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?
Best Answer
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 $n=20$, $k=11$