  We consider linear systems $Ax = b$ where $A \in \mathbb{R}^{m \times n}$
consists of normalized rows, $\|a_i\|_{\ell^2} = 1$, and where up to $\beta m$
entries of $b$ have been corrupted (possibly by arbitrarily large numbers).
Haddock, Needell, Rebrova and Swartworth propose a quantile-based Random
Kaczmarz method and show that for certain random matrices $A$ it converges with
high likelihood to the true solution. We prove a deterministic version by
constructing, for any matrix $A$, a number $\beta_A$ such that there is
convergence for all perturbations with $\beta < \beta_A$. Assuming a random
matrix heuristic, this proves convergence for tall Gaussian matrices with up to
$\sim 0.5\%$ corruption (a number that can likely be improved).
