Uncorrelated binary sequences of lengths 2a3b4c5d7e11f13g based on nested Barker codes and complementary sequences

Certain applications require the use of signals that combine both the capability to operate with low signal-to-noise ratios and the ability to support multiple users without interference. In the case where many users have very different signal-to-noise ratios, it is necessary to consider coding schemes that can be used in a multi-user environment but with different noise immunity levels. Traditional detection systems based on the correlation function and coding sequences have significant limitations in satisfying both objectives, since the cross-correlation between coded signals corresponding with different users is linked to the use of the same coded sequences length. The research topic of binary sequences that have null cross-correlation and different length has not been studied in depth, but it has potential applications in multi-user environments. In this work an algorithm to generate binary sequences completely uncorrelated with certain sets of complementary sequences is presented. The proposed algorithm is based on nested Barker sequences, and it is compared with a previous proposal based on an iterative algorithm. This approach allows to generate more diversity of sequences of different length than the iterative approach, which it makes useful for applications based on binary sequences detection and expand the horizon of many applications.