Abstract : The cross-combined measure (which is a natural extension of crosscorrelation measure) is introduced and important constructions of large families of binary lattices with nearly optimal cross-combined measures are presented. These results are important in the study of large families of pseudorandom binary lattices but they are also strongly related to the one dimensional case: An easy method is showed obtaining strong constructions of families of binary sequences with nearly optimal cross-correlation measures based on the previous constructions of lattices. The important feature of this result is that so far there exists only one type of constructions of very large families of binary sequences with small cross-correlation measure, and this only type of constructions was based on one-variable irreducible polynomials. Since it is very complicated to construct one-variable irreducible polynomials over $\mathbb{F}_p$, it became necessary to show other types of constructions where the generation of sequences are much faster. Using binary lattices based on two-variable irreducible polynomials this problem can be avoided, however a slightly weaker upper bound is obtained for the cross-correlation measure than in the original construction. (But, contrary to one-variable polynomials, using Schöneman-Eisenstein criteria it is very easy to construct two-variable irreducible polynomials over $\mathbb{F}_p$.)

Keywords : pseudorandomness ; cross-combined; cross-correlation; combined measure; correlation measure; binary lattice; binary sequence