10 problems

Levenshtein's conjecture. This property should hold for every $t\geqslant 3$: namely, the relevant boundary case $g(t+1,t)=1$, or equivalently, a PSCA$(t+1,t,1)$ should exist.

Kløve's generalized conjecture. For every positive integer $d$,

Let $M_n$ be the maximum length of a directed cycle of permutations in $\mathcal S_n$ satisfying the Kendall tau distance constraint. For odd $n\geq 5$, Horowitz and Etzion proved…

Nonexistence conjecture. There are no such $D$-diameter perfect codes; equivalently, the answer to the question is no. This is posed as an open problem concerning diameter-perfect…

Let $S_{2n+1}$ be the symmetric group, and let $C$ be a $(2n+1,M,\mathcal{K})$-snake, where $M$ is its number of permutations. Optimal-size conjecture. The optimal size is … The pa…

Let $S_n$ be the symmetric group on $n$ elements, equipped with Kendall's $\tau$-distance. Let $e=[1,2,\ldots,n]$ be the identity permutation, and let ${\cal S}(e,R)$ be the sphere…

Let $S_n$ be the symmetric group on $n$ elements, equipped with Kendall's $\tau$-distance. An anticode is a subset of $S_n$ whose maximum pairwise Kendall's $\tau$-distance is boun…

Conjecture on optimal anticode structure. For any $n$, the optimal anticode of maximal distance $d-1$ has the above form. In particular,

Variant I monotonicity conjecture. If $J>1$, then the optimal $n_i$ for Variant I CPCs increase monotonically with $i\in\mathcal{S}_1$ and decrease monotonically with…

Monotonicity conjecture. If $J>1$ and $E[\eta_\ell]$ is convex in $\ell$, then the optimal multiplicities $n_i$ for Variant II CPCs increase monotonically with $i$.