Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes

Abstract

Belov, Logachev and Sandimirov construct linear codes of minimum distance d for roughly 1/q k/2 of the values of d < q k-1. In this article we shall prove that, for q = p prime and roughly $${\frac{3}{8}}$$ -th's of the values of d < q k-1, there is no linear code meeting the Griesmer bound. This result uses Blokhuis' theorem on the size of a t-fold blocking set in PG(2, p), p prime, which we generalise to higher dimensions. We also give more general lower bounds on the size of a t-fold blocking set in PG(?, q), for arbitrary q and ? ? 3. It is known that from a linear code of dimension k with minimum distance d < q k-1 that meets the Griesmer bound one can construct a t-fold blocking set of PG(k?1, q). Here, we calculate explicit formulas relating t and d. Finally we show, using the generalised version of Blokhuis' theorem, that nearly all linear codes over $${{\mathbb F}_p}$$ of dimension k with minimum distance d < q k-1, which meet the Griesmer bound, have codewords of weight at least d + p in subcodes, which contain codewords satisfying certain hypotheses on their supports.