Nets and sequences obtained from irreducible polynomials over finite fields (Q1277248)

Arrange all monic irreducible polynomials over the finite field \(F_q\) in a sequence \(p_1, p_2,\dots\) according to nondecreasing degrees, and for any positive integer \(s\) put \(T_q(s)= \sum_{i=1}^s (\deg(p_i)-1)\). Then it is shown that for any prime powers \(q_1<q_2\) we have \(T_{q_1}(s)\geq T_{q_2}(s)\) for all \(s\), with strict inequality for \(s> q_1\). This settles a conjecture of \textit{G. L. Mullen, A. Mahalanabis} and \textit{H. Niederreiter} [Lect. Notes. Statist. 106, 58-86 (1995; Zbl 0838.65004)]. The result has implications for the construction of \((t,m,s)\)-nets and \((t,s)\)-sequences due to \textit{H. Niederreiter} [J. Number Theory 30, 51-70 (1988; Zbl 0651.10034)].