On Coding Theory, there's three trivial perfect codes.
They are:
- Binary codes of odd length
- Codes with contains only one codeword
- Codes that are the whole $A^n_q$
So, the $A^n_q$ case, considering the definition that says "A perfect code is a code such $A_q^n$ is the disjoint union of balls of some fixed radius centered on the codewords" is trivial by taking the radius to be 0.
But the other two cases I can't see why. Any hints to demonstrate that? (Could it be that codes with one codeword are analogue to the $A_q^n$ case? Taking the radius to be n or M=|C|)