Iterative Algebraic Soft-Decision List Decoding of Reed-Solomon Codes

In this paper, we present an iterative soft-decision decoding algorithm for Reed-Solomon codes offering both complexity and performance advantages over previously known decoding algorithms. Our algorithm is a list decoding algorithm which combines two powerful soft decision decoding techniques which were previously regarded in the literature as competitive, namely, the Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation based on adaptive parity check matrices, recently proposed by Jiang and Narayanan. Building on the Jiang-Narayanan algorithm, we present a belief-propagation based algorithm with a significant reduction in computational complexity. We introduce the concept of using a belief-propagation based decoder to enhance the soft-input information prior to decoding with an algebraic soft-decision decoder. Our algorithm can also be viewed as an interpolation multiplicity assignment scheme for algebraic soft-decision decoding of Reed-Solomon codes.Comment: Submitted to IEEE for publication in Jan 200

Iterative Soft Input Soft Output Decoding of Reed-Solomon Codes by Adapting the Parity Check Matrix

An iterative algorithm is presented for soft-input-soft-output (SISO) decoding of Reed-Solomon (RS) codes. The proposed iterative algorithm uses the sum product algorithm (SPA) in conjunction with a binary parity check matrix of the RS code. The novelty is in reducing a submatrix of the binary parity check matrix that corresponds to less reliable bits to a sparse nature before the SPA is applied at each iteration. The proposed algorithm can be geometrically interpreted as a two-stage gradient descent with an adaptive potential function. This adaptive procedure is crucial to the convergence behavior of the gradient descent algorithm and, therefore, significantly improves the performance. Simulation results show that the proposed decoding algorithm and its variations provide significant gain over hard decision decoding (HDD) and compare favorably with other popular soft decision decoding methods.Comment: 10 pages, 10 figures, final version accepted by IEEE Trans. on Information Theor

The hybrid list decoding and Chase-like algorithm of Reed-Solomon codes.

Thesis (M.Sc.Eng.)-University of KwaZulu-Natal, 2005Reed-Solomon (RS) codes are powerful error-correcting codes that can be found in a wide variety of digital communications and digital data-storage systems. Classical hard decoder of RS code can correct t = (dmin -1) /2 errors where dmin = (n - k+ 1) is the minimum distance of the codeword, n is the length of codeword and k is the dimension of codeword. Maximum likelihood decoding (MLD) performs better than the classical decoding and therefore how to approach the performance of the MLD with less complexity is a subject which has been researched extensively. Applying the bit reliability obtained from channel to the conventional decoding algorithm is always an efficient technique to approach the performance of MLD, although the exponential increase of complexity is always concomitant. It is definite that more enhancement of performance can be achieved if we apply the bit reliability to enhanced algebraic decoding algorithm that is more powerful than conventional decoding algorithm. In 1997 Madhu Sudan, building on previous work of Welch-Berlekamp, and others, discovered a polynomial-time algorithm for decoding low-rate Reed- Solomon codes beyond the classical error-correcting bound t = (dmin -1) /2. Two years later Guruswami and Sudan published a significantly improved version of Sudan's algorithm (GS), but these papers did not focus on devising practical implementation. The other authors, Kotter, Roth and Ruckenstein, were able to find realizations for the key steps in the GS algorithm, thus making the GS algorithm a practical instrument in transmission systems. The Gross list algorithm, which is a simplified one with less decoding complexity realized by a reencoding scheme, is also taken into account in this dissertation. The fundamental idea of the GS algorithm is to take advantage of an interpolation step to get an interpolation polynomial produced by support symbols, received symbols and their corresponding multiplicities. After that the GS algorithm implements a factorization step to find the roots of the interpolation polynomial. After comparing the reliability of these codewords which are from the output of factorization, the GS algorithm outputs the most likely one. The support set, received set and multiplicity set are created by Koetter Vardy (KV) front end algorithm. In the GS list decoding algorithm, the number of errors that can be corrected increases to tcs = n - 1 - lJ (k - 1) n J. It is easy to show that the GS list decoding algorithm is capable of correcting more errors than a conventional decoding algorithm. In this dissertation, we present two hybrid list decoding and Chase-like algorithms. We apply the Chase algorithms to the KV soft-decision front end. Consequently, we are able to provide a more reliable input to the KV list algorithm. In the application of Chase-like algorithm, we take two conditions into consideration, so that the floor cannot occur and more coding gains are possible. With an increase of the bits that are chosen by the Chase algorithm, the complexity of the hybrid algorithm increases exponentially. To solve this problem an adaptive algorithm is applied to the hybrid algorithm based on the fact that as signal-to-noise ratio (SNR) increases the received bits are more reliable, and not every received sequence needs to create the fixed number of test error patterns by the Chase algorithm. We set a threshold according to the given SNR and utilize it to finally decide which unreliable bits are picked up by Chase algorithm. However, the performance of the adaptive hybrid algorithm at high SNRs decreases as the complexity decreases. It means that the adaptive algorithm is not a sufficient mechanism for eliminating the redundant test error patterns. The performance of the adaptive hybrid algorithm at high SNRs motivates us to find out another way to reduce the complexity without loss of performance. We would consider the two following problems before dealing with the problem on hand. One problem is: can we find a terminative condition to decide which generated candidate codeword is the most likely codeword for received sequence before all candidates of received set are tested? Another one is: can we eliminate the test error patterns that cannot create more likely codewords than the generated codewords? In our final algorithm, an optimality lemma coming from the Kaneko algorithm is applied to solve the first problem and the second problem is solved by a ruling out scheme for the reduced list decoding algorithm. The Gross list algorithm is also applied in our final hybrid algorithm. After the two problems have been solved, the final hybrid algorithm has performance comparable with the hybrid algorithm combined the KV list decoding algorithm and the Chase algorithm but much less complexity at high SNRs

Error-correction coding for high-density magnetic recording channels.

Finally, a promising algorithm which combines RS decoding algorithm with LDPC decoding algorithm together is investigated, and a reduced-complexity modification has been proposed, which not only improves the decoding performance largely, but also guarantees a good performance in high signal-to-noise ratio (SNR), in which area an error floor is experienced by LDPC codes.The soft-decision RS decoding algorithms and their performance on magnetic recording channels have been researched, and the algorithm implementation and hardware architecture issues have been discussed. Several novel variations of KV algorithm such as soft Chase algorithm, re-encoded Chase algorithm and forward recursive algorithm have been proposed. And the performance of nested codes using RS and LDPC codes as component codes have been investigated for bursty noise magnetic recording channels.Future high density magnetic recoding channels (MRCs) are subject to more noise contamination and intersymbol interference, which make the error-correction codes (ECCs) become more important. Recent research of replacement of current Reed-Solomon (RS)-coded ECC systems with low-density parity-check (LDPC)-coded ECC systems obtains a lot of research attention due to the large decoding gain for LDPC-coded systems with random noise. In this dissertation, systems aim to maintain the RS-coded system using recent proposed soft-decision RS decoding techniques are investigated and the improved performance is presented

New Approaches to the Analysis and Design of Reed-Solomon Related Codes

The research that led to this thesis was inspired by Sudan's breakthrough that demonstrated that Reed-Solomon codes can correct more errors than previously thought. This breakthrough can render the current state-of-the-art Reed-Solomon decoders obsolete. Much of the importance of Reed-Solomon codes stems from their ubiquity and utility. This thesis takes a few steps toward a deeper understanding of Reed-Solomon codes as well as toward the design of efficient algorithms for decoding them. After studying the binary images of Reed-Solomon codes, we proceeded to analyze their performance under optimum decoding. Moreover, we investigated the performance of Reed-Solomon codes in network scenarios when the code is shared by many users or applications. We proved that Reed-Solomon codes have many more desirable properties. Algebraic soft decoding of Reed-Solomon codes is a class of algorithms that was stirred by Sudan's breakthrough. We developed a mathematical model for algebraic soft decoding. By designing Reed-Solomon decoding algorithms, we showed that algebraic soft decoding can indeed approach the ultimate performance limits of Reed-Solomon codes. We then shifted our attention to products of Reed-Solomon codes. We analyzed the performance of linear product codes in general and Reed-Solomon product codes in particular. Motivated by these results we designed a number of algorithms, based on Sudan's breakthrough, for decoding Reed-Solomon product codes. Lastly, we tackled the problem of analyzing the performance of sphere decoding of lattice codes and linear codes, e.g., Reed-Solomon codes, with an eye on the tradeoff between performance and complexity.</p

Comparison of code rate and transmit diversity in MIMO systems

A thesis submitted in ful lment of the requirements for the degree of Master of Science in the Centre of Excellence in Telecommunications and Software School of Electrical and Information Engineering, March 2016In order to compare low rate error correcting codes to MIMO schemes with transmit diversity, two systems with the same throughput are compared. A VBLAST MIMO system with (15; 5) Reed-Solomon coding is compared to an Alamouti MIMO system with (15; 10) Reed-Solomon coding. The latter is found to perform signi cantly better, indicating that transmit diversity is a more e ective technique for minimising errors than reducing the code rate. The Guruswami-Sudan/Koetter-Vardy soft decision decoding algorithm was implemented to allow decoding beyond the conventional error correcting bound of RS codes and VBLAST was adapted to provide reliability information. Analysis is also performed to nd the optimal code rate when using various MIMO systems.MT201

Advanced channel coding techniques using bit-level soft information

In this dissertation, advanced channel decoding techniques based on bit-level soft information are studied. Two main approaches are proposed: bit-level probabilistic iterative decoding and bit-level algebraic soft-decision (list) decoding (ASD). In the first part of the dissertation, we first study iterative decoding for high density parity check (HDPC) codes. An iterative decoding algorithm, which uses the sum product algorithm (SPA) in conjunction with a binary parity check matrix adapted in each decoding iteration according to the bit-level reliabilities is proposed. In contrast to the common belief that iterative decoding is not suitable for HDPC codes, this bit-level reliability based adaptation procedure is critical to the conver-gence behavior of iterative decoding for HDPC codes and it significantly improves the iterative decoding performance of Reed-Solomon (RS) codes, whose parity check matrices are in general not sparse. We also present another iterative decoding scheme for cyclic codes by randomly shifting the bit-level reliability values in each iteration. The random shift based adaptation can also prevent iterative decoding from getting stuck with a significant complexity reduction compared with the reliability based parity check matrix adaptation and still provides reasonable good performance for short-length cyclic codes. In the second part of the dissertation, we investigate ASD for RS codes using bit-level soft information. In particular, we show that by carefully incorporating bit¬level soft information in the multiplicity assignment and the interpolation step, ASD can significantly outperform conventional hard decision decoding (HDD) for RS codes with a very small amount of complexity, even though the kernel of ASD is operating at the symbol-level. More importantly, the performance of the proposed bit-level ASD can be tightly upper bounded for practical high rate RS codes, which is in general not possible for other popular ASD schemes. Bit-level soft-decision decoding (SDD) serves as an efficient way to exploit the potential gain of many classical codes, and also facilitates the corresponding per-formance analysis. The proposed bit-level SDD schemes are potential and feasible alternatives to conventional symbol-level HDD schemes in many communication sys-tems