turkmath.org

The strong cylinder conjecture, as posed in a paper by Simeon Ball, following results due to Michel Lavrauw and himself, is a simple-to-state conjecture, but has resisted all attacks so far. For any prime $p$, we can simply put it as: except for the set of points on $p$ parallel lines in $AG(3,p)$, are there other sets of $p^{2}$ points in $AG(3,p)$ such that every plane intersect it in a multiple of $p$ points? This conjecture builds on work of Rédei regarding factorizations of elementary abelian groups and has been extended by Sascha Kurz and myself to the setting of divisible codes. The latter setting provides a natural context to extend the conjecture to prime powers $q$ and higher dimensions. We will discuss the historical context and the new coding-theoretical setting in which it appears. Surprisingly, it turns out that higher-dimensional generalizations are equivalent to the original conjecture. We will see that the latter is true for small primes by combinatorial techniques, but fails for almost all prime powers, starting with $q=8$. This shows that the combinatorial approach quickly reaches its limits and new techniques are necessary to advance us to a full solution.