Discrepancy of generalized $LS$-sequences

The $LS$-sequences are a parametric family of sequences of points in the unit interval. They were introduced by Carbone, who also proved that under an appropriate choice of the parameters $L$ and $S$, such sequences are low-discrepancy. The aim of the present paper is to provide explicit constants in the bounds of the discrepancy of $LS$-sequences. Further, we generalize the construction of Carbone and construct a new class of sequences of points in the unit interval, the generalized $LS$-sequences.