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Low-discrepancy sequences and global function fields with many rational places. (English) Zbl 0893.11029

Digital \((t,s)\)-sequences in a prime power base \(q\) for which the quality parameter \(t\) has the least possible order of magnitude are constructed. The construction uses algebraic function fields over the finite field \(GF(q)\). Let \(g(K/GF(q))\) denote the genus of the global function field \(K/GF(q)\) and set \(V_q(s)= \min q(K/GF(q))\), where the minimum is extended over all global function fields such that the number of places of degree 1 is not less than \(s+1\). The author establishes various bounds for this important quantity \(V_q(s)\). For instance, it is shown that for all prime powers \(q\) the estimate \[ V_q(s)\leq {c\over \log q} s+1 \] holds (with an absolute constant \(c>0)\). The proof follows from an application of J.-P. Serre’s bounds for the number of places of degree 1 [Harvard University, Lecture Notes (1985)].
Reviewer: R.F.Tichy (Graz)

MSC:

11K45 Pseudo-random numbers; Monte Carlo methods
11R58 Arithmetic theory of algebraic function fields
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