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I have generated a LDPC parity check matrix using Gallager approach for n = 20 , j = 3 and k = 4.

The parity check matrix is defined by: enter image description here

The problem is, I don't know how to proceed further to generate the codeword for which I'll need the generator matrix. The Gauss elimination method was of no help. Can u suggest a better method to find the generator matrix. Thanks in advance.

m2016b
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Dhanush
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1 Answers1

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We want to use Gaussian elimination to normalize this parity check matrix H into normalized form [P I].

Since H is a 15x20 matrix, P would be a 15x5 matrix and I would be a 15x15 matrix.

However, observe that the last 15 entries of the first row in H are all zeroes.

Thus, it is impossible to normalize H using Gaussian elimination alone.

We might try to use a slightly different approach to normalize H, namely the method suggested in this StackExchange answer https://math.stackexchange.com/questions/2818445/how-do-i-find-parity-check-matrix-if-generator-matrix-cant-be-written-in-standa :

  1. Convert H to row-echolon form

  2. Perform column-swaps to produce normalized from [P I]

  3. Create the normalized generator matrix G = [I PT]

  4. Undo the column-swaps from step 2 onto the generator matrix G

However, for the matrix H given in your example, this approach does also not work because the rows are linearly dependent and the row-echolon form has all-zeroes as its last two rows.

The issue of how to generate a reasonable generator matrix G even when the parity check matrix H has linear dependencies is discussed here:

Bruno Rijsman
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