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

Cheng Du; Yi Jiang

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arXiv · 2401.15381

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

Cheng Du·Yi Jiang

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Abstract

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

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Cheng Du, Yi Jiang. 2024-01-27. Golay Complementary Sequences of Arbitrary Length and Asymptotic Existence of Hadamard Matrices. https://arxiv.org/abs/2401.15381

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