6. (10%) A matrix \( H \) for a linear block code is defined such that each vector \( v \) is a valid codeword if and only if \( Hv^T = 0 \). Assume that

$$H = \begin{bmatrix} 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 1 \end{bmatrix}$$

i) Is the vector \( v_{1} = [1\ 0\ 0\ 1\ 0\ 1] \) a valid codeword vector? Why or why not?
ii) How many valid codewords can you find?