Numerical cubature using error-correcting codes
Greg Kuperberg
TL;DR
The paper addresses the challenge of constructing high-dimensional, equal-weight t-cubature formulas with as few points as possible. It introduces a thinning method based on convolution of equal-weight product formulas and linear error-correcting codes, notably BCH codes, to produce a smaller, yet valid, t-cubature set. This yields explicit, positive, interior formulas for regions like the n-cube, S^{n-1}, B_n, Δ_n, and Gaussian R^n with point counts such as O(n^{\floor{t/2}}) (cube with q=2), O(n^{t-2}) for odd t on several measures, and O(n^{t-1}) for simplices, among others, often beating the non-constructive Tchakaloff bound and approaching Stroud-type lower bounds. By linking code duals to orthogonal arrays and leveraging product-convolution structures, the work provides asymptotically efficient cubature constructions and connects to independent results by Victoir, while highlighting practical thinning strategies via Hadamard and Kerdock designs for high-dimensional integration.
Abstract
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 $n \to \infty$. 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 $\R^n$ with $O(n^{t-2})$ points when $t$ is odd. When $μ$ is spherically symmetric and $t=5$, we obtain $O(n^2)$ points. For each $t \ge 4$, we also obtain explicit, positive, interior formulas for the $n$-simplex with $O(n^{t-1})$ 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.