Can you design me a single board computer using an ARM processor with a Linux port?

I can create a design for you. I don't know what your goals are. But instead of asking someone to design, why not buy one? I recommend Raspberry Pi. It has an ARM CPU, and supports many Linux distributions such as Raspbian, Ubuntu MATE, and Arch Linux. An alternative to Raspberry Pi is BeagleBone Black. It has a more powerful ARM CPU, and is also compatible with many Linux distributions.