A useful result in searches for good vertex-modified number-theoretic rules

Number-theoretic rules are particularly suited to the evaluation of multiple integrals in which the integrand is periodic. For nonperiodic integrands, an alternative is to use vertex-modified versions of number-theoretic rules. Good vertex-modified number-theoretic rules may be found by doing a computer search based on some criterion of goodness. Such criteria include a variant of the L2 discrepancy and the vertex variance. Here we present a result which may be used to speed up searches for good vertex-modified number-theoretic rules based on these criteria when the generator vectors are of the one-parameter form of Korobov.