The length conjecture for optimal locally recoverable codes

Let $q$ be the alphabet size, and let an optimal locally recoverable code have minimum distance $d$ and locality $r$. Length conjecture for optimal locally recoverable codes. Every optimal locally recoverable code with minimum distance $d$ and locality $r$ has length upper bounded by

$$q^{1+O\left(\frac1d\right)}.$$

The conjecture concerns the asymptotic length of optimal locally recoverable codes. The paper notes that known constructions achieve lengths of order $q^{1+O(1/d)}$, while an extremal-graph-theoretic upper bound for the construction in the paper is $O\left(q^{1+\frac{2}{\lfloor (d-1)/2 \rfloor}}\right)$; the general asserted upper bound remains open.