Balanced ($\mathbb{Z} _{2u}\times \mathbb{Z}_{38v}$, {3, 4, 5}, 1) difference packings and related codes

Let \begin{document}$ m $\end{document} , \begin{document}$ n $\end{document} be positive integers, and \begin{document}$ K $\end{document} a set of positive integers with size greater than 2. An \begin{document}$ (m,n,K,1) $\end{document} optical orthogonal signature pattern code, \begin{document}$ (m,n,K,1) $\end{document} -OOSPC, was introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirement. Let \begin{document}$ G $\end{document} be an additive group, a balanced \begin{document}$ (G, K, 1) $\end{document} difference packing, \begin{document}$ (G, K, 1) $\end{document} -BDP, can be used to construct a balanced \begin{document}$ (m,n,K,1) $\end{document} -OOSPC when \begin{document}$ G = {\mathbb{Z}}_m\times {\mathbb{Z}}_n $\end{document} . In this paper, the existences of optimal \begin{document}$ ( {\mathbb{Z}}_{2u}\times {\mathbb{Z}}_{38v}, \{3,4,5\},1) $\end{document} -BDPs are completely solved with \begin{document}$ u, \ v\equiv 1\pmod2 $\end{document} , and the corresponding optimal balanced \begin{document}$ (2u, 38v,\{3,4,5\},1) $\end{document} -OOSPCs are also obtained.