List decoding

About: List decoding is a research topic. Over the lifetime, 7251 publications have been published within this topic receiving 151182 citations.

Recursive decoding and its performance for low-rate Reed-Muller codes

Abstract: Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length n and fixed order r. An algorithm is designed that has complexity of order nlogn and corrects most error patterns of weight up to n(1/2-/spl epsiv/) given that /spl epsiv/ exceeds n/sup -1/2r/. 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 n grows.

Low-Latency Reweighted Belief Propagation Decoding for LDPC Codes

Abstract: In this paper we propose a novel message passing algorithm which exploits the existence of short cycles to obtain performance gains by reweighting the factor graph. The proposed decoding algorithm is called variable factor appearance probability belief propagation (VFAP-BP) algorithm and is suitable for wireless communications applications with low-latency and short blocks. Simulation results show that the VFAP-BP algorithm outperforms the standard BP algorithm, and requires a significantly smaller number of iterations when decoding either general or commercial LDPC codes.

The random coding bound is tight for the average code (Corresp.)

Abstract: The random coding bound of information theory provides a well-known upper bound to the probability of decoding error for the best code of a given rate and block length. The bound is constructed by upper-bounding the average error probability over an ensemble of codes. The bound is known to give the correct exponential dependence of error probability on block length for transmission rates above the critical rate, but it gives an incorrect exponential dependence at rates below a second lower critical rate. Here we derive an asymptotic expression for the average error probability over the ensemble of codes used in the random coding bound. The result shows that the weakness of the random coding bound at rates below the second critical rate is due not to upperbounding the ensemble average, but rather to the fact that the best codes are much better than the average at low rates.

Population Coding with Correlation and an Unfaithful Model

Abstract: This study investigates a population decoding paradigm in which the maximum likelihood inference is based on an unfaithful decoding model (UMLI). This is usually the case for neural population decoding because the encoding process of the brain is not exactly known or because a simplified decoding model is preferred for saving computational cost. We consider an unfaithful decoding model that neglects the pair-wise correlation between neuronal activities and prove that UMLI is asymptotically efficient when the neuronal correlation is uniform or of limited range. The performance of UMLI is compared with that of the maximum likelihood inference based on the faithful model and that of the center-of-mass decoding method. It turns out that UMLI has advantages of decreasing the computational complexity remarkably and maintaining high-level decoding accuracy. Moreover, it can be implemented by a biologically feasible recurrent network (Pouget, Zhang, Deneve, & Latham, 1998). The effect of correlation on the decoding accuracy is also discussed.

List decoding Reed-Solomon, Algebraic-Geometric, and Gabidulin subcodes up to the Singleton bound.

Abstract: We consider Reed-Solomon (RS) codes whose evaluation points belong to a subfield, and give a linear-algebraic list decoding algorithm that can correct a fraction of errors approaching the code distance, while pinning down the candidate messages to a well-structured affine space of dimension a constant factor smaller than the code dimension. By pre-coding the message polynomials into a subspace-evasive set, we get a Monte Carlo construction of a subcode of Reed-Solomon codes that can be list decoded from a fraction (1-R-e) of errors in polynomial time (for any fixed e > 0) with a list size of O(1/e). Our methods extend to algebraic-geometric (AG) codes, leading to a similar claim over constant-sized alphabets. This matches parameters of recent results based on folded variants of RS and AG codes. but our construction here gives subcodes of Reed-Solomon and AG codes themselves (albeit with restrictions on the evaluation points).Further, the underlying algebraic idea also extends nicely to Gabidulin's construction of rank-metric codes based on linearized polynomials. This gives the first construction of positive rate rank-metric codes list decodable beyond half the distance, and in fact gives codes of rate R list decodable up to the optimal (1-R-e) fraction of rank errors. A similar claim holds for the closely related subspace codes studied by Koetter and Kschischang.We introduce a new notion called subspace designs as another way to pre-code messages and prune the subspace of candidate solutions. Using these, we also get a deterministic construction of a polynomial time list decodable subcode of RS codes. By using a cascade of several subspace designs, we extend our approach to AG codes, which gives the first deterministic construction of an algebraic code family of rate R with efficient list decoding from 1-R-e fraction of errors over an alphabet of constant size (that depends only on e). The list size bound is almost a constant (governed by log* (block length)), and the code can be constructed in quasi-polynomial time.