I've been reading about perfect codes and working on various exercises to get a better understanding about these types of codes. I came across an interesting statement that I am having trouble showing.
A coset of a linear perfect code is also a perfect code.
Can anyone help? Thanks in advance!
Let $C$ be a linear code, let $v+C$ be a coset. Given two elements $x,y$ in the coset, we have $x=v+a$, $y=v+b$ for some $a,b$ in $C$. Can you show that the distance between $x$ and $y$ is the same as the distance between $a$ and $b$? Can you see how to apply that to your question?