definition : Let $w$ be a word in $\mathbb{F}_{q}^{n}$ and $r$ a natural number. The ball of radius $r$ with center $w$, denoted by $B_{r}(w)=\{x \in \mathbb{F}_{q}^{n} : d(w,x) \leq r\}$ .
Now there is an example ( which I don't understand) The ball of radius one with center at $(0,0,0,0,0)$ in $\mathbb{F}_{2}$ consists of $(0,0,0,0,0)$ and all the words weight one. For $w=(1,0,1,1,0)$,
$B_{1}(w)= \left( \begin{array}{c} 1 \\ 0\\ 1\\ 1\\ 0 \end{array} \right)^{tr}$, $ \left( \begin{array}{c} 0\\ 0\\ 1\\ 1\\ 0 \end{array} \right)^{tr}$ $\left( \begin{array}{c} 1\\ 1\\ 1\\ 1\\ 0 \end{array} \right)^{tr}$, $\left( \begin{array}{c} 1\\ 0\\ 0\\ 1\\ 0 \end{array} \right)^{tr}$, $\left( \begin{array}{c} 1\\ 0\\ 1\\ 0\\ 0 \end{array} \right)^{tr}$, $ \left( \begin{array}{c} 1\\ 0\\ 1\\ 1\\ 1 \end{array} \right) $
would someone please explain this example to me. Thank you
Your ball of radius one with center $w$ consists of $w$ and all the words at distance one from $w$ (Hamming distance).