According to this page, there should be a code of length 20 with hamming distance 6 and between 2048-2279 codewords (or in other words, that contains at least $2^{11}$ words, meaning that data with 11 bits could be transmitted within blocks of 20 with HD 6).
Where or how do I find the generator matrix? Would I be able to detect how many errors happened?
The generator matrix is only for linear codes, that page is for general codes.
For linear binary codes, according to this table, $d=5$ is the most you can get for $n=20$, $k=11$