Boolean complementary pairs

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Gobin, Jared
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Golay pairs (GPs) are pairs of (1,-1)-sequences with zero autocorrelation. GPs have numerous applications in engineering and mathematics such as in device synchronization, spectroscopy, and in the construction of Hadamard matrices. As a result, there is significant interest in understanding their mathematical structure. However, the structure of GPs appears to be very mysterious. Ternary complementary pairs (TCPs)—a generalization of GPs—are pairs of (0,1,-1)-sequences with zero autocorrelation. Similar to GPs, the structure of TCPs appears to be quite challenging. In order to gain insight into TCPs and GPs, Craigen introduced Boolean complementary pairs (BCPs), which are pairs of Z_2-sequences with zero autocorrelation. Craigen solved the structure of even-weight BCPs and with Woodford discovered a factorization involving 2x2 matrices in the odd-weight case. A review of Golay pairs, ternary complementary pairs, and Boolean complementary pairs is given. The odd-weight BCP factorization is extended to all pairs of Z_2-sequences with zero autocorrelation. Consequences of this extension are then established, including a connection to affixes.
Complementary pairs of sequences