Optimal threshold-based multi-trial error/erasure decoding with the Guruswami-Sudan algorithm

.We show new and improved list decoding properties of folded Reed–Solomon (RS) codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the source of recent advances in coding theory: folded RS codes were the first known explicit construction of capacity-achieving list decodable codes [V. Guruswami and A. Rudra, IEEE Trans. Inform. Theory, 54 (2008), pp. 135–150], and multiplicity codes were the first construction of high-rate locally decodable codes [S. Kopparty, S. Saraf, and S. Yekhanin, J. ACM, 61 (2014), 28]. In this work, we show that folded RS codes and multiplicity codes are in fact better than previously known in the context of list decoding and local list decoding. Our first main result shows that folded RS codes achieve list decoding capacity with constant list sizes, independent of the block length. Prior work with constant list sizes first obtained list sizes that are polynomial in the block length and relied on pre-encoding with subspace evasive sets to reduce the list sizes to a constant [V. Guruswami and C. Wang, IEEE Trans. Inform. Theory, 59 (2013), pp. 3257–3268], [Z. Dvir and S. Lovett, Proc. 44th STOC, ACM, 2012, 351–358]. The list size we obtain is \((1/\varepsilon )^{O(1/\varepsilon )}\) where \(\varepsilon\) is the gap to capacity, which matches the list size obtained by pre-encoding with subspace evasive sets. For our second main result, we observe that univariate multiplicity codes exhibit similar behavior, and we use this, together with additional ideas, to show that multivariate multiplicity codes are locally list decodable up to their minimum distance. By known reductions, this gives, in turn, capacity-achieving locally list decodable codes with query complexity \(\exp ( \tilde O((\log N)^{5/6}))\) . This improves on the tensor-based construction of [B. Hemenway, N. Ron-Zewi, and M. Wootters, SIAM J. Comput., 49 (2019), pp. 157–195], which gave capacity-achieving locally list decodable codes of query complexity \(N^{\tilde O(1/\log \log N)}\) , and is close to the best known query complexity of \(\exp ( \tilde O(\sqrt{\log N}))\) for high-rate locally (uniquely) decodable codes [S. Kopparty et al., J. ACM, 64 (2017), 11].Keywordserror–correcting codeslist decodinglocal decodingfolded Reed–Solomon codesmultiplicity codesMSC codes11T7194B0594B3594B6094B65

This chapter, as well as the next one, explore the relation between the list decoding radius and minimum distance of a code. Understanding the relation between these parameters is useful for two reasons: (a) for several important families of codes like Reed-Solomon codes, we have precise bounds on the distance, and one can use the relation between list decoding radius and distance to understand the list decoding potential of these codes; and (b) this shows that one approach to construct good list decodable codes is to construct large distance codes, and the latter is a relatively well-studied and better understood problem. Also, historically the most significant algorithmic results on list decoding have been fueled by an attempt to decode codes whose good minimum distance highlighted their good combinatorial list decodability properties.

Reed-Solomon (RS) codes are error-correcting codes used in many of today's communication systems. RS encoding and decoding are typically implemented using dedicated hardware elements used in ADSL modems and digital TV. In this paper we propose using software in lieu of a hardware-based RS de-coder. This is accomplished using the Berlekamp-Massey algorithm, implemented on a programmable DSP. This software-based RS decoder using Berlekamp-Massey is implemented on Motorola's MSC8101 StarCore DSP. In order to evaluate the algorithm we use the following criteria: computation cost; cycle count, critical paths in the decoding scheme, and error location in the codeword. Furthermore, we examine the effect of changing the RS code's k, t parameters and its primitive polynomial in real-time implementation. The Berlekamp-Massey algorithm operates over finite field arithmetic, whose steps, as applied on the StarCore DSP, are discussed in detail and specifically evaluated using the assembly code for the syndrome search. We conclude that the Berlekamp-Massey algorithm, used for RS decoding, should be implemented using long codewords.

This paper presents a high-throughput variable-length eight-parallel Reed-Solomon (RS) decoder for high-rate WPAN systems. The proposed architecture can support not only RS (255,239) code but also various shortened RS codes. Moreover, the variable-length architecture provides low latency for various shortened RS codes and the eight-parallel design also provides high data processing rate. Using 90-nm CMOS standard cell technology, the proposed variable-length RS decoder has been synthesized and measured for performance. The proposed RS decoder can provide a maximum 19-Gbps data rate at clock frequency 300 MHz.

Polar codes are the first provable capacity-achieving channel codes. Despite the splendid performance of long Polar codes, short Polar codes have relatively poor performance compared with other modern channel coding schemes (e.g., Turbo codes and LDPC). In this paper, we explore some practical methods to improve the performance of Polar codes with short to moderate codeword lengths. First, we use Reed Solomon (RS) codes as outer codes. With a specific interleaving strategy, we can concatenate multiple RS codes with one frame of Polar codes. Combining a strategy of allocating unequal RS code rates with the concatenation, different levels of protection are assigned based on the error pattern of successive cancellation list (SCL) decoders. Thus, the finite length performance will certainly be enhanced for this encoding scheme. Meanwhile, the memory size that the original SCL decoding procedure requires is reduced, and the increment of overall decoding complexity is small. Additionally, we propose an intra-frame interleaver to further enhance the performance by dispersing errors. Finally, we designed a list decoding scheme for the proposed multi-RS concatenated Polar codes. Depending on the soft information generated by an SCL decoder, we calculated the reliability of each RS symbol and conducted soft RS decoding. So, the overall performance was enhanced under this joint decoding strategy. Simulation results indicate that the bit error rate (BER) performance of short Polar codes can be well improved.

Techniques are presented for computing upper and lower bounds on the number of errors that can be corrected by list decoders for general block codes and, specifically, for Reed--Solomon (RS) codes. The list decoder of Guruswami and Sudan implies such a lower bound (referred to here as the GS bound) for RS codes. It is shown that this lower bound, given by means of the code's length, the minimum Hamming distance, and the maximal allowed list size, in fact applies to all block codes. Ranges of code parameters are identified where the GS bound is tight for worst-case RS codes, in which case the list decoder of Guruswami and Sudan provably corrects the largest possible number of errors. On the other hand, ranges of parameters are provided for which the GS lower bound can be strictly improved. In some cases the improvement applies to all block codes with a given minimum Hamming distance, while in others it applies only to RS codes.

Let $q$ be a power of 2 and $\gamma\le q$ an integer. Based on the codewords of $[q,2,q-1]$ extended Reed-Solomon (RS) code over the finite field $\mathbb {F}_q$ , we can construct a $(\gamma,q)$ -regular low-density parity-check (LDPC) code, called a full-length RS-LDPC code and denoted by $\mathcal{C}(\gamma,q)$ . In this letter, the minimum distance of these codes is investigated. For any given $q$ and $\gamma\le q$ , an upper bound on $d(\mathcal{C}(\gamma,q))$ , the minimum distance of $\mathcal{C}(\gamma,q)$ , is provided. Furthermore, we determine the values of $d(\mathcal{C}(\gamma,q))$ for $\gamma=2$ , 3, and 4, and present the closed-form expressions on the numbers of minimum-weight codewords in $\mathcal{C}(\gamma,q)$ for $\gamma=2$ and 3.

We construct an explicit family of linear rank-metric codes over any field F that enables efficient list decoding up to a fraction rho of errors in the rank metric with a rate of 1-rho-eps, for any desired rho in (0,1) and eps > 0. Previously, a Monte Carlo construction of such codes was known, but this is in fact the first explicit construction of positive rate rank-metric codes for list decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an F-subspace that evades certain structured subspaces over an extension field of F. These structured spaces arise from the linear-algebraic list decoder for Gabidulin codes due to Guruswami and Xing (STOC'13). Our construction is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which are a collection of subspaces, every pair of which have low-dimensional intersection, and which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order that are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list decoding RS codes reduces to list decoding such folded RS codes. However, as we only list decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list decoding RS codes.

This paper presents a high-speed low-complexity Reed-Solomon (RS) decoder architecture using a novel pipelined recursive modified Euclidean (PrME) algorithm block for very high-speed optical communications. The RS decoder features a low-complexity key equation solver using a PrME algorithm block. The recursive structure enables the novel low-complexity PrME algorithm block to be implemented. Pipelining and parallelizing allow the inputs to be received at very high fiber-optic rates, and outputs to be delivered at correspondingly high rates with minimum delay. This paper presents the key ideas applied to the design of an 80-Gb/s RS decoder architecture, especially that for achieving high throughput and reducing complexity. The 80-Gb/s 16-channel RS decoder has been designed and implemented using 0.13-/spl mu/m CMOS technology in a supply voltage of 1.2 V. The proposed RS decoder has a core gate count of 393 K and operates at a clock rate of 625 MHz.

Reed-Solomon (RS) codes have been widely used in a variety of communication systems to protect digital data against errors occurring in the transmission process. Since the decoding process for RS codes is rather computation-extensive, special-purpose hardware structures are often necessary for it to meet the real-time requirements. In this paper, an area-efficient pipelined very large scale integration (VLSI) architecture is proposed for RS decoding. The architecture is developed based on a time domain algorithm using the remainder decoding concept. A prominent feature of the proposed system is that, for a t-error-correcting RS code with block length n, it involves only 2t consecutive symbols to compute a discrepancy value in the decoding process, instead of n consecutive symbols used in the previous RS decoders based on the same algorithm without using the remainder decoding concept. The proposed RS decoder can process one data block every n clock cycles, i.e., the average decoding rate is one symbol per clock cycle. As compared to a similar pipelined RS decoder with the same decoding rate, it gains significant improvements in hardware complexity and latency.

In this paper, a low-power Reed-Solomon (RS) decoder for STM-16 optical communications is presented. It mainly contains one (255,239) RS decoder and four 2 K-bit embedded memory for correcting the received codewords. Except the novel syndrome calculator reducing half the syndrome computations, our proposal also features a modified Berlekamp-Massey algorithm in the key equation solver and a terminated mechanism in the Chien search circuit. The (255,239) RS decoder is implemented by 0.25 /spl mu/m CMOS 1P5M standard cells with gate counts of 32.9 K and area of 2.03 mm/sup 2/. Simulation results show our approach can work successfully at the data rate of 2.5-Gbps and achieve 80% reduction of power dissipation on the average.

Given an algebraic geometry code $${C_{\mathcal L}(D, \alpha P)}$$ , the Guruswami---Sudan algorithm produces a list of all codewords in $${C_{\mathcal L}(D, \alpha P)}$$ within a specified distance of a received word. The initialization step in the algorithm involves parameter choices that bound the degree of the interpolating polynomial and hence the length of the list of codewords generated. In this paper, we use simple properties of discriminants of polynomials over finite fields to provide improved parameter choices for the Guruswami---Sudan list decoding algorithm for algebraic geometry codes. As a consequence, we obtain a better bound on the list size as well as a lower degree interpolating polynomial.

Reed-Solomon (RS) codes are used as error-correcting codes in numerous digital communication and storage systems. Algebraic soft-decision decoding (ASD) of RS codes can achieve substantial coding gain with polynomial complexity. Among practical ASD algorithms, the iterative bit-level generalized minimum distance (BGMD) decoding can achieve similar or higher coding gain with lower complexity. The interpolation is a major step of ASD. The maximum achievable speed of this step is limited by the inherent serial nature of the interpolation algorithm. In this paper, a novel interpolation scheme that is capable of combining multiple interpolation iterations, as well as sharing interpolation results from previous decoding iterations, is developed for the iterative BGMD decoding. In addition, efficient VLSI architectures are proposed to implement the developed scheme. Based on the proposed architectures, an interpolator for a (255, 239) RS code is implemented on field programmable gate array (FPGA) devices. On a Xilinx Virtex-II device, our interpolator can achieve a throughput of 440 Mbps, which is 64% higher than the fastest previous design, with 51% less FPGA resource.

In 2000 Roth and Ruckenstein proposed an extended key equation for solving the interpolation step in the Sudan decoding algorithm. Generalizing their idea, a sequence of key equations for the Guruswami-Sudan (GS) algorithm, which is able to list decode a Reed-Solomon code with arbitrary rate, is derived. This extension allows a reduction of the number of equations and therefore a reduction of the algorithmpsilas complexity. Furthermore, we indicate how to adapt the fundamental iterative algorithm for block Hankel matrices and thus solving the GS-interpolation step efficiently.

A major challenge in the area of space time (ST) codes is to find codes suitable for efficient decoding, thus overcoming the problem of many existing ST code designs which require maximum-likelihood (ML) decoding. A solution could be to apply single-input single-output (SISO) channel codes and theory over temporal channel fading to the multi-input single-output (MISO) code construction and classical suboptimum decoding methods. For these purposes, an ST code construction which allows the use of efficient decoding algorithms is described. We propose a concatenated code, where the inner code is the diagonal ST Hadamard (D-STH) code with Paley constructions and the outer code is an algebraic block code, such as a Reed-Solomon (RS) code. As a decoding method, we investigate a bounded minimum distance (BMD) with erasure decoding algorithm. A simple method to achieve the optimum threshold for deciding on an erased symbol is derived. Using this, the proposed concatenated scheme is able to exploit almost all of the spatial diversity of the system.