are the natural generalization of dual hyperovals in a projective plane. After proving some general properties of them, we get the classification of two-dimensional dual hyperovals in projective spaces of order 2. A characterization of the only two-dimensional dual hyperoval which is known in PG(5,4) is also given. Finally the classification of 2-transitive two-dimensional dual hyperovals is reached.     

On the duals of certain d-dimensional dual hyperovals in

Let d2. A construction of d-dimensional dual hyperovals in PG(2d+1,2) using quadratic APN functions was discovered by Yoshiara in [S. Yoshiara, Dimensional dual hyperovals associated with quadratic APN functions, Innov. Incidence Geom., in press]. In this note, we prove that the duals of the d-dimensional dual hyperovals in PG(2d+1,2) constructed from quadratic APN functions are also d-dimensional dual hyperovals in PG(2d+1,2) if, and only if, d is even. Some examples are presented.     

Dickson多项式与差集

1998年,Maschietti用超卵形线构造了几个循环差集．R．Evans,H．D．L．Holloman, C．Krattnthaler与Qing Xiang等给出了其对应的二元序列具有良好自相关函数的简单代数证明．在本文中,证明了超卵形线与二对一映射有着紧密的联系,并且推广了Maschietti的结果．     

Dimensional dual hyperovals and APN functions with translation groups

In this paper we develop a theory of translation groups for dimensional dual hyperovals and APN functions. It will be seen that both theories can be treated, to a large degree, simultaneously. For small ambient spaces it will be shown that the translation groups are normal in the automorphism group of the respective geometric object. For large ambient spaces there may be more than one translation group. We will determine the structure of the normal closure of the translation groups in the automorphism group and we will exhibit examples which in fact do admit more than one translation group.     

Notes on APN functions, semibiplanes and dimensional dual hyperovals

Two geometric objects, incidence graphs of semibiplanes and dimensional dual hyperovals, are respectively associated with APN and quadratic APN functions. From Proposition 2 (resp. Proposition 5), two APN (resp. quadratic APN) functions are CCZ (resp. extended affine) equivalent if and only if the associated graphs (resp. dimensional dual hyperovals) are isomorphic. The former graphs for almost bent functions are distance regular graphs by Proposition 4. The structures of automorphism groups of these geometric objects are investigated in Proposition 3 and Lemma 7. In particular, (Edel and Pott, Adv Math Commun 3:59–81 (2009), Question 2) is negatively answered.     

On quadratic APN functions and dimensional dual hyperovals

In this paper we characterize the d-dimensional dual hyperovals in PG(2d + 1, 2) that can be obtained by Yoshiara’s construction (Innov Incid Geom 8:147–169, 2008) from quadratic APN functions and state a one-to-one correspondence between the extended affine equivalence classes of quadratic APN functions and the isomorphism classes of these dual hyperovals.     

Elation Laguerre planes of order 16 are ovoidal

We conduct a computer search for elation Laguerre planes of order 16 and show that each such plane is ovoidal. The search results also confirm the numbers of hyperovals through certain points in the dual translation planes of order 16 given in [15]. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 313–323, 2006     

Strong Duality with Proper Efficiency in Multiobjective Optimization Involving Nonconvex Set-Valued Maps

In this paper, we consider some dual problems of a primal multiobjective problem involving nonconvex set-valued maps. For each dual problem, we give conditions under which strong duality between the primal and dual problems holds in the sense that, starting from a Benson properly efficient solution of the primal problem, we can construct a Benson properly efficient solution of the dual problem such that the corresponding objective values of both problems are equal. The notion of generalized convexity of set-valued maps we use in this paper is that of near-subconvexlikeness.     

Dual pairs and Kostant-Sekiguchi correspondence. II. Classification of nilpotent elements

We classify the homogeneous nilpotent orbits in certain Lie color algebras and specialize the results to the setting of a real reductive dual pair. For any member of a dual pair, we prove the bijectivity of the two Kostant-Sekiguchi maps by straightforward argument. For a dual pair we determine the correspondence of the real orbits, the correspondence of the complex orbits and explain how these two relations behave under the Kostant-Sekiguchi maps. In particular we prove that for a dual pair in the stable range there is a Kostant-Sekiguchi map such that the conjecture formulated in [6] holds. We also show that the conjecture cannot be true in general.     

Equivalence classes of Niho bent functions

Equivalence classes of Niho bent functions are in one-to-one correspondence with equivalence classes of ovals in a projective plane. Since a hyperoval can produce several ovals, each hyperoval is associated with several inequivalent Niho bent functions. For all known types of hyperovals we described the equivalence classes of the corresponding Niho bent functions. For some types of hyperovals the number of equivalence classes of the associated Niho bent functions are at most 4. In general, the number of equivalence classes of associated Niho bent functions increases exponentially as the dimension of the underlying vector space grows. In small dimensions the equivalence classes were considered in detail.

     

Mapping cones and separable states

We study mapping cones and their dual cones of positive maps of the \(n\times n\) matrices into itself. For a natural class of cones there is a close relationship between maps in the cone, super-positive maps, and separable states. In particular the composition of a map from the cone with a map in the dual cone is super-positive, and so the natural state it defines is separable.     

On Automorphisms of Some d-Dimensional Dual Hyperovals in PG(d(d + 3)/2, 2)

In [3], a new d-dimensional dual hyperoval S in PG(d(d + 3)/2, 2) for d ≥ 3 was constructed based on Veronesean dual hyperoval. In this note, we determine the automorphism group of the dual hyperoval S. Received: January 26, 2007. Final Version received: January 7, 2008.     

On the refinement of bounds of heuristic algorithms for the traveling salesman problem

A number of heuristics for the traveling salesman problem (TSP) rely on the assumption that the triangle inequality (TI) is satisfied. When TI does not hold, the paper proposes a transformation such that for the transformed problem the TI holds. Consequently, the bounds obtained for heuristics are valid with appropriate modification. Moreover, for a TSP satisfying TI the same transformation strengthens such bounds. The transformation essentially maps the problem into one that is minimal with respect to the property that TI holds. For the symmetric TSP the transformation is particularly simple. For an application of the transformation in the asymmetric case we need the dual solution of an assignment problem.     

Classification of hyperovals inPG(2,32)

In this paper, we describe an exhaustive computer search that demonstrates that there are precisely 6 isomorphism classes of hyperovals inPG(2,32). The six classes had previously been discovered, and it was known that any further hyperovals would have stabiliser groups of orders 1 or 2. As the techniques for finding hyperovals involved a mixture of group theory and computer search, an exhaustive search was regarded as the only feasible way to eliminate these final cases with small group.     

On hyperovals in small projective planes

In this paper we collate the results of three computer searches for hyperovals in small projective planes, each of which resulted in new hyperovals. The three searches involve finding all hyperovals with non-trivial automorphisms in PG(2,64), all hyperovals with GF(2) o-polynomials in PG(2, 128) and PG(2, 256) and hyperovals stabilised by a particular group in PG(2, 256).