Method for constructing large-girth quasi-cyclic low-density parity-check codes

1. A method for constructing a code, wherein the code is a quasi-cyclic low-density parity-check code, comprising the steps of:

selecting a base matrix for the code;

determining a cost matrix corresponding to the base matrix;

changing repeatedly a single element in the base to maximize a reduction in cost;

constructing a parity check matrix for the code from the base matrix when the cost is zero; and

encoding an information block as a code word using the parity check matrix in an encoder.

2. The method of claim 1 , wherein the changing uses a hill climbing procedure.

3. The method of claim 1 , wherein the base matrix initialized according to parameters.

4. The method of claim 3 , wherein the base matrix is selected randomly based on the parameters.

5. The method of claim 1 , wherein the cost matrix is determine according to a girth of the code and a weight vector based on cycles of the code.

6. The method of claim 5 , wherein the cost is a function of the number of the cycles having a length less than the girth.

7. The method of claim 6 , wherein cost is based on a weighted sum of undesired cycles.

8. The method of claim 1 , further comprising:

generating a gain matrix using the cost matrix and the reduction uses a maximum positive gain in the gain matrix.

9. The method of claim 1 , wherein the cost matrix indicates the cost of changing any value in the base matrix to any other possible value.

10. The method of claim 1 , wherein the channel is a communications channel.

11. The method of claim 1 , wherein the channel includes writing the encoded information block on storage media for later retrieval and decoding.

12. The method of claim 1 , further comprising:

transmitting the code word through a channel; and

decoding the code word in a decoder to recover the information block.