Abstract
New efficient methods are developed for the optimal maximum-likelihood (ML) decoding of an arbitrary binary linear code based on data received from any discrete Gaussian channel. The decoding algorithm is based on monotonic optimization that is minimizing a difference of monotonic (d.m.) objective functions subject to the 0–1 constraints of bit variables. The iterative process converges to the global optimal ML solution after finitely many steps. The proposed algorithm’s computational complexity depends on input sequence length k which is much less than the codeword length n, especially for a codes with small code rate. The viability of the developed is verified through simulations on different coding schemes.
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Berlekamp E.R., McElice R.J., van Tiborg H.C.A.: On the intractibility of certain coding problems. IEEE Trans. Inf. Theory 24, 384–386 (1978)
Feldman J., Wainwright M.J., Karger D.R.: Using linear programming to decode binary linear codes. IEEE Trans. Inf. Theory 51, 954–972 (2005)
Frank M., Wolfe P.: An algorithm for quadratic programming. Naval Res. Log. Q. 3, 95–110 (1956)
Gallager R.G.: Low Density Parity Check Codes. MIT Press, Cambridge (1962)
Hasselberg J., Pardalos P.M., Vairaktarakis G.: Test case generators and computational results for the maximum clique problem. J. Glob. Optim. 3, 463–482 (1993)
Kschischang F.R., Frey B.J., Loeliger H.A.: Factor graphs and sum-product algorithm. IEEE Trans. Inf. Theory 47, 498–519 (2001)
LDPC toolkit for Matlab. http://arun-10.tripod.com/ldpc/ldpc.htm
Mackay D.J.C., Neal R.M.: Near Shannon limit performance of low density parity check codes. Electron. Lett. 32, 1645–1646 (1996)
Mackay D.J.C.: Good error-correcting codes based on very sparse matrices. IEEE Trans. Inf. Theory 45, 399–431 (1999)
Margulis G.A.: Explicit constructions of graphs without short cycles and low density codes. Combinatorica 2, 71–78 (1982)
McEliece R., MacKay D., Cheng J.: Turbo decoding as an instance of Pearl’s belief propagation algorithm. IEEE J. Sel. Areas Commun. 16, 140–152 (1998)
Pardalos P.M., Romeijn E., Tuy H.: Recent developments and trends in global optimization. J. Comput. Appl. Math. 124, 209–228 (2000)
Pearl J.: Probabilistic Reasoning in Intelligent Systems. Margan Kaufmann, Los Altos (1988)
Tanner R.M.: A recursive approach to low complexity codes. IEEE Trans. Inf. Theory 27, 533–547 (1981)
Tuy H.: Convex Analysis and Global Optimization. Kluwer Academic, New York (1999)
Tuy H., Minoux M., Phuong N.T.H.: Discrete monotonic optimization with application to a discrete location problem. SIAM J. Optim. 17, 78–97 (2006)
Tuy H.: Monotonic optimization: problems and solution approaches. SIAM J. Optim. 11, 464–494 (2000)
Yang K., Feldman J., Wang X.: Nonlinear progrramming approachs to decoding low-density parity-check codes. IEEE J. Sel. Areas Commun. 24, 1603–1613 (2006)
Yedidia J.S., Freeman W.T., Weiss Y.: Understanding Belief Propagation and its Generalizations. In: Lakemeyer, G., Nebel, B. (eds) Exploring Artificial Intelligence in the New Millenium chap 8., pp. 239–269. Morgan Kaufmann, CA (2003)
Yedidia J.S., Freeman W.T., Weiss Y.: Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans. Inf. Theory 51, 2282–2312 (2005)
Yuille A.L.: CCCP algorithms to minimize the Bethe and Kikuchi energies: convergent alternatives to belief propagation. Neural Comput. 14, 1691–1722 (2002)
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Tuan, H.D., Son, T.T., Tuy, H. et al. Monotonic optimization based decoding for linear codes. J Glob Optim 55, 301–312 (2013). https://doi.org/10.1007/s10898-011-9816-9
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DOI: https://doi.org/10.1007/s10898-011-9816-9