How to compute generator matrix from a parity check matrix?
With forward-error-correcting coding, one is working in a finite field, typically the field of two elements denoted by GF$(2)$ or $\mathbb F_2$. So, there are no fractional numbers
and no fancy methods such as singular value decomposition: you use bit-by-bit XOR additions of the rows of $H$ and Gauss-Jordan elimination to reduce $H$ to row-echelon form $[P_{(n-k)\times k} \mid I_{(n-k)\times(n-k)}]$. 

Then, set $G = [I_{k\times k} \mid (P^T)_{k\times(n-k)}]$ and you are done. (For 
nonbinary fields, use $[I \mid -P^T]$).  Note
that all arithmetic in the verification $HG^T = 0$ is also finite field arithmetic
with $1\cdot 1 =1$ and $1+1 = 0 = 1\oplus 1$ for the case of GF$(2)$.