The use of blocking sets in Galois geometries and in related research areas

Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems

An empty interval in the spectrum of small weight codewords in the code from points and k-spaces of PG(n, q)

Let \C(n,q) be the $p$-ary linear code defined by the incidence matrix of points and $k$-spaces in \PG(n,q), $q=p^h$, $p$ prime, $h\geq 1$. In this paper, we show that there are no codewords of weight in $]\frac{q^{k+1}-1}{q-1},2q^k[$ in \C(n,q)\setminus\mathrm{C}_{n-k}(n,q)^\bot which implies that there are no codewords with this weight in \C(n,q)\setminus \C(n,q)^{\bot} if $k\geq n/2$. In particular, for the code $\mathrm{C}_{n-1}(n,q)$ of points and hyperplanes of \PG(n,q), we exclude all codewords in $\mathrm{C}_{n-1}(n,q)$ with weight in $]\frac{q^n-1}{q-1},2q^{n-1}[$. This latter result implies a sharp bound on the weight of small weight codewords of $\mathrm{C}_{n-1}(n,q)$, a result which was previously only known for general dimension for $q$ prime and $q=p^2$, with $p$ prime, $p>11$, and in the case $n=2$, for $q=p^3$, $p\geq 7$