  The {\em discrepancy} of a matrix $M \in \mathbb{R}^{d \times n}$ is given by
$\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n}
\|M\boldsymbol{x}\|_\infty$. An outstanding conjecture, attributed to Koml\'os,
stipulates that $\mathrm{DISC}(M) = O(1)$, whenever $M$ is a Koml\'os matrix,
that is, whenever every column of $M$ lies within the unit sphere. Our main
result asserts that $\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2})$ holds
asymptotically almost surely, whenever $M \in \mathbb{R}^{d \times n}$ is
Koml\'os, $R \in \mathbb{R}^{d \times n}$ is a Rademacher random matrix, $d =
\omega(1)$, and $n = \omega(d \log d)$. The factor $d^{-1/2}$ normalising $R$
is essentially best possible and the dependency between $n$ and $d$ is
asymptotically best possible. Our main source of inspiration is a result by
Bansal, Jiang, Meka, Singla, and Sinha (ICALP 2022). They obtained an assertion
similar to the one above in the case that the smoothing matrix is Gaussian.
They asked whether their result can be attained with the optimal dependency $n
= \omega(d \log d)$ in the case of Bernoulli random noise or any other types of
discretely distributed noise; the latter types being more conducive for
Smoothed Analysis in other discrepancy settings such as the Beck-Fiala problem.
For Bernoulli noise, their method works if $n = \omega(d^2)$. In the case of
Rademacher noise, we answer the question posed by Bansal, Jiang, Meka, Singla,
and Sinha. Our proof builds upon their approach in a strong way and provides a
discrete version of the latter. Breaking the $n = \omega(d^2)$ barrier and
reaching the optimal dependency $n = \omega(d \log d)$ for Rademacher noise
requires additional ideas expressed through a rather meticulous counting
argument, incurred by the need to maintain a high level of precision all
throughout the discretisation process.
