Abstract-Recursive list decoding is considered for Reed-Muller (RM)codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed when these recalculations reach the trivial RM codes. In turn, the updated lists of most plausible codewords are used in subsequent decodings. The algorithm is further improved by using permutation techniques on code positions and by eliminating the most error-prone information bits. Simulation results show that for all RM codes of length 256 and many subcodes of length 512, these algorithms approach maximum-likelihood (ML) performance within a margin of 0.1 dB. As a result, we present tight experimental bounds on ML performance for these codes.Index Terms-Maximum-likelihood (ML) performance, Plotkin construction, posterior probabilities, recursive lists, Reed-Muller (RM) codes. I. INTRODUCTIONThe main goal of this correspondence is to design feasible error-correcting algorithms that approach maximum-likelihood (ML) decoding on the moderate lengths ranging from 100 to 1000 bits. The problem is practically important due to the void left on these lengths by the best algorithms known to date. In particular, exact ML decoding has huge decoding complexity even on blocks of 100 bits. On the other hand, Manuscript received June 4, 2004; revised November 15, 2005. This work was supported by the National Science Foundation under Grant CCR-0097125. The material in this correspondence was presented in part at the 38th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, October 2000.I. Dumer is with the College of Engineering, University of California, Riverside, CA 92521 USA (e-mail: dumer@ee.ucr.edu).K. Shabunov is with the XVD Corporation, San Jose, CA 95134 USA (e-mail: kshabunov@xvdcorp.com currently known iterative (message-passing) algorithms have been efficient only on blocks of thousands of bits. To achieve near-ML performance with moderate complexity, we wish to use recursive techniques that repeatedly split an original code into the shorter ones. For this reason, we consider Reed-Muller (RM) codes, which represent the most notable example of recursive constructions known to date. These codes-denoted below asf m r g-have length n = 2 m and Hamming distance d = 2 m0r . They also admit a simple recursive structure based on the Plotkin construction (u u u; u u u + v v v); which splits the original RM code into the two shorter codes of length 2 m01 . This structure was efficiently used in recursive decoding algorithms of [2]- [4], which derive the corrupted symbols of the shorter codes u u u and v v v from the received symbols. These recalculations are then repeated until the process reaches the repetition codes or full spaces, whereupon new information symbols can be retrieved by any powerful algorithm-say, ML decoding. As a result, recursive algorithms achieve bounded distance decoding with a low complexity order of n minfr; m 0 rg, whi...

Abstract-Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length and fixed order . An algorithm is designed that has complexity of order log and corrects most error patterns of weight up to (1 2 ) given that exceeds 1 2 . This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity.To evaluate decoding capability, we develop a probabilistic technique that disintegrates decoding into a sequence of recursive steps. Although dependent, subsequent outputs can be tightly evaluated under the assumption that all preceding decodings are correct. In turn, this allows us to employ second-order analysis and find the error weights for which the decoding error probability vanishes on the entire sequence of decoding steps as the code length grows.

We show that the minimum distance of a linear code is not approximable to within any constant factor in random polynomial time (RP), unless nondeterministic polynomial time (NP) equals RP. We also show that the minimum distance is not approximable to within an additive error that is linear in the block length of the code. Under the stronger assumption that NP is not contained in random quasi-polynomial time (RQP), we show that the minimum distance is not approximable to within the factor 2 log ( ) , for any 0. Our results hold for codes over any finite field, including binary codes. In the process, we show that it is hard to find approximately nearest codewords even if the number of errors exceeds the unique decoding radius 2 by only an arbitrarily small fraction . We also prove the hardness of the nearest codeword problem for asymptotically good codes, provided the number of errors exceeds (2 3) . Our results for the minimum distance problem strengthen (though using stronger assumptions) a previous result of Vardy who showed that the minimum distance cannot be computed exactly in deterministic polynomial time (P), unless P = NP. Our results are obtained by adapting proofs of analogous results for integer lattices due to Ajtai and Micciancio. A critical component in the adaptation is our use of linear codes that perform better than random (linear) codes.Index Terms-Approximation algorithms, bounded distance decoding, computational complexity, dense codes, linear codes, minimum-distance problem, NP-hardness, relatively near codeword problem.

We suggest a technique for constructing lower (existence) bounds for the fault-tolerant threshold to scalable quantum computation applicable to degenerate quantum codes with sublinear distance scaling. We give explicit analytic expressions combining probabilities of erasures, depolarizing errors, and phenomenological syndrome measurement errors for quantum LDPC codes with logarithmic or larger distances. These threshold estimates are parametrically better than the existing analytical bound based on percolation.

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