Reed-Solomon Codes

Non-binary multithreshold decoding (qMTD) for q-ary self-orthogonal codes (qSOC) is considered. The SER performance of qMTD is shown to be close to the results provided by optimum total search methods, which are not realizable for non-binary codes in general. qMTD decoders are compared with different decoders for Reed-Solomon codes. The performance provided with qMTD in some cases is unattainable with classical decoders for arbitrary long Reed-Solomon codes. The result of concatenation of qSOC with simple to decode outer codes is described. Method of improving of qMTD decoder's performance for qSOC is offered. Some simulated results obtained by using these two decoding techniques (the base and modified ways) are presented as well. Comparison of the results showed that the change in the threshold element's algorithm can significantly improve speed of qMTD work. It's shown that for a larger gain this modification qMTD should be used after conventional decoding iterations.

Algebraic soft-decision decoding (ASD) of Reed-Solomon (RS) codes can provide substantial coding gain with polynomial complexity. Among the ASD algorithms with practical multiplicity assignment schemes, the Low-Complexity Chase (LCC) decoding can achieve similar or higher coding gain. Interpolation is a major step in ASD. Since the maximum multiplicity of the interpolation point is only one in LCC, the interpolation over each point has low complexity. However, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η</sup> test vectors are involved in the LCC, and the interpolation needs to be carried out on each of them. In order to reduce the computational complexity of the overall interpolation, intermediate results can be stored and shared. Nevertheless, the storage requires large memory, which accounts for a significant portion of the overall hardware requirement of the interpolator. In this paper, we propose a novel interpolation procedure, in which the 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η</sup> test vectors are mapped to the vertices of a dimension-η hypercube and the vectors mapped to adjacent vertices have only one different entry. In addition, a backward interpolation is proposed to support the traversal from one vertex to its neighbors. Traveling through the entire hypercube, the interpolation over each test vector can be done one after another and the memory requirement is reduced by a factor of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η-1</sup> . Efficient architectures are also developed for the proposed interpolation procedure. With about the same latency and the same number of gates as in prior efforts, our architecture can reduce the memory size to 25% and the number of registers to 57% for a (255, 239) RS code with η = 3. The saving further increases with η.

Fading in mobile satellite communications severely degrades the performance of data transmission. The emphasis in this paper is on the performance of uncoded and coded FCMA (frequency comb multiple access) in the fading environment. The channel is modeled with nonfrequency selective Rician and Rayleigh fading using noncoherent demodulation with Reed-Solomon (RS) codes and hard decision decoding. Berlekamp-Massey decoding of RS codes is used to compensate for the fading. The bit error rate (BER) after decoding is calculated for specific codes and for different values of Rician channel parameters. The results of this paper are obtained by simulation techniques. They show that substantial coding gains are obtained compared to the uncoded reference system. They are also useful as reference for validating the results of simulation studies.

Reed Solomon codes are being widely used for error-correcting methods and in construction of fault tolerant systems such as RAID. Recent studies[2],[7],[8] revealed that these codes have a potential application in cryptography. The hardness in decoding Reed Solomon codes can be exposed to construct new cryptographic applications. It is in this scenario that we talk about the suitability of Reed Solomon codes for Digital Fingerprinting in particular and cryptography in general. We derive results for bounds on Fingerprint length, design tracing method for Boneh.s broadcast encryption system[2]; and discuss collusion attacks on fingerprint. Pirate strategies and performance issues are also presented.KeywordsEncryption SchemeSecret MessageReed Solomon CodeDetectable MarkElliptic Curve Discrete Logarithm ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A new iterative soft decision decoding method for Reed-Solomon (RS) codes is proposed. This method is based on bit level belief propagation (BP) decoding. In order to make BP decoding effective for RS codes, we use an extended binary parity check matrix with a lower density and reduced number of 4-cycles compared to the original binary parity check matrix of the code. In our proposed method, we take advantage of the cyclic structure of RS codes. Based on this property, we can apply the belief propagation algorithm on any cyclically shifted version of the received symbols with the same binary parity check matrix. For each shifted version of received symbols, the geometry of the factor graph will change and deterministic errors can be avoided. Our method results in considerable performance improvement of RS codes compared to hard decision decoding. The performance is also superior to some popular soft decision decoding methods.

Determining deep holes is an important topic in decoding Reed-Solomon codes.\nIn a previous paper [8], we showed that the received word $u$ is a deep hole of\nthe standard Reed-Solomon codes $[q-1, k]_q$ if its Lagrange interpolation\npolynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at\nmost $k-1$. In this paper, we extend this result by giving a new class of deep\nholes of the generalized Reed-Solomon codes.\n

Protecting nonvolatile memory systems in harsh radiation environments encountered in space missions is important and error correcting schemes can extend the lifetime of those memory systems. For example, recent research has shown that LDPC codes can extend the lifetime of nonvolatile memory under space radiation environment more than Bose-Chaudhuri-Hocquenghem (BCH) or Reed-Solomon (RS) codes at fixed codeword error rates. However, conventional memory models assume that bit errors are independent, but multibit errors were reported in satellite experiments. Moreover, memory feature sizes are shrinking and multibit-per-cell structures are becoming standard so radiation will increasingly lead to multibit errors. For these reasons, we can expect that the bit errors in memory systems will be correlated. In this work, we develop a mathematical multibit-per-cell memory model under a radiation environment. In this memory model, bit errors are correlated and the probability of errors depends on radiation parameters and time. For correlated bit errors, nonbinary codes can be more effective than binary codes. We will demonstrate that nonbinary LDPC code can outperform conventional BCH and RS codes in a correlated multibit error environment.

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.

Gao's algorithm similar to Shiozaki's algorithm for decoding nonsystematic Reed-Solomon (RS) codes which is a subclass of the redundant residue polynomial codes is considered here. This algorithm computes the message polynomial directly without explicitly finding the error-locator polynomial and errorevaluator polynomial. In this letter, a simplified decoding algorithm to correct both errors and erasures is used in conjunction with Gao's algorithm for efficiently decoding RS codes. In addition, we show that the extended Gao algorithm similar to the Shiozaki-Truong-Cheung-Reed algorithm significantly reduces the decoding complexity.

A fast decoding algorithm for double error correcting Reed Solomon (RS) codes is presented. A new approach to the RS code decoding is introduced to eliminate the Chien search, to reduce substantially the root search time and hence to increase the decoding speed. It is interesting that the error locating time almost constant as the number of bits per symbol is increased. The algorithm finds suitable for microprocessor-based as well as VLSI implementations. Various RS codes are considered. Complexity, memory space and decoding time are all minimized as compared to the normal Chien search approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The decoding of extended Reed-Solomon codes

A novel scheme is presented for encoding and decoding of Reed-Solomon codes of prime lengths. Encoding is performed on a collection of codewords which are mapped through Galois Fourier transform into a codeword in a low-density parity-check code with a binary parity-check matrix for transmission. Using this matrix, a binary iterative soft-decision decoding algorithm is applied to jointly decode a collection of codewords in the Reed-Solomon code. By allowing information sharing among the received vectors corresponding to the code-words in the collection, the proposed decoding scheme achieves superior performance over algorithms decoding individual Reed-Solomon codewords including maximum likelihood decoding.

A step-by-step decoding algorithm is proposed for double-error-correcting (DEC) Reed-Solomon (RS) codes of length n=2/sup m/-1. Since the decoding algorithm can directly find the error value at the highest-order position in a received vector without determining the error location polynomial, the decoding procedure is very simple and regular, and therefore it is suitable for hardware implementation. Moreover, the new algorithm can also be used for decoding DEC binary BCH codes and DEC nonbinary BCH codes. Finally, based on the decoding algorithm a hardware decoder is also presented.

Symbol-level iterative information set decoding of RS codes

In this paper, we impose a few sparse parity-check (PC) equations to a Reed-Solomon (RS) code as a concatenation of PC and RS codes, which can provide considerable performance gain by an efficient soft-decision decoding for some existing RS coded systems, such as satellite communications. We propose two construction methods for the sub-matrices. The type-1 construction method directly concatenates a specially designed sub-matrix with regular row weight, and the type-2 construction method is developed to achieve a much lower encoding complexity while maintaining the performance gain. With the help of PC concatenation, the quantity of the channel soft information can be refined by the newly added sparse PC equations which significantly improve the performance of Chase-type decoders. In order to reduce the decoding complexity, we propose a threshold-based noise-aided Chase algorithm (TB-NACA) for PC-RS codes, which achieves significant performance gain compared to the hard -decision decoding of the original RS codes. Moreover, the proposed PC-RS code under the TB-NACA decoding can even significantly outperform the conventional Chase-type decoders over the AWGN channels and Rayleigh fading channels with high-order QAM.