Some linear lattices combine very good properties (e.g. highest density, maximum kissing number, etc.) like the hexagonal lattice (in 2-dimensional space R2), the E8 (in R8) or the famous Leech lattice (=E24, in R24). However, not in all dimensions such extraordinary good lattices are known (only in d=1,2,8,24); and this is for several good geometrical reasons, which become clear when inspecting these very good lattices and "good" dimensions. So we know why just 1,2,8,24 are so special.
For Monte-Carlo integration usually random samples are used (usually standard-uniform samples, so X in (0,1)d). However, for variance reduction, we can also use N d-dimensional lattice points X=X1,X2,...Xn). But for MC the discrepancy D matters mostly regarding integration error (not kissing number, etc.). D gets worse for higher dimensions e.g. roughly following (logN)^d/N. So for discrepancy there are almost no known exceptional "good" dimensions; actually only d=1 and 2 are (a bit) exceptional.
So I wonder: Is d=8 or d=24 also "special" regarding discrepancy or MC integration, e.g.:
- In those dimensions it might be easier to construct low-discrepancy sets?
- Or even a extraordinary low D can be achieved in those d?
- Or one can prove that even in these d no special characteristics exist regarding D?
- Actually (by far) not only D matters regarding integration error, but e.g. also the coverage in lower-dimensional projections, so maybe on these d=8 or 24 may give advantages (or not).
Something similar to #1 is often the case for prime numbers N. And in d=2 explicit constructions are available giving optimum D (e.g. using Fibonacci numbers). If #3 can be proven, then #2 is disproved, which would be helful too for me - although still #1 could be possible, offering at least practical advantages. A comment regarding #4: Each variable xi itself (1d-projection) can be made optimum distributed (like delta-x=1/N) by using latin hypercube sets. So if we could extend this to higher sub-dimensions by means of E8 or E24 it would be very helpful too (even if D is non-optimum). Note: Ignoring E8 and E24, one can improve LHS a bit by using methods like orthogonal LHS. In 2D the Fibonacci lattice is giving lowest discrepancy, and it can be created by rotating/shearing a Z2 lattice. I wonder if something similar is possible in d=8 as well?