A more accurate one-dimensional analysis and design of irregular LDPC codes

We introduce a new one-dimensional (1-D) analysis of low-density parity-check (LDPC) codes on additive white Gaussian noise channels which is significantly more accurate than similar 1-D methods. Our method assumes a Gaussian distribution in message-passing decoding only for messages from variable nodes to check nodes. Compared to existing work, which makes a Gaussian assumption both for messages from check nodes and from variable nodes, our method offers a significantly more accurate estimate of convergence behavior and threshold of convergence. Similar to previous work, the problem of designing irregular LDPC codes reduces to a linear programming problem. However, our method allows irregular code design in a wider range of rates without any limit on the maximum variable-node degree. We use our method to design irregular LDPC codes with rates greater than 1/4 that perform within a few hundredths of a decibel from the Shannon limit. The designed codes perform almost as well as codes designed by density evolution.

[1]  Li Ping,et al.  Concatenated tree codes: A low-complexity, high-performance approach , 2001, IEEE Trans. Inf. Theory.

[2]  S. Brink Rate one-half code for approaching the Shannon limit by 0.1 dB , 2000 .

[3]  Daniel A. Spielman,et al.  Improved low-density parity-check codes using irregular graphs and belief propagation , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[4]  David J. C. MacKay,et al.  Good Error-Correcting Codes Based on Very Sparse Matrices , 1997, IEEE Trans. Inf. Theory.

[5]  Radford M. Neal,et al.  Near Shannon limit performance of low density parity check codes , 1996 .

[6]  H. Jin,et al.  Irregular repeat accumulate codes , 2000 .

[7]  Hesham El Gamal,et al.  Analyzing the turbo decoder using the Gaussian approximation , 2001, IEEE Trans. Inf. Theory.

[8]  M. Aminshokrollahi New sequences of linear time erasure codes approaching the channel capacity , 1999 .

[9]  Sae-Young Chung,et al.  On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit , 2001, IEEE Communications Letters.

[10]  Stephan ten Brink,et al.  Design of low-density parity-check codes for modulation and detection , 2004, IEEE Transactions on Communications.

[11]  G. M. Maggio,et al.  An approximate analytical model of the message passing decoder of LDPC codes , 2002, Proceedings IEEE International Symposium on Information Theory,.

[12]  Dariush Divsalar,et al.  Low Complexity Turbo-like Codes , 2000 .

[13]  M. Shokrollahi,et al.  Capacity-achieving sequences , 2001 .

[14]  Sae-Young Chung,et al.  Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation , 2001, IEEE Trans. Inf. Theory.

[15]  Stephan ten Brink,et al.  Design of repeat-accumulate codes for iterative detection and decoding , 2003, IEEE Trans. Signal Process..

[16]  Daniel A. Spielman,et al.  Efficient erasure correcting codes , 2001, IEEE Trans. Inf. Theory.

[17]  Stephan ten Brink Iterative Decoding Trajectories of Parallel Concatenated Codes , 1999 .

[18]  Rüdiger L. Urbanke,et al.  Design of capacity-approaching irregular low-density parity-check codes , 2001, IEEE Trans. Inf. Theory.

[19]  Amin Shokrollahi,et al.  New Sequences of Linear Time Erasure Codes Approaching the Channel Capacity , 1999, AAECC.

[20]  Stephan ten Brink,et al.  Convergence behavior of iteratively decoded parallel concatenated codes , 2001, IEEE Trans. Commun..

[21]  Niclas Wiberg,et al.  Codes and Decoding on General Graphs , 1996 .

[22]  Shlomo Shamai,et al.  On interleaved, differentially encoded convolutional codes , 1999, IEEE Trans. Inf. Theory.

[23]  Rüdiger L. Urbanke,et al.  The capacity of low-density parity-check codes under message-passing decoding , 2001, IEEE Trans. Inf. Theory.

[24]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.