Suppose that we have a block code $C$ which has unique decoding, meaning that for all words $x$ (not necessarily in the code) there exists a unique word $c$ in the code with minimum distance (i.e. the other words in $C$ are at distance strictly greater than $c$ from $x$).
I want to know if under this condition the code is a Hamming code, i.e. the Hamming bound is attained.
For a rather trivial counterxample, consider some perfect code, and augment it by adding a
0in the last position.