Simeon Ball

Maximal arcs in Desarguesian planes of odd order do not exist

Forq an odd prime power, and 1<n<q, the Desarguesian planePG(2,q) does not contain an(nq−q+n,n)-arc.

On the Number of Slopes of the Graph of a Function Defined on a Finite Field

Given a setUof sizeqin an affine plane of orderq, we determine the possibilities for the number of directions of secants ofU, and in many cases characterize the setsUwith given number of secant directions.

On Ovoids of Parabolic Quadrics

It is known that every ovoid of the parabolic quadric Q(4, q), q=ph, p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p=2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points.We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q>3.We conclude with a 1 mod p result for ovoids of Q(6, q), q=ph, p prime.

The number of directions determined by a function over a finite field

A proof is presented that shows that the number of directions determined by a function over a finite field GF(q) is either 1, at least (q + 3)/2, or between q/s + 1 and (q - 1)/(s - 1) for some s where GF(s) is a subfield of GF(q). Moreover, the graph of those functions that determine less than half the directions is GF(s)-linear. This completes the unresolved cases s = 2 and 3 of the main theorem in Blokhuis et al. (J. Combin. Theory Ser. A 86 (1999) 187).

Bounds on (n,r)-arcs and their application to linear codes

This article reviews some of the principal and recently-discovered lower and upper bounds on the maximum size of (n,r)-arcs in PG(2,q), sets of n points with at most r points on a line. Some of the upper bounds are used to improve the Griesmer bound for linear codes in certain cases. Also, a table is included showing the current best upper and lower bounds for q=<19, and a number of open problems are discussed.

An easier proof of the maximal arcs conjecture

It was a long-standing conjecture in finite geometry that a Desarguesian plane of odd order contains no maximal arcs. A rather inaccessible and long proof was given recently by the authors in collaboration with Mazzocca. In this paper a new observation leads to a greatly simplified proof of the conjecture.

On the classification of semifield flocks

Abstract A classical lemma of Weil is used to characterise quadratic polynomials f with coefficients GF ( q n ), q odd, with the property that f ( x ) is a non-zero square for all x ∈ GF ( q ). This characterisation is used to prove the main theorem which states that there are no subplanes of order q contained in the set of internal points of a conic in PG (2, q n ) for q ⩾4 n 2 −8 n +2. As a corollary to this theorem it then follows that the only semifield flocks of the quadratic cone of PG (3, q n ) for those q exceeding this bound are the linear flocks and the Kantor–Knuth semifield flocks.

On Intersection Sets in Desarguesian Affine Spaces

Lower bounds on the size of t -fold blocking sets with respect to hyperplanes or t -intersection sets in AG(n, q) are obtained, some of which are sharp.

Symplectic Spreads

We construct an infinite family of symplectic spreads in spaces of odd rank and characteristic.

Punctured combinatorial Nullstellensätze

In this article we extend Alon’s Nullstellensatz to functions which have multiple zeros at the common zeros of some polynomials g1,g2, …, gn, that are the product of linear factors. We then prove a punctured version which states, for simple zeros, that if f vanishes at nearly all, but not all, of the common zeros of g1(X1), …,gn(Xn) then every residue of f modulo the ideal generated by g1, …, gn, has a large degree.This punctured Nullstellensatz is used to prove a blocking theorem for projective and affine geometries over an arbitrary field. This theorem has as corollaries a theorem of Alon and Füredi which gives a lower bound on the number of hyperplanes needed to cover all but one of the points of a hypercube and theorems of Bruen, Jamison and Brouwer and Schrijver which provides lower bounds on the number of points needed to block the hyperplanes of an affine space over a finite field.