How would having a codeword withweight two affect the coset priority here?

Question

I am trying to understand syndrome decoding. My understanding so far is that we calculate syndrome vector based on what we assume would be the error and that the coset leader vector chosen is to be the one with minimal weight. I don't understand this comment how would having a codeword of weight 2 result in not choosing weight 1 vector as thecoset leader ? [1]: https://i.stack.imgur.com/kiw5h.jpg

Answer 1

Definition coset and coset leader. A coset is a set parametrized by the its syndrome $s\in\mathbb{F}^{n-k}$ and designates a set of vectors with the same syndrome, i.e. $$ A(s) = \{ x \in \mathbb{F}^n : xH^{\mathrm{T}}=s \}. $$ The coset leader of a coset $A(s)$ is the vector with minimum weight in $A(s)$.

By the definition of the coset it is easy to see that for all $x_1,x_2 \in A(s)$, it holds that for any two $x_1-x_2\in \mathcal{C}$ and thus $d(x_1,x_2)\geq d_{min}(\mathcal{C})$. Hence, if the coset leader has weight 1, $\mathrm{wt}(x_1)=1$, this coset leader is unique provided that $d_{min}(\mathcal{C})\geq 3$.