On Legendre Symbol Lattices, Ii

In Part I of this paper we constructed a two dimensional binary lattice by using the Legendre symbol and polynomials of two variables, and we studied its pseudorandom properties. We proved that if the polynomial is non- degenerate then under certain conditions the lattice possesses strong pseudoran- dom properties, while in the degenerate case it may occur that the lattice has only weak pseudorandom properties. In this paper we continue our analysis of the degenerate case and we will give both lower and upper bounds for the pseu- dorandom measures of the lattices. We will also give an algorithm to decide if a polynomial is degenerate. Finally, we shall construct a large family of non- degenerate polynomials satisfying one of the sufficient conditions for which the corresponding lattices have strong pseudorandom properties.