The Ferrers-diagram rank-metric code dimension conjecture

Let ${\cal F}$ be a Ferrers diagram and let $\delta$ be a positive integer. For each $i$ with $0\leq i\leq\delta-1$, let $\nu_i$ be the number of dots in ${\cal F}$ outside the first $i$ rows and the rightmost $\delta-1-i$ columns. The quantity $\min_i\{\nu_i\}$ is the upper bound for the dimension of a Ferrers-diagram rank-metric code with minimum rank distance $\delta$. Ferrers-diagram rank-metric code dimension conjecture. The upper bound of Theorem~ is attainable for any given set of parameters $q$, ${\cal F}$, and $\delta$. This conjecture asserts the existence of an optimal Ferrers-diagram rank-metric code for every Ferrers diagram and minimum rank distance; such codes would meet the general upper bound and are central to constructions of constant-dimension codes.