Golay Complementary Sequences of All Sequence Lengths

Two sequences whose aperiodic autocorrelation functions sum to form a perfect impulse constitute a Golay complementary sequence pair, each being a Golay complementary sequence. These sequences have many applications in wireless communication, ultrasound, and radar systems. When used in the frequency domain of orthogonal frequency division multiplexing signals, the time domain waveform's peak-to-mean envelope power ratio (PMEPR) is upper bounded by 2. We show that the paraunitary construction proposed by Budišin and Spasojević can create 8-QAM Golay complementary sequences for all lengths where BPSK and QPSK Golay complementary sequences were previously unknown. Through recursive and paraunitary constructions, Golay complementary sequences can be generated for all sequence lengths. Furthermore, this result enables the construction of 8-QAM unitary matrices of all even orders, expanding the potential applications of Golay complementary sequences.