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Two popular and often applied methods to obtain two-dimensional point setswith the optimal order of $L_p$ discrepancy are digit scrambling andsymmetrization. In this paper we combine these two techniques and symmetrize$b$-adic Hammersley point sets scrambled with arbitrary permutations. It isalready known that these modifications indeed assure that the $L_p$ discrepancyis of optimal order $\mathcal{O}\left(\sqrt{\log{N}}/N\right)$ for $p\in[1,\infty)$ in contrast to the classical Hammersley point set. We prove anexact formula for the $L_2$ discrepancy of these point sets for specialpermutations. We also present the permutations which lead to the lowest $L_2$discrepancy for every base $b\in\{2,\dots,27\}$ by employing computer searchalgorithms.

$L_2$ discrepancy of symmetrized generalized Hammersley point sets in base $b$