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The packing problem in statistics, coding theory and finite projective spaces: Update 2001. (English) Zbl 1025.51012
Blokhuis, A. (ed.) et al., Finite geometries. Proceedings of the fourth Isle of Thorns conference, Brighton, UK, April 2000. Dordrecht: Kluwer Academic Publishers. Dev. Math. 3, 201-246 (2001).

The packing problem in statistics, coding theory and finite projective spaces regards the determination of the maximal or minimal sizes of given subconfigurations of finite projective spaces. The problem is not only interesting from a geometrical point of view; it also arises when coding theoretical problems and problems from the design of experiments are translated into equivalent geometrical problems. The geometrical interest in the packing problem and the links with problems investigated in other research fields have given this problem a central place in Galois geometries, that is, the study of finite projective spaces.

The paper under review is an excellent survey of recent results concerning the packing problem. That paper updates the authors’ 1998 survey on the same theme that was written for Bose Memorial Conference (Colorado, June 7-11, 1995) [J. Stat. Plann. Inference 72, No. 1-2, 355-380 (1998; Zbl 0958.51013)]. Since then, considerable progress has been made on the following kinds of subconfigurations: n-arcs in PG(2,q), n-arcs in PG(N,q), n-caps in PG(N,q), (n,r)-arcs in PG(2,q), multiple blocking sets in PG(2,q), blocking sets in PG(N,q), n-tracks and almost MDS codes, minihypers.

Many open questions are presented in the paper, and a comprehensive bibliography of 229 references is compiled.

51E23Spreads and packing problems (geometry)
94B05General theory of linear codes
51E22Linear codes and caps in Galois spaces