Let $C$ be a code. Show that $d(c+a, c+b)= d(a,b)$ is true, for all $a,b,c \in C$
I am having the problem to understand how this addition is defined, since there are no information about $C$, only that it is a code. Is the code meant to be a finite field?
I assume that $d$ here, denotes the (Hamming) distance between two code-words of your code and $a,b\in\mathbb{Z_p^n}$.
If so, the statement essentially tells us that if we take two-code words $a,b$ and add another code-word, say $c$, to each one, then the distance between the original pair and the one we take is the same.
An easy way to see why this holds true is by using the-I guess familiar to you-notion of weight, that is, the distance of a code-word from the "zero-word" of your code.
So the weight of a word $a$ is $w(a):=d(a,0)$
That is, the weight of a word represents how many non-zero characters appear in it.
It follows that $d(a,b)=w(a-b)$.
Now we have $d(a+c,b+c)=w(a+c-(b+c))=w(a-b)=d(a,b)$