Gauss-Jordan elimination over GF(2) - matlab

Question

I need to transform a parity-check matrix H (that only consists of ones and zeros) from a non-standard to a standard form, this is, express it as:
Hsys = [A | I]
H and Hsys share the same dimension: (n-k,n). I above corresponds to an identity matrix of dimension (n-k).
Gauss-Jordan elimination comes in handy to solve this problem. Matlab has an specific command, rref, for this purpose, however it is no longer valid while working over GF(2) as in our case. Glancing through the Internet I found in Github a potentially suitable solution to overcome this drawback. However it does not always work out.
I also tried doing HH = mod(rref(H),2), which did not work at all, as many of the output elements weren't binary.
Here below you may find three samples of non-standard parity check matrices in which Gauss-Jordan elimination (over GF(2)) can be applied. As there should always be a way to arrange any matrix to be systematic, I would need a method that works out with matrices of any dimension.
These first sample is taken from sid's post in Stackoverflow, not responded yet:
H=[1 0 1 1 0;
0 0 1 0 1;
1 0 0 1 0;
1 0 1 1 1];
H=[1 1 0 1 1 0 0 1 0 0;
0 1 1 0 1 1 1 0 0 0;
0 0 0 1 0 0 0 1 1 1;
1 1 0 0 0 1 1 0 1 0;
0 0 1 0 0 1 0 1 0 1];
The last one is a matrix of dimension (50x100) and can be found in this link to my Dropbox.
Edit on 21/06/2017
The solution proposed by #Jonas worked out in some cases, but not in most of them, as H matrix seems to be singular. Any other similar way to do this?
Thank you in advance, and best regards.

Answer 1

Here is how I'd do it (using Gauss-Jordan elimination):
H=[1 1 0 1 1 0 0 1 0 0;
0 1 1 0 1 1 1 0 0 0;
0 0 0 1 0 0 0 1 1 1;
1 1 0 0 0 1 1 0 1 0;
0 0 1 0 0 1 0 1 0 1];
rows = size(H, 1);
cols = size(H, 2);
r = 1;
for c = cols - rows + 1:cols
if H(r,c) == 0
% Swap needed
for r2 = r + 1:rows
if H(r2,c) ~= 0
tmp = H(r, :);
H(r, :) = H(r2, :);
H(r2, :) = tmp;
end
end
% Ups...
if H(r,c) == 0
error('H is singular');
end
end
% Forward substitute
for r2 = r + 1:rows
if H(r2, c) == 1
H(r2, :) = xor(H(r2, :), H(r, :));
end
end
% Back Substitution
for r2 = 1:r - 1
if H(r2, c) == 1
H(r2, :) = xor(H(r2, :), H(r, :));
end
end
% Next row
r = r + 1;
end
Let me know if this doesn't solve your issue.

Answer 2

I went through the same issue and #jonas code also produced mostly singular matrix error. you can try the following code which I found to be helpful when searching for the systematic form of H. Its also include the calculation of G.
% Gauss-Jordan elimination
swaps=zeros(m,2);
swaps_count=1;
n=size(H, 2);
m=size(H, 1);
j=1;
index=1;
while index<=m
i=index;
while (HH(i,j)==0)&(i<m)
i=i+1;
end
if HH(i,j)==1
temp=HH(i,:);
HH(i,:)=HH(index,:);
HH(index,:)=temp;
for i=1:m
if (index~=i)&(HH(i,j)==1)
HH(i,:)=mod(HH(i,:)+HH(index,:),2);
end
end
swaps(swaps_count,:)=[index j];
swaps_count=swaps_count+1;
index=index+1;
j=index;
else
j=j+1;
end
end
for i=1:swaps_count-1
temp=HH(:,swaps(i,1));
HH(:,swaps(i,1))=HH(:,swaps(i,2));
HH(:,swaps(i,2))=temp;
end
G=[(HH(:,m+1:n))' eye(n-m)];
for i=swaps_count-1:-1:1
temp=G(:,swaps(i,1));
G(:,swaps(i,1))=G(:,swaps(i,2));
G(:,swaps(i,2))=temp;
end
disp(sum(sum((mod(H*G',2)))));