On the discrepancy of generalized Niederreiter sequences

First, we propose a notion of (t,e,s)-sequences in base bb, where e is an integer vector (e1,…,es)(e1,…,es) with ei≥1ei≥1 for i=1,…,si=1,…,s, which are identical to (t,s)(t,s)-sequences in base bb when e=(1,…,1), and show that a generalized Niederreiter sequence in base bb is a (0,e,s)-sequence in base bb, where eiei is equal to the degree of the base polynomial for the ii-th coordinate. Then, by using the signed splitting technique invented by Atanassov, we obtain a discrepancy bound for a (t,e,s)-sequence in base bb. It follows that a (unanchored) discrepancy bound for the first N>1N>1 points of a generalized Niederreiter sequence in base bb is given as NDN≤(1s!∏i=1s2⌊bei/2⌋eilogb)(logN)s+O((logN)s−1), where the constant in the leading term is asymptotically much smaller than the one currently known.