If we have an \(X_{n}\) with girth 3 then the cubes are defined by the Hamming encoding up until the last 3 digits.

So 110001 in same cube as 110110

We have <cube-identity>??? where cube-identity fixed and unique for each cube, the ? are binary bits.

The strategy would be to partition \(X_{n}\) into cubes. Then want to find (largest?) tree inside of the cubes. Make \(Z_{2}\)