Hamming distance for a linear code

Question

I want to solve this exercise:

"""

Prove the equality $d_{min}(D)=\min\{wt_H(z) | z \in D \} $ for a linear code D.

"""

$wt_H $ denotes the Hamming weight. What is $d_{min}$? The read that it is the mininmum distance of the error? What do I have to calculate then to get this value $d_{min}$?

Does one of you know an ansatz what one has to do here to prive this equality?

Answer 1

Let $x,y\in D$ be such that $d_{min}(D)=d(x,y)$. Note that $d(a,b)=d(a-c,b-c)$ for all $a,b,c\in D$. Hence $d_{min}(D)=d(x,y)=d(x-y,0)=wt_H(x-y)$.