7 problems

Let $r,s$ be positive integers, let $n:=r+s$, and consider the Johnson scheme $J(2n,n)$. A tight symmetric inner distribution code is a code for which equality holds in the relevan…

Diameter-perfect-code conjecture. There are no nontrivial diameter perfect codes in $J(n,w)$, except for Steiner systems and their complements.

Delandtsheer's conjecture. There are no nontrivial perfect codes in $J(n,w)$.

Large-set conjecture. There is a function $L$ such that, if $n\geqslant L(k)$, then the Johnson scheme $J(n,k)$ is non-synchronizing if and only if there exists a large set of Stei…

Divisibility conjecture. There is a function $H$ such that, if $n\geqslant H(k)$, then the Johnson scheme is non-separating if and only if the divisibility conditions for…

Non-separation conjecture. There is a function $G$ such that, if $n\geqslant G(k)$ and the Johnson scheme $J(n,k)$ is non-separating, then there exists a Steiner system $S(t,k,n)$…

The largest 4-distance set conjecture. The set