How to show that, for a linear code C, that the covering radius of the code is the same as the weight of the coset of largest weight.

Question

I am in an independent study using the text by Huffman and Pless titled Fundamentals of Error-Correcting Codes. I am trying to prove that the covering radius $\rho(C)$ for a linear code is the same as the weight of the coset with largest weight. The proof strategy seems to be pretty standard, i.e. we show that each quantity is less than or equal to the other.

So far, by definition, the covering radius is given by

$\rho(\mathcal C)$ = $\underset{x\in \mathbb F_q^n}{max}$ $\underset{c\in \mathcal C}{min}$ $d(x,c)$.

Additionally, the weight of a coset is the smallest weight of a vector in the coset, where any vector of this smallest weight is called a coset leader. Two vectors x and y belong to the same coset if and only if y-x $\in \mathcal C$.

I know this isn't much, but I am not sure how to proceed from here. Any hints to point me in the right direction are very much appreciated!

Answer 1

"Hint":

Can you show that (here $w\in C$) if $$ d(x,w)=\min_{c\in C}d(x,c), $$ then $w-x$ is a leader of the coset $-x+C$?

And conversely :-)