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

Abstract

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