Perfectly equidistributed Quasi-Monte Carlo sequences from Artin-Schreier polynomials

Abstract

To numerically integrate a function, one may resort to Quasi-Monte Carlo estimators, that average integrand values at pseudo-random well-distributed uniform sampling locations. Better uniformity improves the worst-case integration-error bound. A standard measure of uniformity is given by an integer $t$ value, where $t=0$ yields the best uniformity. Producing sequences of samples with bounded $t$ values can be achieved with Sobol' recursive construction, that uses coefficients of irreducible polynomials. While $b$-dimensional sequences with $t=0$ can be obtained by taking $b$ polynomials of degree $1$ over the Galois Field $\mathrm{GF}(b)$, we show conditions that guarantee $t=0$ for specific higher degree polynomials. In particular, we relate the Sobol' construction to tensorized powers of Pascal matrices when the chosen polynomials only differ by a constant and exhibit simple conditions to guarantee $t=0$ in this case. We then focus on Artin-Schreier irreducible polynomials, in the form $p_i(x) = x^b - x + c_i$, where $i \in \{1, \dots, b-1\}$ and $b$ is prime, and we make explicit conditions that always guarantees $t=0$ in $b-1$ dimensions. Combining $b$-dimensional Sobol' of degree $1$ and our $(b-1)$-dimensional Artin-Schreier sequence of degree $b$, we provide a fast greedy procedure that optimizes the $(2b-1)$-dimensional combined $t$ value, while guaranteeing $t=0$ projection in subspaces.