Showing papers on "BCH code published in 1994"

Near optimum decoding of product codes

Abstract: A new iterative decoding algorithm for product codes (block) based on soft decoding and soft decision output of the component codes is described in the paper. Monte Carlo simulations of the bit error rate (BER) after decoding using this new algorithm for different product codes indicate coding gains of up to 8 dB. This new coding scheme is attractive for digital transmission systems requiring powerful coding schemes with a high code rate (R>0.8). In the paper the authors compare their coding scheme with one of the best coding schemes, the "turbo-codes", in terms of BER performance.

Introduction to Error Correcting Codes

Abstract: Part 1 Introduction: bit strings and codes codes and error correction erasures and soft decision decoding hamming distance and sphere-packing Shannon's theorem. Part 2 Linear codes: matrix representation the standard array the null-matrix or parity-check matrix the syndrome the columns of the null matrix perfect codes further bounds on linear codes the Varsharmov-Gilbert bound the Plorkin bound bounds in practice nonbinary linear codes nonbinary codes with characteristic 2. Part 3 Cyclic codes: the generating polynomial systematic cyclic codes the roots of g(x) and the null-matrix error detection with cycle codes weight distribution shortened cyclic codes and feedback shift registers error correction with cyclic codes nonbinary cyclic codes. Part 4 BCH codes: minimum polynomials the roots of BCH codes some examples of BCH codes error correction of binary BCH codes practical procedures for solving the equations an example of BCH error correction error correction of nonbinary BCH codes Reed Solomon (RS) codes a worked RS example an example of practical use of RS codes other aspects of RS codes. Part 5 Convolutional codes: tree and trellis codes the Viterbi algorithm linear convolutional codes control of decoding errors.

Lee-metric BCH codes and their application to constrained and partial-response channels

Abstract: Shows that each code in a certain class of BCH codes over GF(p), specified by a code length n/spl les/p/sup m/-1 and a runlength r/spl les/(p-1)/2 of consecutive roots in GF(p/sup m/), has minimum Lee distance /spl ges/2r. For the very high-rate range these codes approach the sphere-packing bound on the minimum Lee distance. Furthermore, for a given r, the length range of these codes is twice as large as that attainable by Berlekamp's (1984) extended negacyclic codes. The authors present an efficient decoding procedure, based on Euclid's algorithm, for correcting up to r-1 errors and detecting r errors, that is, up to the number of Lee errors guaranteed by the designed minimum Lee distance 2r. Bounds on the minimum Lee distance for r/spl ges/(p+1)/2 are provided for the Reed-Solomon case, i.e., when the BCH code roots are in GF(p). The authors present two applications. First, Lee-metric BCH codes can be used for protecting against bitshift errors and synchronization errors caused by insertion and/or deletion of zeros in (d, k)-constrained channels. Second, the code construction with its decoding algorithm can be formulated over the integer ring, providing an algebraic approach to correcting errors in partial-response channels where matched spectral-null codes are used. >

Generalized Berlekamp-Massey decoding of algebraic-geometric codes up to half the Feng-Rao bound

Abstract: Summary form only given, as follows. Efficient decoding of BCH- and Reed-Solomon codes can be done using the Berlekanp-Massey (1969) algorithm, and it is natural to try to use the extension of this to N dimensions of Sakata (see Inform. Computat., vol.84, no.2, p.207, 1990) to decode algebraic geometry codes. We treat a general class of algebraic geometry codes and show how to decode these up to half the Feng-Rao (see IEEE Trans. Inform. Theory, vol.IT 39, no.1 p.37-45, 1993) bound, using an extension and modification of the Sakata algorithm. >

Use of Grobner bases to decode binary cyclic codes up to the true minimum distance

Abstract: A general algebraic method for decoding all types of binary cyclic codes is presented. It is shown that such a method can correct t=[(d-1)/2] errors, where d is the true minimum distance of the given cyclic code. The key idea behind this decoding technique is a systematic application of the algorithmic procedures of Grobner bases to obtain the error-locator polynomial L(z). The discussion begins from a set of syndrome polynomials F and the ideal T(F) generated by F. It is proved here that the process of transforming F to the normalized reduced Grobner basis of I(F) with respect to the "purely lexicographical" ordering automatically converges to L(z). Furthermore, it is shown that L(z) can be derived from any normalized Grobner basis of I(F) with respect to any admissible total ordering. To illustrate this new approach, the procedures for decoding certain BCH codes and quadratic residue codes are demonstrated. >

Idempotents and the BCH bound

Abstract: Using a characterization of the idempotents of a narrow-sense primitive binary BCH code, the authors are able to give classes of such codes whose minimum distance does not exceed the BCH bound. Their results are compiled in a table. >

Layered image transmission over cellular radio channels

Abstract: In the paper, a layered image transmission system is developed over cellular radio channels. The image coder is based on a multiresolution representation coupled with the ISO JPEG baseline coding standard. The multiresolution representation forms a pyramid in which information is layered in terms of different priorities that facilitates a more efficient use of forward error correction coding (FEC). After an error sensitivity analysis for different layers in an equalized mobile radio channel, an error control scheme based on block-interleaving and Bose-Chaudhuri-Hocquenghem (BCH) codes is proposed to minimize the effect of bursty channel errors on the picture quality degradation. Application of the new technique to the proposed Telecommunications Industry Association (TIA) Northern American digital cellular standard (IS-54) under different vehicle speeds with a signal-to-noise ratio of 22 dB is also investigated. Simulation results using a standard test image are presented to show the significant advantage of the proposed approach. >

Generalized Hamming weights of trace codes

Abstract: Linear codes over F/sub p/ often admit a natural representation as trace codes of codes that are defined over an extension field F/sub p/m. In the paper, the authors obtain estimates for the weights of subcodes of such trace codes. Their main result is a far-reaching generalization of the Carlitz-Uchiyama bound for the duals of binary BCH codes. In particular, they prove sharp bounds for the generalized Hamming weights of a large class of codes, including duals of BCH codes, classical Goppa codes, Melas codes, and arbitrary cyclic codes of length n=p/sup m/-1. The main tool is the theory of algebraic functions over finite fields, in particular the Hasse-Weil bound for the number of places of degree one. >

Subspace subcodes of Reed-Solomon codes

Abstract: We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed /spl nu/-dimensional vector subspace S of GF (2/sup m/). SSRS codes are constructed using properties of the Galois field GF(2/sup m/). They are not linear over the field GF(2/sup /spl nu//), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2/sup /spl nu//) is a subfield of GF(2/sup m/), it may not be the best /spl nu/-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems.

On generalized Hamming weights of BCH codes

Abstract: Introduces methods from algebraic geometry to determine generalized Hamming weights of BCH codes. As an application of these methods the authors determine for primitive 2- (resp., 3-) error correcting BCH codes the third (resp., the second) generalized Hamming weight. >

Capture probability and throughput analysis of slotted ALOHA and unslotted np-ISMA in a Rician/Rayleigh environment

Abstract: The performance of mobile radio communication systems is analyzed, in terms of the capture probability and throughput, for the slotted ALOHA and the unslotted np-ISMA (nonpersistent inhibit sense multiple access) protocol with receiver capture. The influence of Rayleigh faded interference and additive white Gaussian noise (AWGN) is studied on the performance of the two protocols considering BPSK (binary phase shift keying) modulated desired test signal, which suffers from Rician fading and the near-far effect. BCH (Bose Chaudhuri Hocquenghem) error correction codes consisting of packets with equal numbers of bits are introduced to improve the performance of the system, considering both fast and slow multipath fading. >

Bounds on the minimum distance of the duals of BCH codes

Abstract: We consider primitive cyclic codes of length p/sup m/-1 over F/sub p/. The codes of interest here are duals of BCH codes. For these codes, a lower bound on their minimum distance can be found via the adaptation of the Weil bound to cyclic codes. However, this bound is of no significance for roughly half of these codes. We shall fill this gap by giving, in the first part of the correspondence, a lower bound for an infinite class of duals of BCH codes. Since this family is a filtration of the duals of BCH codes, the bound obtained for it induces a bound for all duals. In the second part we present a lower bound obtained by implementing an algorithmic method due to Massey and Schaub (1988)-the rank-bounding algorithm. The numerical results are surprisingly higher than all previously known bounds.

Weight distributions of cosets of two-error-correcting binary BCH codes, extended or not

Abstract: Let B be the binary two-error-correcting BCH code of length 2/sup m/-1 and let B/spl circ/ be the extended code of B. We give formal expressions of weight distributions of the cosets of the codes B/spl circ/ only depending on m. We can then deduce the weight distributions of the cosets of B. When m is odd, it is well known that there are four distinct weight distributions for the cosets of B. So our main result is about the even case. Camion, Courteau, and Montpetit (see ibid., vol.38, no.7, p.1353, 1992) observe that for the lengths 15, 63, and 255 there are eight distinct weight distributions. We prove that this property holds for the codes B/spl circ/ and B for all even m. >

A fractal video communicator

Abstract: The image quality and compression ratio trade-offs of five different 176/spl times/144 pels quarter common intermediate format (QCIF) fractal image codecs are investigated by simulation. The average peak signal-to-noise ratio (PSNR) ranges from 29 dB to 37 dB, while the coding rate ranges from 0.24 bit/pel (bpp) to 1.22 bit/pel. Two of the candidate codecs, a 0.28 bit/pel and a 1.1 bit/pel codec, were subjected to bit-sensitivity analysis and protected by the source-sensitivity matched shortened binary BCH (122,80,6) and BCH (122,52,11) codes and transmitted using coherent pilot symbol assisted (PSAM) square-constellation 16-level quadrature amplitude modulation (16-QAM). The proposed fractal video communicators required a channel signal-to-noise ratio (SNR) and signal-to-interference ratio (SIR) of about 15 dB in order to maintain a video peak SNR (PSNR) of 31 dB and 35 dB at signaling rates of 39 kBaud and 156 kBaud, respectively, over Rayleigh-fading channels having a propagation frequency of 1800 MHz and a pedestrian speed of 2 mph. >

Generation of matrices for determining minimum distance and decoding of cyclic codes

Abstract: A simple method based on Newton's identities and their extensions is presented for determining the actual minimum distance of cyclic codes. More significantly, it is shown that this method also provides a mechanism for generating the type of syndrome matrices needed by Feng and Tzeng's (see ibid., vol.40, p.1364-1374, Sept. 1994) new procedure for decoding cyclic and BCH codes up to their actual minimum distance. Two procedures for generating such matrices are given. With these procedures, we have generated syndrome matrices having only one class of conjugate syndromes on the minor diagonal for all binary cyclic codes of length n<63 and many codes of length 63/spl les/n/spl les/99. A listing of such syndrome matrices for selected codes of length n<63 is included. An interesting connection of the method presented to the shifting technique of van Lint (1986) and Wilson is also noted.

Method and apparatus for a reduced iteration decoder

Abstract: The present invention is a reduced iteration decoder circuit and method for computing error-locator sequence values (λ at 132) for use in the correction of bit errors in Reed-Solomon or BCH coded information (r at 111). By utilizing special properties of Reed-Solomon code and BCH codes, the decoder circuit of the present invention can detect n symbol errors using approximately n mathematical iterations (in 120) with substantially reduced decoding processing time. A further reduction of decoding time is achieved by the performance of a substantial portion of the decoding processing in a parallel manner. The present invention may be utilized in digital communication systems and data storage systems or other information systems where Reed-Solomon or BCH encoding is utilized.

A dual-rate algebraic CELP-based speech transceiver

Abstract: A novel high-quality, low-complexity dual-rate 4.7/6.5 kbits/s algebraic code excited linear predictive (ACELP) codec is proposed for adaptive speech communicators, which can drop their source rate and speech quality under network control in order to invoke a more error resilient modem amongst less favourable channel conditions. Source-matched binary Bose-Chaudhuri-Hoequenghem (BCH) codecs combined with unequal protection diversity- and pilot-assisted 16and 64-level quadrature amplitude modulation (18-QBM, 64-QAM) are employed In order to accommodate both the 4.7 and the 6.5 kbits/s coded speech bits at a signalling rate of 3.1 kBd. In a bandwidth of 200 kHz 32 time slots can be created, which allows to support in excess of 50 users, when employing packet reservation multiple access (PRMA). Good communications quality speech is delivered in an equivalent bandwidth of 4 kHz, if the channel signal-to-noise ratio (SNR) and signal-to-interference ratio (SIR) are in excess of about 15 and 25 dB for the lower and higher speech quality 16-QAM and 64-QAM systems, respectively. >

Malfunction in the Peterson-Gorenstein-Zierler decoder

Abstract: Most versions of the Peterson-Gorenstein-Zierler (PGZ) decoding algorithm are not true bounded distance decoding algorithms in the sense that when a received vector is not in the decoding sphere of any codeword, the algorithm does not always declare a decoding failure. For a t-error-correcting BCH code, if the received vector is at distance i, i/spl les/t from a codeword in a supercode with BCH distance t+i+1, the decoder will output that codeword from the supercede. If that codeword is not a member of the t-error-correcting code, then decoder malfunction is said to have occurred. We describe the necessary and sufficient conditions for decoder malfunction, and show that malfunction can be avoided in the PGZ decoder by checking t-/spl nu/ equations, where /spl nu/ is the number of errors hypothesized by the decoder. A formula for the probability of decoder malfunction is also given, and the significance of decoder malfunction is considered for PGZ decoders and high-speed Berlekamp-Massey decoders. >

Divisibility Properties and New Bounds for Cyclic Codes and Exponential Sums in One and Several Variables

Abstract: Serre has obtained sharp estimates for the number of rational points on an algebraic curve over a finite field. In this paper we supplement his technique with divisibility properties for exponential sums to derive new bounds for exponential sums in one and several variables. The new bounds give us an improvement on previous bounds for the minimum distance of the duals of BCH codes. The divisibility properties also imply the existence of gaps in the weight distribution of certain cyclic codes, and in particular gives us that BCH codes are divisible (in the sense of H. N. Ward).

On automorphism groups of binary primitive BCH codes

Abstract: It is proved that, for a given m/spl ges/5, the automorphism group Aut(C) of a binary primitive BCH code C of length 2/sup m/-1 with Bose distance d/sub 0/, 3/spl les/d/sub 0/-1 >

Novel simulation technique for assessing coding system performance

Abstract: A novel simulation technique for evaluating the performance of coded communication systems at low bit error rates (BERs) is presented. To illustrate the generic technique, the performance of a soft decision decoding algorithm for block codes is evaluated. Realistic assessments of residual BER at 10/sup -9/ and below are achieved.

New time domain errors and erasures decoding algorithm for BCH codes

Abstract: A new algorithm is presented for solving the key equation that simultaneously computes the error locator polynomial and the errata evaluator polynomial. The algorithm is similar to the Berlekamp algorithm but is more symmetrical in its treatment of the iterated pairs of polynomials making it particularly well suited to a highly parallel hardware implementation.

A hybrid-ARQ protocol with optimal adaptive error control for multidestination satellite communications

Abstract: In this paper, a go-back-N hybrid-ARQ protocol, which is a BCH code adaptively concatenated with a rate repetition convolutional code with the Viterbi decoding for satellite communications over broadcast channels, is proposed The BCH code is used for error correction and simultaneously for error detection, and the convolutional code with the Viterbi decoding is used only for error correction A dynamic programming technique is applied to adaptively select the optimal encoding method for a frame to be transmitted Analyses and simulations show that the proposed protocol provides satisfactory throughput efficiency under mean and high error rate conditions, especially in environments with a large number of receivers and a large link round trip delay >

A mobile speech, video and data transceiver scheme

Abstract: The complexity robustness, image and speech quality as well as packet multiplexing issues of a re-configurable multi-media mobile communicator are addressed. The proposed moderate complexity motion-compensated discrete cosine transform (DCT) based image communicator provides an image peak signal-to-noise ratio (PSNR) of 38 dB at an average bit rate of about 25 kbits/s. The speech codec used is a low-complexity 32 kbit/s CCITT G721 standard scheme. Bandwidth efficient 16 or 64-level quadrature amplitude modulation (QAM) combined with embedded low-complexity binary Bose-Chaudhuri-Hocquenghem (BCH) forward error correction (FEC) coding is deployed. The 2O-slot packet reservation multiple access (PRMA) scheme used supports an extra 2.4 kbit/s low-rate data channel for each speech user, in addition to providing 5-6 videophone channels. The ADPCM/DCT/BCH/16-QAM and ADPCM/DCT/BCH/64-QAM schemes provide nearly unimpaired speech and image quality for channel SNRs in excess of 30 dB and 38 dB, respectively. >

Performance analysis of hybrid forward error correction schemes in a fast and slow Rician fading wideband land-mobile satellite channel with BPSK and path diversity

Abstract: The performance of combined FEC/ARQ techniques for a direct-sequence CDMA system using coherent binary phase shift keying (BPSK) modulation is investigated. Throughput and delay are analyzed for the cases of slow and fast fading, considering the application of maximal ratio combining in light and heavy shadowing environments. By calculating the packet success probability, numerical results are presented for the throughput and delay. FEC with selective-repeat ARQ is implemented using BCH codes of varying lengths.

Information bit series transmission system

Abstract: PURPOSE:To provide an information bit series transmission system for realizing more efficient error correction decoding while suppressing the increase of the probability of error correction at the time of transmitting an information bit series. CONSTITUTION:In coding and decoding techniques in this information bit series transmission system, a CRC error detection code processing 41 is performed to the information bit series A and its processing result is BCH code processed 43 and transmitted to a transmission part 45. The random error correction decoding processing 47 of cyclic codes and a burst error correction decoding processing 48 are performed to a code series received by the transmission part 45. The processing result is CRC error detection decoding processed 50.

Modified Berlekamp-Massey algorithm for decoding BCH codes defined over rings

Abstract: We propose an extension of the Berlekamp-Massey (1969) algorithm for decoding BCH codes defined over integer residue rings and for synthesizing the shortest linear feedback shift register (LFSR) that will generate a given finite sequence of elements lying also in an integer residue ring. >

Unequal error protection capabilities of some binary primitive BCH codes

Abstract: Using a recent result on the span of their minimum weight vectors, it is shown that binary primitive BCH codes, containing second-order punctured Reed-Muller (RM) codes of the same minimum distance, are binary cyclic unequal error protection (UEP) codes. The values of the error correction levels for this class of binary cyclic UEP codes are determined. >

On fault tolerant matrix decomposition

Abstract: We present a fault tolerant algorithm for matrix factorization in the presence of multiple hardware faults which can be used for solving the linear systemAx=b without determining the correctZU decomposition ofA. HereZ is eitherL for ordinary Gaussian decomposition with partial pivoting,X for pairwise or neighbor pivoting (motivated by the Gentleman-Kung systolic array structure), orQ for the usualQR decomposition. Our algorithm generalizes that of Luk and Park whose method allows for the correction of a single error in a single iterate of the matrixU. Using ideas from the theory of error correcting codes we prove that the algorithm of Luk and Park can in fact tolerate multiple errors in multiple iterates ofU provided these are all confined to a single column. We then generalize the algorithm to one that tolerates multiple errors in multiple iterates ofU provided they are confined to two columns. Our procedure for identifying the erroneous columns is based on the extended Euclidean algorithm and it analogous to the decoding algorithms for BCH codes. We indicate how our methods may be adapted to apply to any number of columns and finally we show how to compute a correct factorization ofA.