We present a construction for improving numerical cubature formulas with
equal weights and a convolution structure, in particular equal-weight product
formulas, using linear error-correcting codes. The construction is most
effective in low degree with extended BCH codes. Using it, we obtain several
sequences of explicit, positive, interior cubature formulas with good
asymptotics for each fixed degree t as the dimension nββ. Using a
special quadrature formula for the interval [arXiv:math.PR/0408360], we obtain
an equal-weight t-cubature formula on the n-cube with O(n^{\floor{t/2}})
points, which is within a constant of the Stroud lower bound. We also obtain
t-cubature formulas on the n-sphere, n-ball, and Gaussian Rn with
O(ntβ2) points when t is odd. When ΞΌ is spherically symmetric and
t=5, we obtain O(n2) points. For each tβ₯4, we also obtain explicit,
positive, interior formulas for the n-simplex with O(ntβ1) points; for
t=3, we obtain O(n) points. These constructions asymptotically improve the
non-constructive Tchakaloff bound.
Some related results were recently found independently by Victoir, who also
noted that the basic construction more directly uses orthogonal arrays.Comment: Dedicated to Wlodzimierz and Krystyna Kuperberg on the occasion of
their 40th anniversary. This version has a major improvement for the n-cub