The b-adic Diaphony as a Tool to Study Pseudo-randomness of Nets

We consider the b-adic diaphony as a tool to measure the uniform distribution of sequences, as well as to investigate. pseudo-random properties of sequences. The study of pseudo-random properties of uniformly distributed nets is extremely important. for quasi-Monte Carlo integration. It is known that the error of the quasi-Monte Carlo integration depends on the distribution of the points of the net. On the other hand, the b-adic diaphony gives information about the points distribution of the net. Several particular constructions of sequences (x(i)) are considered. The b-adic diaphony of the two dimensional nets {y(i) = x(i), x(i-1))} is calculated numerically. The numerical results show that if the two dimensional net {y(i)} is uniformly distributed and the sequence (x(i)) has good pseudo-random properties, then the value of the b-adic diaphony decreases with the increase of the number if the points. The analysis of the results shows a direct relation between pseudo-randomness of the points of the constructed sequences and nets and the b-adic diaphony as well as the discrepancy.