How to prove that for any $c∈C$, $HW = HD(c)$

Question

If there's a Binary Linear Block Code.
Let $HW$ be the set of all distinct Hamming weights that codewords of $C$ may have.
Let $HD(c)$ be the set of all distinct Hamming distances between $c$ and any codeword.

Prove: $HW = HD(c)$ for any $c∈C$.

I think it results from the generator matrix, but still haven't managed anything out. Any ideas? Thanks in advance!

Answer 1

Fix $c \in C$.

$$ HD(c) = \{d_H(x, c) \mid x \in C\} = \{w_H(x - c) \mid x \in C\} $$

A vector space is closed under addition and scalar multiplication, so $x - c \in C$ if and only if $x \in C$. Thus

$$ \{w_H(x - c) \mid x \in C\} = \{w_H(x) \mid x \in C\} = HW. $$