Blocking sets in finite projective spaces and coding theory

A small minimal k-blocking set is a point set B in the finite projective space PG(n,q), meeting every (n-k)-space, where |B| < 3(q^k+1)/2, and such that no proper subset of B is a k-blocking set. The first chapter provides an extensive overview of the notions that will be used in the dissertation. In the second chapter, we derive a new bound on the size of a minimal k-blocking set and show a unique reducibility result for k-blocking sets of size at most 2q^k. The linearity conjecture for blocking sets states that all small minimal blocking sets are linear sets. In the third chapter, we investigate linear sets on a line, and prove a result on the intersection of a linear set with a subline, that will be used in the fourth chapter to prove the linearity conjecture for k-blocking sets in PG(n,p^3), where p is prime. In the fifth chapter, we turn our attention to partial covers of PG(n,q). These are sets of hyperplanes, covering almost all points of the projective space; dually, they are almost 1-blocking sets. Assuming that there is at least one hyperplane that is not blocked by an almost 1-blocking set D, we deduce the minimum number of hyperplanes that are not blocked by D and extend this result to almost k-blocking sets. In the last two chapters, we deal with the codes arising from finite geometric structures. We first study the code of points and hyperplanes in PG(n,q), and show, using the theory of blocking sets, that there are no codewords with weight larger than the weight of a hyperplane and smaller that the difference of two hyperplanes. We extend this result for the code of points and k-spaces in PG(n,q) and investigate the dual code, deducing a new upper bound on the minimum weight. Finally, we investigate the LDPC codes arising from linear representations and polar spaces; we deduce bounds on the minimum weight, and characterise codewords of small and large weight using geometric techniques