On the discrepancy of generalized Niederreiter sequences

First, we propose a notion of (t,e,s)-sequences in base b, where e is an integer vector (e₁,.,e_s) with e_i≥1 for i=1,.,s, which are identical to (t,s)-sequences in base b when e=(1,.,1), and show that a generalized Niederreiter sequence in base b is a (0,e,s)-sequence in base b, where ^ei is equal to the degree of the base polynomial for the i-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 b. It follows that a (unanchored) discrepancy bound for the first N>1 points of a generalized Niederreiter sequence in base b is given as ND_N≤(1/ s!aπi=1s2Š^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.