Coding Theory (coset leader) If $C$ is a $[n,k,d]-$ code and $u\in\mathbb B^n$. Show that $\omega(u)\le\lfloor \frac{d-1}{2}\rfloor$ then $u$ is a unique coset leader in $u+C$. I have to prove this statement, but how to show that $u$ is a coset leader. If $u$ is a coset leader I can prove uniqueness.

Suposse that a word ${\bf y}$ is received. Minimum distance decoding requires tha we decode ${\bf y}$ as a codeword ${\bf x}$ for wich ${\bf e}={\bf y}-{\bf x}$ has smallest weight, where ${\bf e}\in \mathbb{B}^{n}$.

So, if ${\bf e}\in \mathbb{B}^{n} $ is of weight at most $\lfloor{\frac{d-1}{2}\rfloor}$, then ${\bf e}$ is a unic coset leader in ${\bf e}+C$. Suposse we have two coset leader in ${\bf e}+C$. Let be ${\bf e}_{1}$ and ${\bf e}_{2}$ two coset leader in ${\bf e}+C$ with $w({\bf e}_{1})\leq \lfloor{ \frac{d-1}{2}}\rfloor$ and $w({\bf e}_{2})\lfloor{\leq \frac{d-1}{2}\rfloor}$. Then, their difference

${ \bf e}_{1}-{\bf e}_{2}={\bf y}-{\bf x}_{1}-({\bf y}-{\bf x}_{2})={\bf x}_{2}-{\bf x}_{1}\in C$ will be a wordcode, which is not possible.

$w({\bf e}_{1}-{\bf e}_{2})=d({\bf e}_{1},{\bf e}_{2})\leq d({\bf e}_{1},0)+d(0,{\bf e}_{2})=d-1$