Exact Minimum Extension Degrees for Reed Solomon Codes under Linear and Nonlinear Repair at the Cut Set Bound

We study the minimum extension degree required for a Reed--Solomon code to admit exact $\mathbb{F}_q$-linear repair at the cut-set bound from every prescribed set of $d$ helpers. Let $s=d-k+1$, and let $Q_s(n)$ be the minimum product of $n$ pairwise coprime integers, each at least $s$. For every pri