I want to expand on a question I got in my home exercise. Although I did manage to solve this problem I wondered if the statement is true to be "if and only if" statement.
The exercise was: $C$ is a linear code $[n,k,d]$, if $\omega(u)\le\lfloor\frac{d-1}{2}\rfloor$ then u is an unique coset leader in $u+C$.
As I said, I did menege to solve this and there is even an answer here, but is the opposite way is true?
i.e. if $u$ is an unique coset leader in $u+C$ then $\omega(u)\le\lfloor\frac{d-1}{2}\rfloor$.
Or with equivalent rephrasing: if $u$ is a coset leader of $C$ that satisfy $\omega(u)>\lfloor\frac{d-1}{2}\rfloor$ then the coset leader is not unique.
I have tried to prove this statement or to find a counter example but I have have not succeeded yet.
Any help would be appreciated.
The converse is not true. Though the easy examples are uninterestingly trivial. For example, consider the linear 1-dimensional code $$\mathcal{C}=\{000000, 000011\}.$$ Its minimal distance is $d=2$. Yet even the word of weight three, $u=111000$, is a coset leader because $u+\mathcal{C}=\{111000,111011\}.$
On the other hand if $\mathcal{C}$ is a maximal code, by which I mean that there is no supercode $\mathcal{C}'$ containing $\mathcal{C}$ and with the same minimum distance $d$, then no word $u$ of weight $\ge d$ is a coset leader. This is because otherwise $\mathcal{C}'=\mathcal{C}\cup(u+\mathcal{C})$ would be a larger code with minimum distance $d$.
In between, it is more difficult to say. The good codes often cover the entire space reasonably well, and then this won't happen, but the arguments are a bit more delicate.