Abstract
The cross-combined measure (which is a natural extension of the cross-correlation measure) is introduced and important constructions of large families of binary lattices with optimal or nearly optimal cross-combined measures are presented. These results are also strongly related to the one-dimensional case: An easy method is shown obtaining strong constructions of families of binary sequences with nearly optimal cross-correlation measures based on the previous constructions of families of lattices. The important feature of this result is that so far there exists only one type of construction of very large families of binary sequences with small cross-correlation measure, and this only type of construction was based on one-variable irreducible polynomials. However there are relatively fast algorithms to construct one-variable irreducible polynomials, still in certain applications these algorithms are too complicated or are not fast enough, thus it became necessary to show other types of constructions where the generation of sequences is much faster. Using binary lattices based on two-variable irreducible polynomials this problem can be avoided. (Since, contrary to one-variable polynomials, using the Schöneman-Eisenstein criteria it is possible to generate two-variable irreducible polynomials over \(\mathbb F_p\) easily and very fast.)
Research partially supported by Hungarian National Research Development and Innovation Funds NK 104183 and K 119528.
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I would like to thank the referee for his careful reading and valuable advice concerning Theorem 1.
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Gyarmati, K. (2018). On the Cross-Combined Measure of Families of Binary Lattices and Sequences. In: Kaczorowski, J., Pieprzyk, J., Pomykała, J. (eds) Number-Theoretic Methods in Cryptology. NuTMiC 2017. Lecture Notes in Computer Science(), vol 10737. Springer, Cham. https://doi.org/10.1007/978-3-319-76620-1_13
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