Golay Complementary Sequences of Arbitrary Length and Asymptotic Existence of Hadamard Matrices

In this work, we construct 4-phase Golay complementary sequence (GCS) set of cardinality 2^3+⌈log_2 r ⌉ with arbitrary sequence length n, where the 10^13-base expansion of n has r nonzero digits. Specifically, the GCS octets (eight sequences) cover all the lengths no greater than 10^13. Besides, based on the representation theory of signed symmetric group, we construct Hadamard matrices from some special GCS to improve their asymptotic existence: there exist Hadamard matrices of order 2^t m for any odd number m, where t = 6⌊1/40log_2m⌋ + 10.