We give a direct construction of a $\left({\mathbb{Z}}_p\times G,\left\{0\right\}\times G,4,1\right)$ relative difference family for $G\in \left\{{\mathbb{Z}}_6\times {\mathbb{Z}}_6,{\mathbb{Z}}_2\times {\mathbb{Z}}_{18},{\mathbb{Z}}_6\times {\mathbb{Z}}_{18},{\mathbb{Z}}_2\times {\mathbb{Z}}_{54}\right\}$ and every prime $p\equiv 3\left(\mathrm{mod}4\right)$ with $p>3$. These allow us to construct an optimal $\left({\mathbb{Z}}_{6m}\times {\mathbb{Z}}_{6n},4,1\right)$ difference packing and an optimal $\left({\mathbb{Z}}_{2m}\times {\mathbb{Z}}_{18n},4,1\right)$ difference packing for every pair of positive integers $\left(m,n\right)$. The corresponding optimal optical orthogonal signature pattern codes are also obtained.

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关于最优( Z 6 m × Z 6 n , 4 , 1 ) 和( Z 2 m × Z 18 n , 4 , 1 ) 差分包装及其相关代码

我们直接构造一个 $\left({\mathbb{Z}}_p\times G,\left\{0\right\}\times G,4,1\right)$ 相对差异族 $G\in \left\{{\mathbb{Z}}_6\times {\mathbb{Z}}_6,{\mathbb{Z}}_2\times {\mathbb{Z}}_{18},{\mathbb{Z}}_6\times {\mathbb{Z}}_{18},{\mathbb{Z}}_2\times {\mathbb{Z}}_{54}\right\}$ 和每个素数 $p\equiv 3\left(\mathrm{模组}4\right)$ 和 $p>3$. 这些使我们能够构建一个最优$\left({\mathbb{Z}}_{6米}\times {\mathbb{Z}}_{6n},4,1\right)$ 差异包装和最优 $\left({\mathbb{Z}}_{2米}\times {\mathbb{Z}}_{18n},4,1\right)$ 每对正整数的差分打包 $\left(米,n\right)$. 还获得了相应的最优光学正交特征码。