  In this paper we improve the best known constant for the discrepancy
formulated in the Komlos Conjecture. The result is based on the improvement of
the subgaussian bound for the random vector constructed in the Gram-Schmidt
Random Walk algorithm. Moreover, we present detailed argument for the smoothed
analysis of this random vector. The analysis concerns a modification of a given
matrix in the conjecture by a Gaussian type perturbation. Our result improves
the recent paper in this direction.
