List decoding

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

Codes for Computationally Simple Channels: Explicit Constructions with Optimal Rate

Abstract: In this paper, we consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter p and (b) the process which adds the errors can be described by a sufficiently "simple" circuit. Codes for such channel models are attractive since, like codes for standard adversarial errors, they can handle channels whose true behavior is unknown or varying over time. For three classes of channels, we provide explicit, efficiently encodable/decodable codes of optimal rate where only inefficiently decodable codes were previously known. In each case, we provide one encoder/decoder that works for every channel in the class. Unique decoding for additive errors: We give the first construction of a poly-time encodable/decodable code for additive (a.k.a. oblivious) channels that achieve the Shannon capacity 1-H(p). List-decoding for online log-space channels: A space-S(N) bounded channel reads and modifies the transmitted codeword as a stream, using at most S(N) bits of workspace on transmissions of N bits. For constant S, this captures many models from the literature, including "discrete channels with finite memory" and "arbitrarily varying channels". We give an efficient code with optimal rate (arbitrarily close to 1-H(p)) that recovers a short list containing the correct message with high probability for channels which read and modify the transmitted codeword as a stream, using at most O(\log N) bits of workspace on transmissions of N bits. List-decoding for poly-time channels: For any constant c we give a similar list-decoding result for channels describable by circuits of size at most N^c, assuming the existence of pseudorandom generators.

Joint source/channel coding and MAP decoding of arithmetic codes

Abstract: In this paper, a novel maximum a posteriori (MAP) estimation approach is employed for error correction of arithmetic codes with a forbidden symbol. The system is founded on the principle of joint source channel coding, which allows one to unify the arithmetic decoding and error correction tasks into a single process, with superior performance compared to traditional separated techniques. The proposed system improves the performance in terms of error correction with respect to a separated source and channel coding approach based on convolutional codes, with the additional great advantage of allowing complete flexibility in adjusting the coding rate. The proposed MAP decoder is tested in the case of image transmission across the additive white Gaussian noise channel and compared against standard forward error correction techniques in terms of performance and complexity. Both hard and soft decoding are taken into account, and excellent results in terms of packet error rate and decoded image quality are obtained.

Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes

Abstract: Folded Reed-Solomon (RS) codes are an explicit family of codes that achieve the optimal tradeoff between rate and list error-correction capability: specifically, for any e > 0, Guruswami and Rudra presented an nO(1/ e) time algorithm to list decode appropriate folded RS codes of rate R from a fraction 1-R-e of errors. The algorithm is based on multivariate polynomial interpolation and root-finding over extension fields. It was noted by Vadhan that interpolating a linear polynomial suffices for a statement of the above form. Here, we give a simple linear-algebra-based analysis of this variant that eliminates the need for the computationally expensive root-finding step over extension fields (and indeed any mention of extension fields). The entire list-decoding algorithm is linear-algebraic, solving one linear system for the interpolation step, and another linear system to find a small subspace of candidate solutions. Except for the step of pruning this subspace, the algorithm can be implemented to run in quadratic time. We also consider a closely related family of codes, called (order m) derivative codes and defined over fields of large characteristic, which consist of the evaluations of f as well as its first m-1 formal derivatives at N distinct field elements. We show how our linear-algebraic methods for folded RS codes can be used to show that derivative codes can also achieve the above optimal tradeoff. The theoretical drawback of our analysis for folded RS codes and derivative codes is that both the decoding complexity and proven worst-case list-size bound are nΩ(1/ e). By combining the above idea with a pseudorandom subset of all polynomials as messages, we get a Monte Carlo construction achieving a list-size bound of O(1/ e2) which is quite close to the existential O(1/ e) bound (however, the decoding complexity remains nΩ(1/ e)). Our work highlights that constructing an explicit subspace-evasive subset that has small intersection with low-dimensional subspaces-an interesting problem in pseudorandomness in its own right-could lead to explicit codes with better list-decoding guarantees.

Efficient message passing scheduling for terminated LDPC convolutional codes

Abstract: Message passing schedules that reduce the decoding complexity of terminated LDPC convolutional code ensembles are analyzed. Considering the AWGN channel, various schedules are compared by means of density evolution. The results of the analysis together with computer simulations for some (3,6)-regular codes confirm that sliding window decoding is an attractive practical solution for low-latency and low-complexity decoding.

Large Scale LP Decoding with Low Complexity

Abstract: Linear program (LP) decoding has become increasingly popular for error-correcting codes due to its simplicity and promising performance. Low-complexity and efficient iterative algorithms for LP decoding are of great importance for practical applications. In this paper we focus on solving the binary LP decoding problem by using the alternating direction method of multipliers (ADMM). Our main contribution is that we propose a linear-complexity algorithm for the projection onto a parity polytope (having a computational complexity of small O(d), where small d is the check-node degree), as compared to recent work , which has a computational complexity of small O(d log d). In particular, we show that the projection onto the parity polytope can be transformed to a projection onto a simplex.