Showing papers on "BCH code published in 2000"
26 Oct 2000
TL;DR: A polynomial-time soft-decision decoding algorithm for Reed-Solomon codes is developed and it is shown that the asymptotic performance can be approached as closely as desired with a list size that does not depend on the length of the code.
Abstract: A polynomial-time soft-decision decoding algorithm for Reed-Solomon codes is developed. This list-decoding algorithm is algebraic in nature and builds upon the interpolation procedure proposed by Guruswami and Sudan(see ibid., vol.45, p.1757-67, Sept. 1999) for hard-decision decoding. Algebraic soft-decision decoding is achieved by means of converting the probabilistic reliability information into a set of interpolation points, along with their multiplicities. The proposed conversion procedure is shown to be asymptotically optimal for a certain probabilistic model. The resulting soft-decoding algorithm significantly outperforms both the Guruswami-Sudan decoding and the generalized minimum distance (GMD) decoding of Reed-Solomon codes, while maintaining a complexity that is polynomial in the length of the code. Asymptotic analysis for alarge number of interpolation points is presented, leading to a geo- metric characterization of the decoding regions of the proposed algorithm. It is then shown that the asymptotic performance can be approached as closely as desired with a list size that does not depend on the length of the code.
672 citations
01 Sep 2000
TL;DR: The first lower bound on the peak-to-average power ratio (PAPR) of a constant energy code of a given length n, minimum Euclidean distance and rate is established and there exist asymptotically good codes whose PAPR is at most 8 log n.
Abstract: The first lower bound on the peak-to-average power ratio (PAPR) of a constant energy code of a given length n, minimum Euclidean distance and rate is established. Conversely, using a nonconstructive Varshamov-Gilbert style argument yields a lower bound on the achievable rate of a code of a given length, minimum Euclidean distance and maximum PAPR. The derivation of these bounds relies on a geometrical analysis of the PAPR of such a code. Further analysis shows that there exist asymptotically good codes whose PAPR is at most 8 log n. These bounds motivate the explicit construction of error-correcting codes with low PAPR. Bounds for exponential sums over Galois fields and rings are applied to obtain an upper bound of order (log n)/sup 2/ on the PAPRs of a constructive class of codes, the trace codes. This class includes the binary simplex code, duals of binary, primitive Bose-Chaudhuri-Hocquenghem (BCH) codes and a variety of their nonbinary analogs. Some open problems are identified.
288 citations
Book•
01 Sep 2000
TL;DR: Appendices: the Euclidean algorithm factorization of 1 + xn example of compact disc encoding solutions to selected exercises of classicalcryptography topics in algebra and number theory public-key cryptography.
Abstract: Coding theory: introduction to coding theory linear codes perfect and related codes cyclic linear codes BCH codes Reed-Solomon codes burst error-correcting codes convolutional codes Reed-Muller and Preparata codes. Cryptography: classicalcryptography topics in algebra and number theory public-key cryptography. Appendices: the Euclidean algorithm factorization of 1 + xn example of compact disc encoding solutions to selected exercises.
56 citations
TL;DR: This correspondence generalizes the construction of Bose-Chaudhuri-Hocquenghem and Reed-Solomon codes and applies the generalized Chinese remainder theorem to construct codes.
Abstract: In this correspondence, we study maximum distance with respect to rank (MDR) codes over the ring Z/sub k/. We generalize the construction of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon codes and apply the generalized Chinese remainder theorem to construct codes.
47 citations
TL;DR: This work considers the minimum cyclotomic coset representatives and derives some of their properties, which allow more precise estimates of the dimension of Bose-Chaudhuri-Hocquenghem (BCH) and classical Goppa codes of a given designed minimum distance.
Abstract: We consider the minimum cyclotomic coset representatives and derive some of their properties. The results allow more precise estimates of the dimension of Bose-Chaudhuri-Hocquenghem (BCH) and classical Goppa codes of a given designed minimum distance, and more precise estimates of the true designed distance of BCH codes and the minimum distance of classical Goppa codes.
38 citations
Patent•
01 Nov 2000TL;DR: In this paper, a set of permuted generator polynomials, each representing shifted and exclusive-ored (XORed) versions of the cyclic code generator, are presented.
Abstract: Methods and apparatus for cyclic code codeword creation, error detection, and error correction are disclosed. The methods and apparatus utilize a set of permuted generator polynomials, each representing shifted and exclusive-ored (XORed) versions of the cyclic code generator polynomial according to a specific input bit pattern. The permuted generator polynomial may be provided by look-up table, hardware, or a software equivalent of this hardware. Use of the permuted generator polynomial greatly reduces the number of calculations required to calculate syndromes and trap errors in codewords. The permuted generator polynomial can be used to replace m iterations of a polynomial division operation with a single XOR operation. The bit pattern used to select a permuted generator polynomial is derived from the m high-order bits of the dividend at each step. Using the disclosed embodiments, the number of clock cycles or instruction cycles required to perform cyclic code computations can be significantly reduced for both hardware and software implementations.
37 citations
TL;DR: It is proved that the parity-check subcodes of Goppa codes and the extended Goppa Codes are both Alternant codes, and some families of non-cyclic Goppa code families with a cyclic parity- check subcode are constructed.
37 citations
10 Sep 2000
TL;DR: This work adapts the classical Chase (1972) decoding algorithm to the watermarking context, and proposes a new soft decoding algorithm based on tree exploration that enables to approach optimal decoding at lower cost.
Abstract: Watermarking is often modeled as the transmission of a message over a very noisy channel. Indeed, the watermark power must be very low to ensure invisibility, and the modifications suffered by an image can be rather strong (JPEG or MPEG compression for instance), leading to a high noise level. To ensure a reliable transmission on a channel with such a low SNR, channel coding is thus mandatory. However, as standard codes are not adapted to such noisy channels, we propose a coding scheme based on concatenated codes. We show that, for these codes, optimal decoding is far too complex, and that hard decoding algorithms used in practice lead to poor performance. As soft decoding enables to approach optimal decoding at lower cost, we adapt the classical Chase (1972) decoding algorithm to the watermarking context, and propose a new soft decoding algorithm based on tree exploration.
29 citations
27 Nov 2000
TL;DR: A Bose-Chaudhuri-Hocquenghem coder in the field of the real numbers is investigated for simultaneous source coding and impulse noise cancellation of satellite images and the issue of robust transmission of wavelet compressed images through the noisy channel is examined.
Abstract: In this paper, a Bose-Chaudhuri-Hocquenghem (BCH) coder in the field of the real numbers is investigated for simultaneous source coding and impulse noise cancellation of satellite images. Our channel is a binary symmetric channel (BSC). Our approach is to make a carefully designed interpolation of the subband images prior to quantization and transmission. Compared to a classical tandem source and channel coding (TSC) scheme, in which a binary BCH coder would take place after quantization, our approach makes use of BCH coding prior to quantization, thus, allowing joint source and channel decoding. We also examine the issue of robust transmission of wavelet compressed images through the noisy channel: simulations show that we obtain a 3.7 dB improvement in PSNR over the classical entropy coder for a global rate of 3.25 transmitted bits per pixel and small BSC crossover probability.
29 citations
21 Aug 2000
TL;DR: A trade-off between the quantity of hidden data and the quality of the watermarked image is achieved by varying the number of quantization levels for the signature, the codeword length, and the scale factor for embedding.
Abstract: This paper describes a robust data embedding scheme, which uses a source and channel coding framework for data hiding. The data to be embedded, referred to as the signature data, is source coded by vector quantization and the indices obtained in the process are embedded in the wavelet transform coefficients of the host image. Transform coefficients of the host are grouped into vectors and perturbed using error-correcting codes derived from BCH codes. Compared to prior work in digital watermarking, the proposed scheme can handle a significantly large quantity of data such as a gray scale image. A trade-off between the quantity of hidden data and the quality of the watermarked image is achieved by varying the number of quantization levels for the signature, the codeword length, and the scale factor for embedding. Experimental results on signature recovery from JPEG compressed watermarked images are included.
27 citations
TL;DR: It is shown that certain binary sequences having a simple trace description possess the ideal autocorrelation property and that each such sequence is balanced and that the dual of the linear cyclic code generated by the sequence and its cyclic shifts, is a triple-error correcting code.
Abstract: In a previous paper, No, Golomb, Gong, Lee and Gaal (see ibid., vol.44, p.814-17, 1998) conjectured that certain binary sequences having a simple trace description possess the ideal autocorrelation property. In the present paper it is shown that each such sequence is balanced and, moreover, that the dual of the linear cyclic code generated by the sequence and its cyclic shifts, is a triple-error correcting code having the same weight distribution as the triple-error correcting Bose-Chaudhuri-Hocquenghem (BCH) code. This cyclic code also contains a cyclic subcode that yields a new family of sequences having the same size and correlation parameters as does the family of Gold sequences.
18 Sep 2000
TL;DR: This contribution concatenates space-time codes with three turbo coding schemes, namely turbo BCH (TBCH) codes, turbo convolutional (TC) codes and turbo trellis coded modulation (TTCM) schemes for the sake of achieving significant coding gain.
Abstract: Space-time block codes provide substantial diversity advantages in multiple transmit antenna assisted systems at a low decoding complexity. In this contribution, we concatenate space-time codes with three turbo coding schemes, namely turbo BCH (TBCH) codes, turbo convolutional (TC) codes and turbo trellis coded modulation (TTCM) schemes for the sake of achieving significant coding gain. The issues of mapping coded bits of the TBCH and TC schemes to different protection classes of various multilevel modulation schemes is also addressed. Finally, the performance and associated complexity of the three turbo schemes is compared.
TL;DR: All affine-invariant codes of length 2/sup m/ over Z/sub 4/, thus generalizing the corresponding result of Kasami, Lin, and Peterson (1967) and giving an alternative proof to Abdukhalikov.
Abstract: Berger and Charpin (see ibid., vol.42, p.2194-2209, 1996 and Des., Codes Cuyptogr., vol.18, no.1/3, p.29-53, 1999) devised a theoretical method of calculating the permutation group of a primitive cyclic code over a finite field using permutation polynomials and a transform description of such codes. We extend this method to cyclic and extended cyclic codes over the Galois ring GR (p/sup a/, m), developing a generalization of the Mattson-Solomon polynomial. In particular, we classify all affine-invariant codes of length 2/sup m/ over Z/sub 4/, thus generalizing the corresponding result of Kasami, Lin, and Peterson (1967) and giving an alternative proof to Abdukhalikov. We give a large class of codes over Z/sub 4/ with large permutation groups, which include generalizations of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Muller (RM) codes.
TL;DR: In this article, it was shown that crooked functions can be used to construct distance regular graphs with parameters of a Kasami distance regular graph, symmetric 5-class association schemes similar to those recently constructed by de Caen and van Dam from Kasami graphs, and uniformly packed codes with the same parameters as the double errorcorrecting BCH codes and Preparata codes.
Abstract: Let i>V and i>W be i>n-dimensional vector spaces over GF(2). A function i>Q : i>V → i>W is called i>crooked (a notion introduced by Bending and Fon-Der-Flaass) if it satisfies the following three properties:
We show that crooked functions can be used to construct distance regular graphs with parameters of a Kasami distance regular graph, symmetric 5-class association schemes similar to those recently constructed by de Caen and van Dam from Kasami graphs, and uniformly packed codes with the same parameters as the double error-correcting BCH codes and Preparata codes.
27 Nov 2000
TL;DR: This work concatenate space-time block codes with convolutional codes (CC), turbo convolutionAL codes (TC), turbo Convolutional Code (TC) , turbo BCH codes (TBCH), trellis coded modulation (TCM) and turboTrelliscoded modulation (TTCM), for achieving a high coding gain.
Abstract: Space-time block codes provide substantial diversity advantages for multiple transmit antenna systems at a low decoding complexity. We concatenate space-time codes with convolutional codes (CC), turbo convolutional codes (TC), turbo BCH codes (TBCH), trellis coded modulation (TCM) and turbo trellis coded modulation (TTCM) schemes for achieving a high coding gain. The associated performance and complexity of the the coding schemes is compared.
TL;DR: Algorithms are shown that simplify the solution and lead to further improvements of the latter bound, by factors ranging from order of log n, for c=0 and cn (in which case the overhead constant drops dramatically), to order of min (c, log n), for 2?c?n; the algorithms use Las Vegas type randomization in the case of 2.
24 Sep 2000
TL;DR: ATwo-dimensional channel estimation scheme for coherent detection of multicarrier modulated signals is presented which is based on the a posteriori probability (APP) calculation algorithm and outperforms the reference system which applies conventional two-dimensional Wiener filters for channel estimation.
Abstract: A two-dimensional channel estimation scheme for coherent detection of multicarrier modulated signals is presented which is based on the a posteriori probability (APP) calculation algorithm. An iterative estimation and decoding loop over inner estimator/product code and outer soft in/soft out APP decoder allows to further reduce the bit error rate. Heavily punctured recursive systematic convolutional codes are introduced as high-rate component codes of the inner product code. The proposed system outperforms the reference system which applies conventional two-dimensional Wiener filters for channel estimation.
27 Nov 2000
TL;DR: Simulations show that BTC associated with transmit and receive diversity perform as well as CTC in terms of the coding gains while achieving higher spectral efficiencies.
Abstract: Advances in the field of information theory have shown that wireless systems using multiple transmitters and multiple receivers can significantly improve the spectral efficiency. Systems using transmit and receive diversity associated with coding such as space time trellis codes have since been developed. A scheme using convolutional turbo codes (CTC) with two transmitters and two receivers has been proposed previously. It exhibits significant gains over space time trellis codes (up to 6 dB at a BER of 10/sup -5/). Our purpose is to extend that study to block turbo codes (BTC). We use the (32, 26, 4) extended BCH product code with two kinds of modulation (QPSK and 16-QAM). Simulations show that BTC associated with transmit and receive diversity perform as well as CTC in terms of the coding gains while achieving higher spectral efficiencies.
18 Jun 2000
TL;DR: Numerical results indicate that MSD is a sub-optimal decoding method of MLC for both channels, while PDL is most robust to varying channels if block partitioning (BP) is used.
Abstract: Multilevel coding (MLC) schemes based on channel capacity with multistage decoding (MSD) and parallel decoding on levels (PDL) are considered and compared. The channel models AWGN and Rayleigh fading are used in order to study the performance of MLC systems under different conditions. The investigation is done for 8ASK modulation and three set partitioning strategies. In each scheme BCH codes with different code lengths are used as component codes. Numerical results indicate that MSD is a sub-optimal decoding method of MLC for both channels, while PDL is most robust to varying channels if block partitioning (BP) is used. For Ungerboeck partitioning (UP) and mixed partitioning (MP) strategy, the MSD method is strongly recommended to use for the MLC system, while for the BP strategy, PDL is suggested to use as a simple decoding method compared with MSD.
29 May 2000
TL;DR: A soft decision decoding based on Chase's (1972) algorithm applied to Reed-Muller codes for peak-to-MEan Power Ratio (PMEPR) reduction and error correction for OFDM modulated signal is presented.
Abstract: This paper presents a soft decision decoding based on Chase's (1972) algorithm applied to Reed-Muller codes for peak-to-MEan Power Ratio (PMEPR) reduction and error correction. This scheme is applied to the OFDM (Orthogonal Frequency Division Multiplexing) modulated signal to reduce its PMEPR and to have correction capacities. We first obtain a 2 dB coding gain at BER=10/sup -4/ between hard and soft decoding and about 4 dB between the soft decoding and the uncoded system. Next, by modeling nonlinear TWTA power amplifiers with AM/AM and AM/PM distortions, we obtain similar results and note improving BER performance compared to BCH codes with the same parameters.
25 Jun 2000
TL;DR: A simple method using a single-error correcting BCH outer code in order to improve the performance of turbo codes and it is shown that this method reduces the errors dramatically at moderate-to-high E/sub b//N/sub o/.
Abstract: In this paper, we present a simple method using a single-error correcting BCH outer code in order to improve the performance of turbo codes. It is shown that this method reduces the errors dramatically at moderate-to-high E/sub b//N/sub o/.
TL;DR: This work proposes three methods which give tighter estimates of the weight distribution of a binary linear code and finds the distance of the dual of BCH [255,199,15].
Abstract: To estimate the weight distribution of a binary linear code, a linear programming approach has been proposed, which leads to the best known values for BCH (255,k). Following that approach, we propose three methods which give tighter estimates. At the same time, we find the distance of the dual of BCH [255,199,15].
01 Jan 2000
TL;DR: The Maximum A-Posteriori (MAP) algorithm is modified in order to incorporate the parity check bit of extended BCH codes into calculating the Log Likelihood Ratio (LLR) of the coded bits.
Abstract: The Maximum A-Posteriori (MAP) algorithm is modified in order to incorporate the parity check bit of extended BCH codes into calculating the Log Likelihood Ratio (LLR) of the coded bits. The algorithm is then used jointly with channel equalisation – in a scheme referred to as turbo equalisation – in order to mitigate the effects of Inter-Symbol Interference (ISI) introduced by dispersive channels. The performance of the extended BCH codes is evaluated and compared to that of conventional BCH codes. For example, the extended BCH(32,26) code was found to outperform the conventional BCH(31, 26) code by approximately 0.5 – 1.0 dB at a bit error reate (BER) of $10^{-5}$ over dispersive Rayleigh-fading channels.
15 May 2000
TL;DR: The effects of various concatenated component codes, interleaver size, minimum free-distances, weight-distributions as well as puncturing schemes are investigated using both union-bounding and simulations.
Abstract: Turbo codes using short binary BCH codes as component codes are investigated. The bit error rate (BER) performance of the turbo codes is evaluated using binary phase shift keying (BPSK) over additive white Gaussian noise (AWGN) channels. The effects of various concatenated component codes, interleaver size, minimum free-distances, weight-distributions as well as puncturing schemes are investigated using both union-bounding and simulations.
03 Aug 2000
TL;DR: This paper outlines the steps taken to reduce the processing time of a Reed-Solomon encoding and the insight into modern optimization techniques gained from the experience.
Abstract: Computers transfer data in a number of different ways. Whether through a serial port, a parallel port, over a modem, over an ethernet cable, or internally from a hard disk to memory, some data will be lost. To compensate for that loss, numerous error detection and correction algorithms have been developed. One of the most common error correction codes is the Reed-Solomon code, which is a special subset of BCH (Bose-Chaudhuri-Hocquenghem) linear cyclic block codes. In the AURA project, an unmanned aircraft sends the data it collects back to earth so it can be analyzed during flight and possible flight modifications made. To counter possible data corruption during transmission, the data is encoded using a multi-block Reed-Solomon implementation with a possibly shortened final block. In order to maximize the amount of data transmitted, it was necessary to reduce the computation time of a Reed-Solomon encoding to three percent of the processor's time. To achieve such a reduction, many code optimization techniques were employed. This paper outlines the steps taken to reduce the processing time of a Reed-Solomon encoding and the insight into modern optimization techniques gained from the experience.
21 Aug 2000
TL;DR: Numerical results indicate that when code rates of component codes in the MLC scheme are designed based on "capacity rule", the performance of the system with UP is optimum for AWGN fading channels.
Abstract: Based on [1,2,3], a novel criterion, the "capacity rule" and "mapping strategy", for the design of optimal MLC schemes over AWGN channels is proposed. According to this theory, the performance of multilevel coding with multistage decoding schemes (MLC/MSD) in AWGN channels is investigated, in which BCH codes are chosen as component codes, and three mapping strategies with 8ASK modulation are used. Numerical results indicate that when code rates of component codes in the MLC scheme are designed based on "capacity rule", the performance of the system with UP is optimum for AWGN fading channels.
Patent•
31 Oct 2000Abstract: The present invention relates generally to an improved decoder circuit used for
decoding Reed-Solomon or BCH codes. According to one aspect of the invention, a step of
performing solving an error locator polynomial is performed concurrently with a step of
computing an error pattern. In particular, a decoder is provided which performs concurrent
execution of a Chien search that determines the error locator polynomial for a received code
word and a Forney algorithm that computes the error pattern.
Patent•
01 Aug 2000TL;DR: In this article, a method for the transmission of logic channels (BCH) in a commonly used physical transmission system (PCCPCH) of at least two radio communication systems (TDSCDMA@GSM, TDS CDMA@UTRAN) was proposed.
Abstract: The invention relates to a method for the transmission of logic channels (BCH) in a commonly used physical transmission system (PCCPCH) of at least two radio communication systems (TDSCDMA@GSM, TDSCDMA @ UTRAN) wherein the respective logic channel (BCH) and another transmission channel (PILOT) can be differentiated.
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01 Jan 2000
TL;DR: This correspondence generalizes the construction of Bose-Chaidhhuri-Hocquenghem (BCH) and Reed-Solomon codes and applies the generalized Chinese remainder theorem to construct codes.
Abstract: In this correspondence, we study maximum distance with respect to rank codes over the ring . We generalize the construction of Bose-Chaidhhuri-Hocquenghem (BCH) and Reed-Solomon codes and apply the generalized Chinese remainder theorem to construct codes. Index Terms—Codes over rings, , maximum distance with respect to rank codes .