The Halton, Sobol, and Faure sequences and the Braaten-Weller construc
tion of the generalized Halton sequence are studied in order to assess
their applicability for the quasi Monte Carlo integration with large
number of variates. A modification of the Halton sequence (the Halton
sequence leaped) and a new construction of the generalized Halton sequ
ence are suggested for unrestricted number of dimensions and are shown
to improve considerably on the original Halton sequence. Problems ass
ociated with estimation of the error in quasi Monte Carlo integration
and with the selection of test functions are identified. Then an estim
ate of the maximum error of the quasi Monte Carlo integration of nine
test functions is computed for up to 400 dimensions and is used to eva
luate the known generators mentioned above and the two new generators.
An empirical formula for the error of the quasi Monte Carlo integrati
on is suggested.