A construction of low-discrepancy sequences using global function fields

1. Introduction. The idea of using global function fields for the construction of s-dimensional low-discrepancy sequences was first sketched by Niederreiter [9, Section 5], [10, Section 5], and the details were worked out in an improved form by Niederreiter and Xing [11]. In the method of [11] one chooses a suitable global function field which contains s elements satisfying special properties. This global function field can, for instance, be a rational function field, in which case one obtains an earlier construction of low-discrepancy sequences due to Niederreiter [7]. If one chooses certain el-liptic function fields, then it was shown in [11] that one gets improvements on the construction in [7]. Important progress in the construction of low-discrepancy sequences was achieved in the paper [12] of the authors. The key idea of the construction in [12] is to work with global function fields containing many rational places, i.e., places of degree 1. This method yields significantly better results than all previous methods. The only essential condition in this construction is that the global function field contains at least s + 1 rational places. In the present paper we describe a different construction which is also based on global function fields, but which is somewhat more explicit than the construction in [12]. This new construction is also more flexible than that in [12], since we can now use not only rational places, but also places of larger degree. Just like the method in [12], the present construction produces low-discrepancy sequences that are in a sense asymptotically optimal. The new construction produces the best results if places of small degree are used. In the case where we work only with rational places, we get the same results as in [12], but by a different method. Some examples in Section 4 demonstrate that in certain cases the new construction yields improvements on the results in [12] if we use also places of degree greater than 1.