How hard would it be to port the code to Luminary Micro Stelalris chip?
Since it is a similar ARM processor, I assume most of the HAL would be similar or identical. The main attraction to me is the on board ethernet Mac & Phi. It opens the door to a whole lot of new opptunites with web servers, web services, etc. I just don't know enough about the porting kit to know how much work it would be and whetther some of the TI ethernet (lwIP) can be used with HAL.
How hard would it be to port the code to Luminary Micro Stelalris chip?
What code?
Morse code should be pretty easy to implement.
The main attraction to me is the on board ethernet Mac & Phi.
Is that like mac and cheese?
Not even close. The ATmega328P is an AVR architecture, the Stellaris devices are ARM (Cortex-M3). Different toolchains, different word sizes (8 bit vs. 32 bit), different hardware peripherals, etc.
In short, it would be hard 
Interesting, useful, and doable....but hard.
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