Please anyone explain me how Gauss-Jordan elimination (over GF(2)) can be applied to a matrix of any dimension. I have searched over the Internet but i am not clear with concept of Gauss-Jordan elimination (over GF(2)). I need fruitful help from anyone
I am working with KINTEX board for LDPC encoding and decoding.Through Vivado HLS, I need to transform a parity-check matrix H (that only consists of ones and zeros) from a non-standard to a standard form. Here below you may find sample of non-standard parity check matrix in which Gauss-Jordan elimination (over GF(2)) can be applied.
This is my example H matrix. How Gauss-Jordan elimination (over GF(2)) can be applied to example H matrix. Explain me with logic.
H=[1 1 0 0 1 0; 1 0 0 1 0 1; 1 1 1 0 0 1 ];
Expected Systematic H matrix. Given a particular parity-check matrix, rearrange by Gauss-elimination so that H = [I_r|h], that is, so that H begins with an r by r identity matrix.
Hsys= [1 0 0 1 0 1; 0 1 0 1 1 1; 0 0 1 0 1 1];
Gauss-Jordan elimination over any field is exactly the same as Gauss-Jordan elimination over the reals, except that you do the arithmetic in the field. In the case of $GF(2)$ it's even easier because you never have to worry about division.