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 $2^{3+\lceil {\log }_{2}r\rceil }$ 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 $t=6\lfloor \frac{1}{40}{\log }_{2}m\rfloor +10$.