I've a Generator matrix in Gf(2) in non systematic form( no seperate, data bits, and no separate parity). It is a full rank matrix. Now to find H[7x20] from this matrix I used the property G*H_transpose=Zero matrix[13x7]. I found 127 vectors which are orthogonal to these 13 vectors.
out of these i have to pick 7 vectors, which constitute the H matrix. So i made use of the property, that R*H_transpose=0. (R is the codeword [1x20]). But i couldnt find any vector in the 127 vectors satisfying all the valid codewords.
From this it is evident that syndrome should not be zero for error free decoding. Now how to pick my vectors out of my 127 vectors..
Thanks.