Minimum L2-star discrepancy in the unit hypercube

Formal definition

• ContainerThe d-dimensional unit hypercube [0,1]ᵈ, coordinates in the closed interval
• SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
• L2-star discrepancy𝟙 is 1 when every coordinate of the point is at most the corresponding coordinate of u, and 0 otherwise; the integral ranges over all origin-anchored boxes$${{D_2}^{\ast }\left(P\right)}^2=\underset{{\left[0,1\right]}^d}{\int }{\left(\frac{1}{n}\sum _{i=1}^n𝟙\left[x_{ir}\le u_r,\forall r\right]-\prod _{r=1}^d{u}_r\right)}^{2}du$$
• Closed formThe verifier uses Warnock's formula to compute D₂*(P)² exactly on the nine-decimal grid$${{D_2}^{\ast }\left(P\right)}^2=\frac{1}{3^d}-\frac{2^{1-d}}{n}\sum _{i=1}^n\prod _{r=1}^d\left(1-{x_{ir}}^2\right)+\frac{1}{n^2}\sum _{i,j=1}^n\prod _{r=1}^d\left(1-\max (x_{ir},x_{jr})\right)$$
• ScoringThe record is the exact integer M·D² with the common denominator cleared; the page shows D, rounded up at the twelfth decimal
• ObjectiveMake the L2-star discrepancy as small as possible over all legal point sets P$$\underset{P}{\min }{D_2}^{\ast }\left(P\right)$$