Leo Storme

The packing problem in statistics, coding theory and finite projective spaces

In the last few decades, nite projective spaces or, equivalently, Galois geometries have been studied intensively. Apart from being an interesting and exciting area in combinatorics with beautiful results, this eld has many connections with statistics and coding theory. Indeed, several problems have equivalent formulations in the di erent areas. A problem, rst studied in statistics by Fisher [56, 57], has proved to be equivalent to a problem in geometry [25]. In [22, 24], Bose generalized this application of nite projective geometry for the design of experiments and called it the packing problem. He also presented, in 1961, connections between the design of experiments and coding theory [24, 25]. The central problem posed in these articles is the determination of m(n; r; s;N; q), the largest size of a point set, as de ned in Section 1.2. After initial consideration by Bose and his followers as a statistical problem, and further interest by Kustaanheimo [94] and other Finnish astronomers, the topic was taken up by Segre [128, 132] and his followers, perhaps because of Segre’s interest in algebraic geometry over nite elds. Using geometric methods, many fundamental results were obtained. Coding theory provides a second motivation for these problems, which have equivalent formulations in nite projective spaces and coding theory. This amounts in coding theory to studying the row space of a generator matrix of a code and in Galois geometry to studying the column space. The classical example, that is, the equivalence of linear maximum distance separable (MDS) codes and arcs in projective spaces, has

The packing problem in statistics, coding theory and finite projective spaces: Update 2001

This article updates the authors’ 1998 survey [134] on the same theme that was written for the Bose Memorial Conference (Colorado, June 7–11, 1995). That article contained the principal results on the packing problem, up to 1995. Since then, considerable progress has been made on different kinds of subconfigurations.

Generating the group of reversible logic gates

Reversible logic plays a fundamental role both in ultra-low power electronics and in quantum computing. It is therefore important to have an insight into the structure of the group formed by the reversible logic gates and their cascading into reversible circuits. Such insight is gained from constructing chains of maximal subgroups. The subgroup of control gates plays a prominent role, as it is a Sylow 2-subgroup.

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.

Lacunary polynomials, multiple blocking sets and Baer subplanes.

New lower bounds are given for the size of a point set in a Desarguesian projective plane over a finite field that contains at least a prescribed number s of points on every line. These bounds are best possible when q is square and s is small compared with q. In this case the smallest set is shown to be the union of disjoint Baer subplanes. The results are based on new results on the structure of certain lacunary polynomials, which can be regarded as a generalization of Redeis results in the case when the derivative of the polynomial vanishes.

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.

On a Particular Class of Minihypers and Its Applications

A particular class of minihypers was studied previously by the authors (in press, Des. Codes Cryptogr.). For q square, this paper improves the results of that work, under the assumption that no weights occur in the minihyper. Using the link between these minihypers and maximal partial s-spreads of PG(N, q), (s+1) ?(N+1), the findings on minihypers translate immediately into results on the extendability of partial s-spreads with small positive deficiency. Other applications of this characterisation of minihypers are given by P. Govaerts et al. (in press, European J. Combin.).

Small Complete Caps in Spaces of Even Characteristic

In 1959, Segre constructed a complete (3q+2)-cap inPG(3, q),qeven. This showed that the size of the smallest completek-cap inPG(3, q),qeven, is almost equal to the trivial lower bound which is of orderformula]. Generalizing the construction of Segre, complete (qn+3(qn?1+?+q)+2)-caps inPG(2n, q),qeven,q?4, and complete (3(qn+?+q)+2)-caps inPG(2n+1, q),qeven,q?4, are constructed. This shows that in all spacesPG(2n+1, q),qeven, the size of the smallest completek-cap is almost equal to the trivial lower bound which is of orderformula].

On a Particular Class of Minihypers and Its Applications. I. The Result for General q

This article is the first in a series of three articles that discuss a particular class of minihypers and its applications. Proving that for small δ and μ < N, a {δvμ + 1, δvμ; N, q}-minihyper consists of a sum of δ μ-spaces, we show that the excess points of an s-cover with excess δ of PG(N, q), (s + 1)|(N + 1), form a sum of δ s-spaces, and that no maximal partial s-spreads with deficiency δ of PG(N, q), (s + 1)|(N + 1), exist. The case q square will be studied in greater detail in [7] and further applications of these classification results on this class of minihypers will be published in [8].

Small Minimal Blocking Sets inPG(2, q3)

We extend the results of Polverino (1999, Discrete Math., 208/209, 469?476; 2000, Des. Codes Cryptogr., 20, 319?324) on small minimal blocking sets in PG(2,p3 ), p prime, p? 7, to small minimal blocking sets inPG (2, q3), q=ph, p prime, p? 7, with exponent e?h. We characterize these blocking sets completely as being blocking sets of Redei-type.