A sampling design uniform in every pair of columns

Formal definition

• ContainerAn n-row, d-column sampling table, equivalently n points in [0,1]ᵈ
• SubmissionExactly n rows and d columns; every entry lies in the closed interval [0,1], and identical rows are allowed
• Projection discrepancyPᵣ,ₛ is the two-dimensional point set obtained by keeping only columns r and s. The indicator 𝟙 is 1 when a point lies in the origin-anchored rectangle [0,u] × [0,v], and 0 otherwise$${{D_2}^{\ast }\left(P_{r,s}\right)}^2=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\left(\frac{1}{n}\sum _{i=1}^n𝟙\left[x_{ir}\le u,x_{is}\le v\right]-uv\right)}^{2}dudv$$
• ScoreCompute the L2-star discrepancy of all C(d,2) column pairs; the largest one is the submission's score$$\mathrm{score}\left(P\right)=\underset{1\le r<s\le d}{\max }{D_2}^{\ast }\left(P_{r,s}\right)$$
• ObjectiveMake this worst-pair score as small as possible. The verifier computes the squared discrepancy exactly and displays it rounded up at the twelfth decimal$$\underset{P\in {\left({\left[0,1\right]}^d\right)}^n}{\min }\mathrm{score}\left(P\right)$$