Extensions of Atanassov’s methods for Halton sequences

We extend Atanassov’s methods for Halton sequences in two different directions: (1) in the direction of Niederreiter (t,s)−sequences, (2) in the direction of generating matrices for Halton sequences. It is quite remarkable that Atanassov’s method for classical Halton sequences applies almost “word for word” to (t,s)−sequences and gives an upper bound quite comparable to those of Sobol’, Faure and Niederreiter. But Atanassov also found a way to improve further his bound for classical Halton sequences by means of a clever scrambling producing sequences which he named modified Halton sequences. We generalize his method to nonsingular lower triangular matrices in the last part of this article.