Gallian Introduction to Algebraic Coding Theory Ch. 31 Prob. 6

This was an easy problem to grasp mentally (just write out a few examples to see that it works), but for me it was hard to prove. Then again coding theory is not my favorite subject - waaay too dry.

Anyways yeah resorting to strong induction to prove the second half feels desperate but that's all I could come up with ¯\_(ツ)_/¯

***EDIT: Typo - should read "Case 2 (w=1):" instead of "Case 1 (w=1):" in the last part of the proof.