equilateral triangle and coding theory

Question

Let $F=\{0,1\}$; $F$ is a field and let $x$, $y$, $z$ be words in $F^n$ that form an "equilateral triangle" that is: $d(x,y)=d(x,z)=d(y,z)=2t$. Show that there is exactly one word $v$ that belongs to $F^n$ such that $d(x,v)=d(y,v)=d(z,v)=t$.

Answer 1

look at the $k^{th}$ bit for each of $x,y,z$.

choose the $k^{th}$ bit for the equidistant point $v$ as follows:

if all three vertices have the same bit value, then take this value. if they differ, take the majority value.

do this for each of the $n$ bits