Let $x=(x\_1x\_2...x\_n)$ be a binary sequence of length $n$. The Varshamov-Tenengolts code $VT\_0(n)$ consists of all binary vectors $(x\_1, . . . , x\_n)$ satisfying $\Sigma\_{i=1}^n i\*x\_i \equiv0 \pmod {n+1} $.


Prove that $\forall$ $x,y \in VT\_0(n)$ which has equal hamming weight the Hamming distance between $x$ and $y$ is exactly 4. For a binary vectore $x$ the hamming weight is $w$ if $\Sigma\_{i=1}^n x\_i= w $. 



Fixed Hamming distance property of binary deletion correcting codes
Take $n=11$. 11000000100 and 00111000000 are both in the code and they both have Hamming weight 3 but the Hamming distance between them is 6. 


