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 $t = 6\lfloor \frac{1}{40}\log_{2}m\rfloor + 10$.

翻译：本文构建了基数为$2^{3+\lceil \log_2 r \rceil}$的$4$相Golay互补序列集，其序列长度$n$为任意值，其中$n$的$10^{13}$进制展开具有$r$个非零数字。特别地，该Golay互补序列八元组（八条序列）覆盖所有不超过$10^{13}$的长度。此外，基于符号对称群的表示理论，我们从特定Golay互补序列构造阿达玛矩阵以改进其渐近存在性：对任意奇数$m$，存在阶数为$2^t m$的阿达玛矩阵，其中$t = 6\lfloor \frac{1}{40}\log_{2}m\rfloor + 10$。