‐Relative Difference Sets and Their Representations

We show that every ‐relative difference set D in relative to can be represented by a polynomial , where is a permutation for each nonzero a. We call such an f a planar function on . The projective plane Π obtained from D in the way of M. J. Ganley and E. Spence (J Combin Theory Ser A, 19(2) (1975), 134–153) is coordinatized, and we obtain necessary and sufficient conditions of Π to be a presemifield plane. We also prove that a function f on with exactly two elements in its image set and is planar, if and only if, for any .     

Degree 8 Maximal Arcs in PG(2,2), h Odd

In a recent paper R. Mathon gave a new construction method for maximal arcs in finite Desarguesian projective planes that generalised a construction of Denniston. He also gave several instances of the method to construct new maximal arcs. In this paper, the structure of the maximal arcs is examined to give geometric and algebraic methods for proving when the maximal arcs are not of Denniston type. New degree 8 maximal arcs are also constructed in PG(2,2^h), h5, h odd. This, combined with previous results, shows that every Desarguesian projective plane of (even) order greater that 8 contains a degree 8 maximal arc that is not of Denniston type.     

Ovals and Hyperovals in Desarguesian Nets

We determine the Desarguesian planes which hold r-nets with ovals and those which hold r-nets with hyperovals for every r7.     

On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order

The sporadic complete 12‐arc in PG(2, 13) contains eight points from a conic. In PG(2,q) with q>13 odd, all known complete k‐arcs sharing exactly ½(q+3) points with a conic 𝒞 have size at most ½(q+3)+2, with only two exceptions, both due to Pellegrino, which are complete (½(q+3)+3) arcs, one in PG(2, 19) and another in PG(2, 43). Here, three further exceptions are exhibited, namely a complete (½(q+3)+4)‐arc in PG(2, 17), and two complete (½(q+3)+3)‐arcs, one in PG(2, 27) and another in PG(2, 59). The main result is Theorem 6.1 which shows the existence of a (½(q^r+3)+3)‐arc in PG(2,q^r) with r odd and q≡3 (mod 4) sharing ½(q^r+3) points with a conic, whenever PG(2,q) has a (½(q^r+3)+3)‐arc sharing ½(q^r+3) points with a conic. A survey of results for smaller q obtained with the use of the MAGMA package is also presented. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 25–47, 2010     

A group extensions approach to affine relative difference sets of even order

It is shown that a group extensions approach to central relative (k+1,k-1,k,1)-difference sets of even order leads naturally to the notion of an “affine” planar map; a notion analogous to the well-known planar map corresponding to a splitting relative (m,m,m,1)-difference set. Basic properties of affine planar maps are derived and applied to give some new results regarding abelian relative (k+1,k-1,k,1)-difference sets of even order and to give new proofs, in the even order case, for some known results. The paper concludes with computational non-existence results for 10,000<k?100,000.     

Difference Sets and Hyperovals

We construct three infinite families of cyclic difference sets, using monomial hyperovals in a desarguesian projective plane of even order. These difference sets give rise to cyclic Hadamard designs, which have the same parameters as the designs of points and hyperplanes of a projective geometry over the field with two elements. Moreover, they are substructures of the Hadamard design that one can associate with a hyperoval in a projective plane of even order.     

On Small Complete Arcs and Transitive ‐Invariant Arcs in the Projective Plane

Let q be an odd prime power such that q is a power of 5 or (mod 10). In this case, the projective plane admits a collineation group G isomorphic to the alternating group A₅. Transitive G‐invariant 30‐arcs are shown to exist for every . The completeness is also investigated, and complete 30‐arcs are found for . Surprisingly, they are the smallest known complete arcs in the planes , and . Moreover, computational results are presented for the cases and . New upper bounds on the size of the smallest complete arc are obtained for .     

A Correction to “Incidence Matrices and Collineations of Finite Projective Planes” by Chat Yin Ho, Designs, Codes and Cryptography, 18 (1999), 159–162

In this note we consider the set of incidence matrices of a cyclic projective planes whose set of points has cardinality a prime number p. We provide a correct proof of a result of Ho, showing that there exists an incidence matrix that possesses exactly p distinct eigenvalues.     

Spherical Designs and Generalized Sum-Free Sets in Abelian Groups

We extend the concepts of sum-freesets and Sidon-sets of combinatorial number theory with the aimto provide explicit constructions for spherical designs. We calla subset S of the (additive) abelian group G t-free if for all non-negative integers kand l with k+l t, the sum of k(not necessarily distinct) elements of S does notequal the sum of l (not necessarily distinct) elementsof S unless k=l and the two sums containthe same terms. Here we shall give asymptotic bounds for thesize of a largest t-free set in Z _n,and for t 3 discuss how t-freesets in Z _n can be used to constructspherical t-designs.     

An extension of building sets and relative difference sets

Building sets are a successful tool for constructing semi‐regular divisible difference sets and, in particular, semi‐regular relative difference sets. In this paper, we present an extension theorem for building sets under simple conditions. Some of the semi‐regular relative difference sets obtained using the extension theorem are new in the sense that their ambient groups have smaller ranks than previously known. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 50–57, 2000     

Some remarks on orders of projective planes,planar difference sets and multipliers

In this paper we investigate how finite group theory, number theory, together with the geometry of substructures can be used in the study of finite projective planes. Some remarks concerning the function v(x)= x ² + x + 1are presented, for example, how the geometry of a subplane affects the factorization of v(x). The rest of this paper studies abelian planar difference sets by multipliers.Partially supported by NSA grant MDA904-90-H-1013.     

Constructions of Partial Difference Sets and Relative DifferenceSets Using Galois Rings

We use Galois rings to construct partial difference sets and relative difference sets in non-elementary abelianp-groups. As an example, we also use Galois ringG R(4, 2) to construct a (96,20,4) difference set in Z₄ × Z₄ × Z₆.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

On the Non-existence of Semiregular Relative Difference Sets in Groups with All Sylow Subgroups Cyclic

We show that a group with all Sylow subgroups cyclic (other than ) cannot contain a normal semiregular relative difference set (RDSs). We also give a new proof that dihedral groups cannot contain (normal) semiregular RDSs either.     

Coprime ordering of cyclic planar difference sets

Motivated by a question about uniform dessins d’enfants, it is conjectured that every cyclic planar difference set of prime power order m≠4 can be cyclically ordered such that the difference of every pair of neighbouring elements is coprime to the module v?m²+m+1. We prove that this is the case whenever the number ω(v) of different prime divisors of v is less than or equal to 3. To achieve this we consider a graph related to the difference set and show that it is Hamiltonian.     

Minimal Blocking Sets in PG(2, 8) and Maximal Partial Spreads in PG(3, 8)

We prove that PG(2, 8) does not contain minimal blocking sets of size 14. Using this result we prove that 58 is the largest size for a maximal partial spread of PG(3, 8). This supports the conjecture that q ²–q+ 2 is the largest size for a maximal partial spread of PG(3, q), q>7.     

Criterion for Invertibility of Endomorphisms and Test Rank of Metabelian Products of Abelian Groups

Let G be a finitely generated torsion-free group, which is a metabelian product of Abelian groups. We establish a criterion for endomorphisms satisfying some extra restriction to be invertible in G. Test sets for IA-endomorphisms of G are described and its test rank computed.     

On Complete Arcs Arising from Plane Curves

We point out an interplay between -Frobenius non-classical plane curves and complete -arcs in . A typical example that shows how this works is the one concerning an Hermitian curve. We present some other examples here which give rise to the existence of new complete -arcs with parameters and being a power of the characteristic. In addition, for q a square, new complete -arcs with either and or and are constructed by using certain reducible plane curves.     

Cyclic Relative Difference Sets and their p-Ranks

By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q ^3k –1)/(q–1), q–1, q ^3k–1, q ^3k–2) relative difference sets, where q=3 ^e . These relative difference sets are liftings of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.