Error bounds for repeat-accumulate codes decoded via linear programming

Previous Article
Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes
AMC Home
This Issue
Next Article

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
Related Papers

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 "Proc. 1993 IEEE International Conference on Communications,'' Geneva, Switzerland, (1993), 1064-1070. doi: 10.1109/ICC.1993.397441.
[2]	N. Biggs, Constructions for cubic graphs with large girth, Electronic J. Combinatorics, 5 (1998), #A1.
[3]	D. Burshtein, Iterative approximate linear programming decoding of LDPC codes with linear complexity, IEEE Trans. Inform. Theory, 55 (2009), 4835-4859. 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), 3565-3578. doi: 10.1109/TIT.2008.926452.
[5]	D. Divsalar, H. Jin and R. J. McEliece, Coding theorems for Turbo-like codes, in "36th Allerton Conference on Communications, Control, and Computing,'' (1998), 201-210.
[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), 251-257.
[7]	J. Feldman, "Decoding Error-Correcting Codes via Linear Programming'', Ph.D thesis, MIT, 2003.
[8]	J. Feldman and D. R. Karger, Decoding turbo-like codes via linear programming, in "Proc. 43rd annual IEEE Symposium on Foundations of Computer Science (FOCS),'' (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), 82-89. 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), 954-972. doi: 10.1109/TIT.2004.842696.
[11]	R. G. Gallager, "Low-Density Parity-Check Codes,'' MIT Press, Cambridge, MA, USA, 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), 265-280. doi: 10.1109/TIT.2004.839509.
[13]	H. Jin and R. J. McEliece, Irregular repeat-accumlate codes, in "Proceedings Second International Conference on Turbo Codes and Related Topics,'' Brest, France, (2000), 1-8.
[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), 2352-2379. 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), 2376-2393. doi: 10.1109/TIT.2007.899563.
[16]	P. O. Vontobel and R. Koetter, Lower bounds on the minimum pseudoweight of linear codes, in "Proceedings of the 2004 International Symposium on Information Theory,'' (2004).
[17]	P. O. Vontobel and R. Koetter, Towards low-complexity linear-programming decoding, in "Proc. 4th Int. Symposium on Turbo Codes and Related Topics,'' Munich, Germany, (2006); preprint, arXiv:cs/0602088v1

show all references

References:

[1]	C. Berrou, A. Glavieux and P. Thitimajshima, Near Shannon limit errorcorrecting coding and decoding: turbo codes, in "Proc. 1993 IEEE International Conference on Communications,'' Geneva, Switzerland, (1993), 1064-1070. doi: 10.1109/ICC.1993.397441.
[2]	N. Biggs, Constructions for cubic graphs with large girth, Electronic J. Combinatorics, 5 (1998), #A1.
[3]	D. Burshtein, Iterative approximate linear programming decoding of LDPC codes with linear complexity, IEEE Trans. Inform. Theory, 55 (2009), 4835-4859. 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), 3565-3578. doi: 10.1109/TIT.2008.926452.
[5]	D. Divsalar, H. Jin and R. J. McEliece, Coding theorems for Turbo-like codes, in "36th Allerton Conference on Communications, Control, and Computing,'' (1998), 201-210.
[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), 251-257.
[7]	J. Feldman, "Decoding Error-Correcting Codes via Linear Programming'', Ph.D thesis, MIT, 2003.
[8]	J. Feldman and D. R. Karger, Decoding turbo-like codes via linear programming, in "Proc. 43rd annual IEEE Symposium on Foundations of Computer Science (FOCS),'' (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), 82-89. 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), 954-972. doi: 10.1109/TIT.2004.842696.
[11]	R. G. Gallager, "Low-Density Parity-Check Codes,'' MIT Press, Cambridge, MA, USA, 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), 265-280. doi: 10.1109/TIT.2004.839509.
[13]	H. Jin and R. J. McEliece, Irregular repeat-accumlate codes, in "Proceedings Second International Conference on Turbo Codes and Related Topics,'' Brest, France, (2000), 1-8.
[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), 2352-2379. 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), 2376-2393. doi: 10.1109/TIT.2007.899563.
[16]	P. O. Vontobel and R. Koetter, Lower bounds on the minimum pseudoweight of linear codes, in "Proceedings of the 2004 International Symposium on Information Theory,'' (2004).
[17]	P. O. Vontobel and R. Koetter, Towards low-complexity linear-programming decoding, in "Proc. 4th Int. Symposium on Turbo Codes and Related Topics,'' Munich, Germany, (2006); preprint, arXiv:cs/0602088v1

[1]	Beniamin Mounits, Tuvi Etzion, Simon Litsyn. New upper bounds on codes via association schemes and linear programming. Advances in Mathematics of Communications, 2007, 1 (2) : 173-195. doi: 10.3934/amc.2007.1.173
[2]	Sun-Sig Byun, Hongbin Chen, Mijoung Kim, Lihe Wang. L^p regularity theory for linear elliptic systems. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 121-134. doi: 10.3934/dcds.2007.18.121
[3]	Yael Ben-Haim, Simon Litsyn. A new upper bound on the rate of non-binary codes. Advances in Mathematics of Communications, 2007, 1 (1) : 83-92. doi: 10.3934/amc.2007.1.83
[4]	Jean Creignou, Hervé Diet. Linear programming bounds for unitary codes. Advances in Mathematics of Communications, 2010, 4 (3) : 323-344. doi: 10.3934/amc.2010.4.323
[5]	Ali Tebbi, Terence Chan, Chi Wan Sung. Linear programming bounds for distributed storage codes. Advances in Mathematics of Communications, 2020, 14 (2) : 333-357. doi: 10.3934/amc.2020024
[6]	Mohammed Mesk, Ali Moussaoui. On an upper bound for the spreading speed. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3897-3912. doi: 10.3934/dcdsb.2021210
[7]	Kwankyu Lee. Decoding of differential AG codes. Advances in Mathematics of Communications, 2016, 10 (2) : 307-319. doi: 10.3934/amc.2016007
[8]	J. De Beule, K. Metsch, L. Storme. Characterization results on weighted minihypers and on linear codes meeting the Griesmer bound. Advances in Mathematics of Communications, 2008, 2 (3) : 261-272. doi: 10.3934/amc.2008.2.261
[9]	Elisa Gorla, Felice Manganiello, Joachim Rosenthal. An algebraic approach for decoding spread codes. Advances in Mathematics of Communications, 2012, 6 (4) : 443-466. doi: 10.3934/amc.2012.6.443
[10]	Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505
[11]	Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433
[12]	Min Ye, Alexander Barg. Polar codes for distributed hierarchical source coding. Advances in Mathematics of Communications, 2015, 9 (1) : 87-103. doi: 10.3934/amc.2015.9.87
[13]	W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349
[14]	Yuntao Zang. An upper bound of the measure-theoretical entropy. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022052
[15]	Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385
[16]	Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83
[17]	Carla Mascia, Giancarlo Rinaldo, Massimiliano Sala. Hilbert quasi-polynomial for order domains and application to coding theory. Advances in Mathematics of Communications, 2018, 12 (2) : 287-301. doi: 10.3934/amc.2018018
[18]	Jianqin Zhou, Wanquan Liu, Xifeng Wang, Guanglu Zhou. On the $ k $-error linear complexity for $ p^n $-periodic binary sequences via hypercube theory. Mathematical Foundations of Computing, 2019, 2 (4) : 279-297. doi: 10.3934/mfc.2019018
[19]	Hannes Bartz, Antonia Wachter-Zeh. Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases. Advances in Mathematics of Communications, 2018, 12 (4) : 773-804. doi: 10.3934/amc.2018046
[20]	Sara D. Cardell, Joan-Josep Climent. An approach to the performance of SPC product codes on the erasure channel. Advances in Mathematics of Communications, 2016, 10 (1) : 11-28. doi: 10.3934/amc.2016.10.11

2020 Impact Factor: 0.935

Tools

Metrics

Other articles
by authors

on AIMS
Idan Goldenberg David Burshtein
on Google Scholar
Idan Goldenberg David Burshtein

[Back to Top]