Complementary arrays — New directions

Summary form only given. Complementary sequence pairs were introduced by Golay in 1951, and have found application to many areas of signal processing and communications, such as to radar, tomography, and to power control for multicarrier wireless transmission. They are attractive because the sum of their aperiodic autocorrelations has zero sidelobes, and therefore the sum of their Fourier power spectrums is completely flat. Some generalisations of the original binary complementary sequence pairs have been to complementary sets, richer alphabets, arrays, near-complementarity, and complete complementary codes. In this talk I give a brief overview of the basic construction and some of these generalisations. I then focus on complementary sets of arrays (which are also complementary sequences sets), and propose new variants on the complementary principle. The aim is two-fold, firstly to look at the conventional complementary problem in new ways, and secondly to establish new types of complementarity that are mathematically interesting in their own right, and that may also have some practical implications. The arguments exploit the characterisation of complementarity in terms of unitary matrices. I develop Boolean constructions for different types of bipolar complementary 2 × 2 × ... × 2 array, and make connections with quantum information and graph theory, and I also mention some cryptographic interpretations.