Error bounds for repeat-accumulate codes decoded via linear programming

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November 2011, 5(4): 555-570. doi: 10.3934/amc.2011.5.555

Error bounds for repeat-accumulate codes decoded via linear programming

Idan Goldenberg ^1, and David Burshtein ^1,

1.	School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel, Israel

Received January 2010 Revised September 2011 Published November 2011

Abstract
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We examine regular and irregular repeat-accumulate (RA) codes with repetition degrees which are all even. For these codes and with a particular choice of an interleaver, we give an upper bound on the decoding error probability of a linear-programming based decoder which is an inverse polynomial in the block length. Our bound is valid for any memoryless, binary-input, output-symmetric (MBIOS) channel. This result generalizes the bound derived by Feldman et al., which was for regular RA($2$) codes.

Keywords: error performance., linear-programming (LP) decoding, repeat-accumulate codes, upper bound, Coding theory.

Mathematics Subject Classification: 68P30, 94B70, 90C0.

Citation: Idan Goldenberg, David Burshtein. Error bounds for repeat-accumulate codes decoded via linear programming. Advances in Mathematics of Communications, 2011, 5 (4) : 555-570. doi: 10.3934/amc.2011.5.555

References:

[1]	C. Berrou, A. Glavieux and P. Thitimajshima, Near Shannon limit errorcorrecting coding and decoding: turbo codes,, in, (1993), 1064. doi: 10.1109/ICC.1993.397441.
[2]	N. Biggs, Constructions for cubic graphs with large girth,, Electronic J. Combinatorics, 5 (1998).
[3]	D. Burshtein, Iterative approximate linear programming decoding of LDPC codes with linear complexity,, IEEE Trans. Inform. Theory, 55 (2009), 4835. doi: 10.1109/TIT.2009.2030477.
[4]	C. Daskalakis, A. G. Dimakis, R. M. Karp and M. J. Wainwright, Probabilistic analysis of linear programming decoding,, IEEE Trans. Inform. Theory, 54 (2008), 3565. doi: 10.1109/TIT.2008.926452.
[5]	D. Divsalar, H. Jin and R. J. McEliece, Coding theorems for Turbo-like codes,, in, (1998), 201.
[6]	P. Erdős and H. Sachs, Reguläre Graphen gegebene Taillenweite mit minimaler Knotenzahl,, Wiss. Z. Univ. Hall Martin Luther Univ. Halle-Wittenberg Math. Natur. Reine, 12 (1963), 251.
[7]	J. Feldman, "Decoding Error-Correcting Codes via Linear Programming'',, Ph.D thesis, (2003).
[8]	J. Feldman and D. R. Karger, Decoding turbo-like codes via linear programming,, in, (2002). doi: 10.1109/SFCS.2002.1181948.
[9]	J. Feldman, T. Malkin, R. A. Servedio, C. Stein and M. J. Wainwright, LP decoding corrects a constant fraction of errors,, IEEE Trans. Inform. Theory, 53 (2007), 82. doi: 10.1109/TIT.2006.887523.
[10]	J. Feldman, M. J. Wainwright and D. R. Karger, Using linear programming to decode binary linear codes,, IEEE Trans. Inform. Theory, 51 (2005), 954. doi: 10.1109/TIT.2004.842696.
[11]	R. G. Gallager, "Low-Density Parity-Check Codes,'', MIT Press, (1963).
[12]	N. Halabi and G. Even, Improved bounds on the word error probability of RA($2$) codes with linear-programming-based decoding,, IEEE Trans. Inform. Theory, 51 (2005), 265. doi: 10.1109/TIT.2004.839509.
[13]	H. Jin and R. J. McEliece, Irregular repeat-accumlate codes,, in, (2000), 1.
[14]	H. D. Pfister, I. Sason and R. Urbanke, Capacity-achieving ensembles for the binary erasure channel with bounded complexity,, IEEE Trans. Inform. Theory, 51 (2005), 2352. doi: 10.1109/TIT.2005.850079.
[15]	R. Smarandache and P. O. Vontobel, Pseudo-codeword analysis of Tanner fraphs from projective and Euclidean planes,, IEEE Trans. Inform. Theory, 53 (2007), 2376. doi: 10.1109/TIT.2007.899563.
[16]	P. O. Vontobel and R. Koetter, Lower bounds on the minimum pseudoweight of linear codes,, in, (2004).
[17]	P. O. Vontobel and R. Koetter, Towards low-complexity linear-programming decoding,, in, (2006).

show all references

References:

[1]	C. Berrou, A. Glavieux and P. Thitimajshima, Near Shannon limit errorcorrecting coding and decoding: turbo codes,, in, (1993), 1064. doi: 10.1109/ICC.1993.397441.
[2]	N. Biggs, Constructions for cubic graphs with large girth,, Electronic J. Combinatorics, 5 (1998).
[3]	D. Burshtein, Iterative approximate linear programming decoding of LDPC codes with linear complexity,, IEEE Trans. Inform. Theory, 55 (2009), 4835. doi: 10.1109/TIT.2009.2030477.
[4]	C. Daskalakis, A. G. Dimakis, R. M. Karp and M. J. Wainwright, Probabilistic analysis of linear programming decoding,, IEEE Trans. Inform. Theory, 54 (2008), 3565. doi: 10.1109/TIT.2008.926452.
[5]	D. Divsalar, H. Jin and R. J. McEliece, Coding theorems for Turbo-like codes,, in, (1998), 201.
[6]	P. Erdős and H. Sachs, Reguläre Graphen gegebene Taillenweite mit minimaler Knotenzahl,, Wiss. Z. Univ. Hall Martin Luther Univ. Halle-Wittenberg Math. Natur. Reine, 12 (1963), 251.
[7]	J. Feldman, "Decoding Error-Correcting Codes via Linear Programming'',, Ph.D thesis, (2003).
[8]	J. Feldman and D. R. Karger, Decoding turbo-like codes via linear programming,, in, (2002). doi: 10.1109/SFCS.2002.1181948.
[9]	J. Feldman, T. Malkin, R. A. Servedio, C. Stein and M. J. Wainwright, LP decoding corrects a constant fraction of errors,, IEEE Trans. Inform. Theory, 53 (2007), 82. doi: 10.1109/TIT.2006.887523.
[10]	J. Feldman, M. J. Wainwright and D. R. Karger, Using linear programming to decode binary linear codes,, IEEE Trans. Inform. Theory, 51 (2005), 954. doi: 10.1109/TIT.2004.842696.
[11]	R. G. Gallager, "Low-Density Parity-Check Codes,'', MIT Press, (1963).
[12]	N. Halabi and G. Even, Improved bounds on the word error probability of RA($2$) codes with linear-programming-based decoding,, IEEE Trans. Inform. Theory, 51 (2005), 265. doi: 10.1109/TIT.2004.839509.
[13]	H. Jin and R. J. McEliece, Irregular repeat-accumlate codes,, in, (2000), 1.
[14]	H. D. Pfister, I. Sason and R. Urbanke, Capacity-achieving ensembles for the binary erasure channel with bounded complexity,, IEEE Trans. Inform. Theory, 51 (2005), 2352. doi: 10.1109/TIT.2005.850079.
[15]	R. Smarandache and P. O. Vontobel, Pseudo-codeword analysis of Tanner fraphs from projective and Euclidean planes,, IEEE Trans. Inform. Theory, 53 (2007), 2376. doi: 10.1109/TIT.2007.899563.
[16]	P. O. Vontobel and R. Koetter, Lower bounds on the minimum pseudoweight of linear codes,, in, (2004).
[17]	P. O. Vontobel and R. Koetter, Towards low-complexity linear-programming decoding,, in, (2006).

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