Blocking subspaces with points and hyperplanes

In this paper, we characterise the smallest sets $B$ consisting of points and hyperplanes in $\text{PG}(n,q)$, such that each $k$-space is incident with at least one element of $B$. If $k > \frac {n-1} 2$, then the smallest construction consists only of points. Dually, if $k < \frac{n-1}2$, the smallest example consists only of hyperplanes. However, if $k = \frac{n-1}2$, then there exist sets containing both points and hyperplanes, which are smaller than any blocking set containing only points or only hyperplanes.Comment: 7 pages. UPDATE: After publication of this paper, we found out that in case $k = \frac{n-1}2$, the correct lower bound and a classification of the smallest examples was already established by Blokhuis, Brouwer, and Sz\H{o}nyi [A. Blokhuis, A. E. Brouwer, T. Sz\H{o}nyi. On the chromatic number of $q$-Kneser graphs. Des. Codes Crytpogr. 65:187-197, 2012

Maximal cocliques in the Kneser graph of point-line flags in PG(4,q)

We determine the maximal cocliques of size >_ 4q2 + 5q + 5 in the Kneser graph on point-plane ags in PG(4; q). The maximal size of a coclique in this graph is (q2 + q + 1)(q3 + q2 + q + 1)

The largest Erd\H{o}s-Ko-Rado sets in 2-(v,k,1) designs

An Erd\H{o}s-Ko-Rado set in a block design is a set of pairwise intersecting blocks. In this article we study Erd\H{o}s-Ko-Rado sets in 2-(v,k,1) designs, Steiner systems. The Steiner triple systems and other special classes are treated separately. For unitals we also determine an upper bound on the size of the second-largest maximal Erdos-Ko-Rado sets

Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author