论文标题
$ \ bf 3^n u^1 $的holeyschröder设计
Holey Schröder Designs of Type $\bf 3^n u^1$
论文作者
论文摘要
类型$ h_1^{n_1} h_4^{n_2} \ cdots h^{n_k} _k $（hsd $（h_1^{n_1} h_4^h_4^h_4^） (FISQ$(h_1^{n_1}h_4^{n_2}\cdots h^{n_k}_k))$ of order $n$ with $n_i$ missing subquasigroups (holes) of order $h_i, 1 \le i \le k$, which are disjoint and spanning (i.e., $\sum_{1\le i \le k} n_ih_i = n $）。 $ H = 1、2、4 $的HSD $（H^nu^1）$的存在。在本文中，我们考虑存在HSD $（3^nu^1）$，并表明，如果$ 0 \ le u \ le 15 $，HSD $（3^nu^1）$存在，并且仅在$ n（n + 2u -1）\ equiv 0〜（mod〜4）$，$ n \ 4）$，$ n \ ge 4 $ n $ n $ and $ n $ and $ n \ ge 1 + 2u $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 1 + 2U。对于$ 0 \ le u \ le n $，当时存在HSD $（3^nu^1）$，仅当$ n（n + 2u -1）\ equiv 0〜（mod〜4）$和$ n \ ge 4 $，除可能$ n = 29，43 $的情况下。我们还发现了六个新的HSD $（4^nu^1）$。
A holey Schröder design of type $h_1^{n_1}h_4^{n_2}\cdots h^{n_k}_k$ (HSD$(h_1^{n_1}h_4^{n_2}\cdots h^{n_k}_k))$ is equivalent to a frame idempotent Schröder quasigroup (FISQ$(h_1^{n_1}h_4^{n_2}\cdots h^{n_k}_k))$ of order $n$ with $n_i$ missing subquasigroups (holes) of order $h_i, 1 \le i \le k$, which are disjoint and spanning (i.e., $\sum_{1\le i \le k}n_ih_i = n$). The existence of HSD$(h^nu^1)$ for $h=1, 2, 4$ has been known. In this paper, we consider the existence of HSD$(3^nu^1)$ and show that for $0\le u \le 15$, an HSD$(3^nu^1)$ exists if and only if $n(n + 2u -1) \equiv 0~(mod~4)$, $n\ge 4$ and $n\ge 1+2u/3$. For $0 \le u \le n$, an HSD$(3^nu^1)$ exists if and only if $n(n + 2u -1) \equiv 0~(mod~4)$ and $n \ge 4$, with possible exceptions of $n = 29, 43$. We have also found six new HSDs of type $(4^nu^1)$.
