Results on punctured LDPC codes

In this paper we study some fundamental properties of punctured LDPC codes. We first prove that for any ensemble of LDPC codes, there exists a puncturing threshold p*. We then find lower bounds on the achievable rates of punctured codes over general MBIOS channels. These bounds are satisfied by using only one encoder and decoder for all rates. We then prove that for any rates R/sub 1/ and R/sub 2/ satisfying 0 < R/sub 1/ < R/sub 2/ < 1, there exists an ensemble of LDPC codes with the following property. The ensemble can be punctured from rate R/sub 1/ to R/sub 2/ resulting in asymptotically good codes for all rates R/sub 1/ /spl les/ R /spl les/ R/sub 2/. Specifically, this implies that rates arbitrarily close to one are achievable via puncturing. We also show that punctured LDPC codes are as good as ordinary LDPC codes. For binary erasure channel (BEC) and arbitrary positive numbers R/sub 1/ < R/sub 2/ < 1, we prove the existence of the sequences of punctured LDPC codes that are capacity achieving for all rates R/sub 1/ /spl les/ R /spl les/ R/sub 2/. Based on the above observations, we then propose a method to design good punctured LDPC codes over a broad range of rates. The method is very simple and does not suffer from the performance degradation at high rates. Finally, we show that punctured codes might be useful for proof of the existence of capacity-achieving LDPC codes over memoryless binary-input output-symmetric channels.

[1]  Amin Shokrollahi,et al.  Capacity-achieving sequences for the erasure channel , 2002, IEEE Trans. Inf. Theory.

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

[3]  Nazanin Rahnavard,et al.  Nonuniform error correction using low-density parity-check codes , 2005, IEEE Transactions on Information Theory.

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

[5]  Steve McLaughlin,et al.  Optimal puncturing distributions for rate-compatible low-density parity-check codes , 2003, IEEE International Symposium on Information Theory, 2003. Proceedings..

[6]  T. Richrdson,et al.  Finite-length analysis of various low-density parity-check ensembles for the binary erasure channel , 2002, Proceedings IEEE International Symposium on Information Theory,.

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

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

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

[10]  Nazanin Rahnavard,et al.  Results on non-uniform error correction using low-density parity-check codes , 2003, GLOBECOM '03. IEEE Global Telecommunications Conference (IEEE Cat. No.03CH37489).

[11]  Emre Telatar,et al.  Finite-length analysis of low-density parity-check codes on the binary erasure channel , 2002, IEEE Trans. Inf. Theory.