# equilateral triangle and coding theory

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 $d(x,v)=d(y,v)=d(z,v)=t$.

• This is a part of Problem 20 in Chapter 1 of MacWilliams and Sloane, Theory of Error-Correcting Codes, North-Holland Elsevier 1978. – Dilip Sarwate Dec 16 '13 at 0:14

look at the $k^{th}$ bit for each of $x,y,z$.
choose the $k^{th}$ bit for the equidistant point $v$ as follows:
do this for each of the $n$ bits
• Let's hope so! ${}$ – Jyrki Lahtonen Dec 15 '13 at 12:58