符号群正交设计及其应用
Signed group orthogonal designs and their applications
摘要：
Craigen在\cite{Craigenthesis, Craigen}中对{ \it符号群Hadamard矩阵}进行了广泛的介绍和研究。Livinskyi \cite{Ivan}在Craigen的基础上，研究并给出了符号群Hadamard矩阵渐近存在性的一个较好的估计，从而改进了Hadamard矩阵的渐近存在性.本文介绍并研究了符号群正交设计。主要结果包括一种方法，找到任何$k$元组的正整数，然后应用程序，以获得正交设计的符号组正交设计，即任何$k$元组$\big(u_1, u_2, ..., u_{k}\big)$的正整数，我们表明，有一个整数$N=N(u_1, u_2, ..., u_k)$，使每个$n\ge N$，一个完整的正交设计（无零条目）的类型$\big(2^nu_1,2^nu_2,...,2^nu_{k}\big)$存在。这是在\cite{EK}中获得的结果的替代方法。
英文摘要：
Craigen introduced and studied {\it signed group Hadamard matrices} extensively in \cite{Craigenthesis, Craigen}. Livinskyi \cite{Ivan}, following Craigen's lead, studied and provided a better estimate for the asymptotic existence of signed group Hadamard matrices and consequently improved the asymptotic existence of Hadamard matrices. In this paper, we introduce and study signed group orthogonal designs. The main results include a method for finding signed group orthogonal designs for any $k$-tuple of positive integer and then an application to obtain orthogonal designs from signed group orthogonal designs, namely, for any $k$-tuple $\big(u_1, u_2, ..., u_{k}\big)$ of positive integers, we show that there is an integer $N=N(u_1, u_2, ..., u_k)$ such that for each $n\ge N$, a full orthogonal design (no zero entries) of type $\big(2^nu_1,2^nu_2,...,2^nu_{k}\big)$ exists . This is an alternative approach to the results obtained in \cite{EK}.
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