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

arXiv:2610.34377v1 Announce Type: new Abstract: 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 prime power $q$ and all $3\le k2d-2$, we obtain strictly smaller extension degrees, showing that the range restriction cannot be omitted altogether. Finally, we prove that a nonlinear repair scheme attaining the cut-set bound induces an $\mathbb{F}_p$-linear repair scheme on the same code, where $p$ is the characteristic. Consequently, the extension-degree lower bounds over the prime field remain valid without a linearity assumption.