Nov\'{a}k's conjecture on cyclic Steiner triple systems and its generalization

Nov'{a}k conjectured in 1974 that for any cyclic Steiner triple systems of order $v$ with $v\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}6)$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic $(v,k,\lambda )$-designs with $1\le \lambda \le k-1$. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all $v$, but we conjecture that it holds whenever $v$ is sufficiently large compared to $k$. We confirm that the generalization of the conjecture holds when $v$ is prime and $\lambda =1$ and also when $\lambda \le (k-1)\mathrm{/}2$ and $v$ is sufficiently large compared to $k$. As a corollary, we show that for any $k \geq 3$,withthepossibleexceptionoffinitelymanycompositeorders$v$, every cyclic $(v,k,1)$-design without short orbits is generated by a $(v,k,1)$-disjoint difference family.