Showing papers on "BCH code published in 1999"
TL;DR: An improved list decoding algorithm for decoding Reed-Solomon codes and alternant codes and algebraic-geometry codes is presented and a solution to a weighted curve-fitting problem is presented, which may be of use in soft-decision decoding algorithms for Reed- Solomon codes.
Abstract: Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following "curve-fitting" problem over a field F: given n points ((x/sub i//spl middot/y/sub i/))/sub i=1//sup n/, x/sub i/, y/sub i//spl isin/F, and a degree parameter k and error parameter e, find all univariate polynomials p of degree at most k such that y/sub i/=p(x/sub i/) for all but at most e values of i/spl isin/(1,...,n). We give an algorithm that solves this problem for e 1/3, where the result yields the first asymptotic improvement in four decades. The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometry codes. In both cases, we obtain a list decoding algorithm that corrects up to n-/spl radic/(n(n-d')) errors, where n is the block length and d' is the designed distance of the code. The improvement for the case of algebraic-geometry codes extends the methods of Shokrollahi and Wasserman (see in Proc. 29th Annu. ACM Symp. Theory of Computing, p.241-48, 1998) and improves upon their bound for every choice of n and d'. We also present some other consequences of our algorithm including a solution to a weighted curve-fitting problem, which may be of use in soft-decision decoding algorithms for Reed-Solomon codes.
1,108 citations
TL;DR: It is shown that a classical error correcting code C, which contains its dual, C/sup /spl perp///spl sube/C, and which can be enlarged to C'=[n,k'>k+1,d'], can be converted into a quantum code of parameters [[ n,k+k'-n,min(d,[3d'/2])]].
Abstract: It is shown that a classical error correcting code C=[n,k,d] which contains its dual, C/sup /spl perp///spl sube/C, and which can be enlarged to C'=[n,k'>k+1,d'], can be converted into a quantum code of parameters [[n,k+k'-n,min(d,[3d'/2])]] This is a generalization of a previous construction, it enables many new codes of good efficiency to be discovered Examples based on classical Bose-Chaudhuri-Hocquenghem (BCH) codes are discussed
276 citations
TL;DR: Using a new parity-check matrix, a class of convolutional codes with a designed free distance is introduced, which has many characteristics of BCH block codes, therefore, these codes are called BCH convolutionAL codes.
Abstract: Using a new parity-check matrix, a class of convolutional codes with a designed free distance is introduced. This new class of codes has many characteristics of BCH block codes, therefore, we call these codes BCH convolutional codes.
142 citations
TL;DR: Using the Gilbert-Elliott model to study the performance of block-coded transmission over the land mobile channel, a new analytical expression illustrating the effect of various parameters, e,g.
Abstract: By using the Gilbert-Elliott (1960, 1963) model to study the performance of block-coded transmission over the land mobile channel, a new analytical expression illustrating the effect of various parameters, e,g., mobile speed, delay constraint, and parameters for the error correcting code, is found. Comparisons between the results obtained by this analytical expression and results obtained by computer simulations show that the analytical results are accurate for a broad range of channel parameters. The Gilbert-Elliott model is then used to compare the performance of different binary BCH codes when the delay constraint does not allow the assumption of infinite interleaving. In contrast to the memoryless case, where the performance typically is improved with increased block length, short codes are found to be as good, or even superior, due to the fact that the interleaver works better for shorter codes.
127 citations
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TL;DR: This work shows how to construct quantum error-correcting codes based on classical BCH codes, which can exploit additional information about the position of errors with this error model - the quantum erasure channel.
Abstract: After a brief introduction to both quantum computation and quantum error correction, we show how to construct quantum error-correcting codes based on classical BCH codes. With these codes, decoding can exploit additional information about the position of errors. This error model - the quantum erasure channel - is discussed. Finally, parameters of quantum BCH codes are provided.
103 citations
TL;DR: A new efficient serial architecture to implement the Berlekamp-Massey (1968, 1969) algorithm, which is frequently used in BCH and Reed-Solomon (1960) decoders is proposed and significantly reduces the latency.
Abstract: We propose a new efficient serial architecture to implement the Berlekamp-Massey (1968, 1969) algorithm, which is frequently used in BCH and Reed-Solomon (1960) decoders. An inversionless Berlekamp-Massey algorithm is adopted which not only eliminates the finite-field inverter but also introduces additional parallelism. We discover a clever scheduling of three finite-field multipliers to implement the algorithm very efficiently. Compared to a previously proposed serial Berlekamp-Massey architecture, our technique significantly reduces the latency.
45 citations
TL;DR: Values for the minimum weight of the geometric codes are obtained in the binary case, using geometric constructions in the associated geometries, and the BCH bound from coding theory.
Abstract: The geometric codes are the duals of the codes defined by the designs associated with finite geometries. The latter are generalized Reed–Muller codes, but the geometric codes are, in general, not. We obtain values for the minimum weight of these codes in the binary case, using geometric constructions in the associated geometries, and the BCH bound from coding theory. Using Hamada‘s formula, we also show that the dimension of the dual of the code of a projective geometry design is a polynomial function in the dimension of the geometry.
34 citations
TL;DR: A new estimate for the error term in the binomial approximation to the distance distribution of BCH codes is derived, an improvement on the earlier bounds by Kasami-Fujiwara-Lin, Vladuts-Skorobogatov, and Krasikov-Litsyn.
Abstract: We derive a new estimate for the error term in the binomial approximation to the distance distribution of BCH codes. This is an improvement on the earlier bounds by Kasami-Fujiwara-Lin (1985), Vladuts-Skorobogatov (1991), and Krasikov-Litsyn (1995).
33 citations
TL;DR: The proposed contribution offers a choice to opt a worthy BCH code concerning code rate and error correction capability and extends the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequence of codes at once.
32 citations
05 Dec 1999
TL;DR: A solution is proposed to adapt the decoding algorithm to the characteristics of the encoder, the modulation and the number of decoding steps, which can be interesting in terms of decoding latency and power consumption.
Abstract: This paper deals with iterative decoding of product codes for high spectral efficiency transmissions. A solution is proposed to adapt the decoding algorithm to the characteristics of the encoder, the modulation and the number of decoding steps. A coding gain of up to 2 dB is then obtained for short codes. For long codes, the number of decoding steps to achieve a specified bit error rate (BER) can be reduced by 1 or 2, which is interesting in terms of decoding latency and power consumption.
22 citations
TL;DR: The merits of good FPGA designs must parallelize at every opportunity, minimize timing delays through intelligent floor planning, and use each logic block to its fullest are demonstrated by comparing the performance of popular finite field multiplier designs.
Abstract: Among the various tasks performed by software radios is the reconfiguration of the error control coding algorithm to match the requirement of the radio personality. In the digital radio processor, proper assignment of tasks between DSPs and FPGAs provides performance improvements over the use of DSPs alone. Error control coding functions are good candidates to reside on the FPGA side of this functional partition. Unfortunately, good VLSI designs for codes using BCH or Reed-Solomon codes do not map well to FPGAs. Good FPGA designs must parallelize at every opportunity, minimize timing delays through intelligent floor planning, and use each logic block to its fullest. We demonstrate the merits of these concepts by comparing the performance of popular finite field multiplier designs.
TL;DR: In this paper, two infinite series of imprimitive 5-class association schemes are constructed from two edge-disjoint copies of the coset graph of a binary Kasami code (double error-correcting BCH code).
Abstract: Two new infinite series of imprimitive 5-class association schemes are constructed. The first series of schemes arises from forming, in a special manner, two edge-disjoint copies of the coset graph of a binary Kasami code (double error-correcting BCH code). The second series of schemes is formally dual to the first. The construction applies vector space duality to obtain a fission scheme of a subscheme of the Cameron-Seidel 3-class scheme of linked symmetric designs derived from Kerdock sets and quadratic forms over GF(2).
Journal Article•
TL;DR: Simulation results show that the early termination condition combined with a ruling-out condition proposed previously is considerably effective in reducing the number of test error patterns, especially as the total number oftest error patterns concerned grows.
Abstract: SUMMARY In this paper, we consider sufficient conditions for ruling out some useless iteration steps in a class of softdecision iterative decoding algorithms for binary block codes used over the AWGN channel using BPSK signaling. Sufficient conditions for ruling out the next single decoding step, called ruling-out conditions and those for ruling out all the subsequent iteration steps, called early termination conditions, are formulated in a unified way without degradation of error performance. These conditions are shown to be a type of integer programming problems. Several techniques for reducing such an integer programming problem to a set of subprograms with smaller computational complexities are presented. As an example, an early termination condition for Chase-type decoding algorithm is presented. Simulation results for the (64, 42, 8) Reed-Muller code and (64, 45, 8) extended BCH code show that the early termination condition combined with a ruling-out condition proposed previously is considerably effective in reducing the number of test error patterns, especially as the total number of test error patterns concerned grows.
TL;DR: The proposed scheme gives a throughput of 0.9, even at high BER conditions, for any IP packet size under an AWGN channel while, an 8-dB improvement is achieved, when using BCH(63, 51, 2), over selective repeat+stutter scheme 2 (SR+ST 2) under a flat Rayleigh fading channel.
Abstract: This paper proposes and analyzes the performance of a hybrid selective repeat (SR)/multicopy (MC) automatic repeat request (ARQ) scheme to transmit fragmented Internet protocol (IP) packets. The ARQ scheme works in the SR mode until the last IP packet fragment is transmitted. If a fragment is negatively acknowledged after the last fragment is transmitted, then the system goes in the MC mode. In the MC mode, multiple copies of the erroneous fragment are transmitted. After all IP fragments are received without error, the system goes back to the SR mode. The performance of the proposed scheme is evaluated in terms of the bit error rate (BER), IP packet size, and fragmentation size with and without Bose Chaudhuri Hocquenghem (BCH) error correction codes. Both the results are obtained under additive white Gaussian noise (AWGN) as well as flat Rayleigh fading channels. The proposed scheme gives a throughput of 0.9, even at high BER conditions, for any IP packet size under an AWGN channel while, an 8-dB improvement is achieved, when using BCH(63, 51, 2) for throughput of 0.9, over selective repeat+stutter scheme 2 (SR+ST 2) under a flat Rayleigh fading channel.
19 Sep 1999
TL;DR: This paper compares the performance of partial-response GMSK turbo equalisers using two different encoders, namely block BCH turbo codes and convolutional codes, over non-dispersive Gaussian channels and dispersive Rayleigh fading channels.
Abstract: In contrast to previously proposed turbo equalisers, where typically non-iterative channel decoders were used, this paper compares the performance of partial-response GMSK turbo equalisers using two different encoders, namely block BCH turbo codes and convolutional codes. The BER performance is assessed over non-dispersive Gaussian channels, and dispersive Rayleigh fading channels.
TL;DR: The automorphism groups of extended narrow-sense BCH codes defined over any extension field are given, giving the automorphist groups of affine-invariant codes berg, bech1.
Abstract: Affine-invariant codes are extended cyclic codes of length p^m invariant under the affine-group acting on {\F}_{p^m} This class of codes includes codes of great interest such as extended narrow-sense BCH codes In recent papers, we classified the automorphism groups of affine-invariant codes berg, bech1 We derive here new results, especially when the alphabet field is an extension field, by expanding our previous tools In particular we complete our results on BCH codes, giving the automorphism groups of extended narrow-sense BCH codes defined over any extension field
TL;DR: An automatic request (ARQ) protocol is described that allows the soft decision decoders to outperform their hard decision counterparts in both reliability and throughput.
Abstract: A methodology is presented for the design and development of efficient trellis-based soft decision decoders for extended Hamming and BCH codes. A new metric for noncoherent discriminator detection is proposed that substantially improves the performance of trellis-based decoders over additive white Gaussian noise (AWGN) channels. Minimal edge trellises are then presented for the class of extended Hamming codes and the (32, 21) extended BCH code. The latter is in extensive use in narrow-band wireless data systems. An automatic request (ARQ) protocol is described that allows the soft decision decoders to outperform their hard decision counterparts in both reliability and throughput.
TL;DR: This paper presents a new low-weight trellis-based soft-decision iterative decoding algorithm for binary linear block codes based on a set of optimality conditions and the generation of a sequence of candidate codewords for an optimality test.
Abstract: This paper presents a new low-weight trellis-based soft-decision iterative decoding algorithm for binary linear block codes. The algorithm is devised based on a set of optimality conditions and the generation of a sequence of candidate codewords for an optimality test. The initial candidate codeword is generated by a simple decoding method. The subsequent candidate codewords, if needed, are generated by a chain of low-weight trellis searches, one at a time. Each search is conducted through a low-weight trellis diagram centered around the latest candidate codeword and results in an improvement over the previous candidate codewords that have been already tested. This improvement is then used as the next candidate codeword for a test of optimality. The decoding iteration stops whenever a candidate codeword is found to satisfy a sufficient condition on optimality or the latest low-weight trellis search results in a repetition of a previously generated candidate codeword. A divide-and-conquer technique is also presented for codes that are not spanned by their minimum-weight codewords. The proposed decoding algorithm has been applied to some well-known codes of lengths 48, 64, and 128. Simulation results show that the proposed algorithm achieves either practically optimal error performance for the example codes of length 48 and 64 or near optimal error performance for the (128, 29, 32) RM code with a significant reduction in computational decoding complexity.
TL;DR: It is shown that when the basis q is odd, BCH codes can be lengthened to obtain new codes with covering radius R=2 to give new infinite families of linear covering codes with codimension r=2k+1.
Abstract: On the way of generalizing recent results by Cock and the second author, it is shown that when the basis q is odd, BCH codes can be lengthened to obtain new codes with covering radius R=2. These constructions (together with a lengthening construction by the first author) give new infinite families of linear covering codes with codimension r=2k+1 (the case q=3, r=4k+1 was considered earlier). New code families with r=4k are also obtained. An updated table of upper bounds on the length function for linear codes with r\leq 24, R=2, and q=3,5 is given.
Patent•
15 Nov 1999TL;DR: In this article, a system for memory word error correction that enables correction of burst errors in memory words is presented, which is based on an adaptation of two-error correction BCH code which yields burst error correction without increasing the number of error correction bits in the memory words over prior two error BCH error correction schemes.
Abstract: A system for memory word error correction that enables correction of burst errors in memory words. The system is based on an adaptation of two-error correction BCH code which yields burst error correction without increasing the number of error correction bits in the memory words over prior two-error BCH code error correction schemes. The adaptation of two-error correction BCH code when combined with additional techniques for detecting columns of burst errors enables the correction of burst errors and additional random bit errors in memory words.
Patent•
16 Mar 1999
TL;DR: In this article, the authors propose a testing method for the storage device capable of improving productivity by providing a storage device generating no reading error even though two or more errors are generated within a code.
Abstract: PROBLEM TO BE SOLVED: To provide a storage device generating no reading error even though two or more errors are generated within a code and a testing method for the storage device capable of improving productivity. SOLUTION: Abbreviation BCH coded writing data s2, wherein two errors within the code of data s1 inputted into a flash memory 10 can be corrected with an encoder 1, are written into a cell array 3. Data s3 read from the cell array 3, whose error is corrected with an error corrector 5, are decoded to obtain output data s4. In a test at the time of manufacture (delivery), previously written testing data are read for one block at a time and the number of errors within respective codes is counted. The error is corrected for the code whose number of errors is one or less and the block including the codes whose number of errors are two or more is made to be a defective block. When the ratio of the number of the defective block to the number of total blocks is e.g. 1% or less, the block is accepted. When the ratio exceeds 1%, the block is made defective.
01 Apr 1999
TL;DR: A Grobner basis algorithm for errors-and-erasures decoding of BCH codes that avoids computation of the modified syndrome polynomial is given.
Abstract: A Grobner basis algorithm for errors-and-erasures decoding of BCH codes that avoids computation of the modified syndrome polynomial is given. The decoding problem is viewed as an instance of a more general interpolation problem. The algorithm has the same computational complexity as the Berlekamp–Massey algorithm but is more efficient in hardware.
01 Oct 1999
TL;DR: The authors present a novel near maximum likelihood (ML) soft decision decoding (SDD) algorithm for linear block codes that is both general in nature and able to offer significant performance improvements.
Abstract: The authors present a novel near maximum likelihood (ML) soft decision decoding (SDD) algorithm for linear block codes. The proposed algorithm can be subdivided into three distinct algorithms, each achieving a specific objective. The first algorithm achieves near ML decoding performance by utilising channel measurement information. The second and third algorithms maintain the improved decoding performance achieved, while at the same time reducing both the number of decodings and complexity required. The resultant algorithm is both general in nature and able to offer significant performance improvements. The theoretical results are verified by computer simulation.
15 Mar 1999
TL;DR: A novel class of forward error correcting codes constructed using the discrete Walsh transform are presented, defined on the field of real numbers, and compared to those of the well-known BCH and RS codes.
Abstract: We present a novel class of forward error correcting codes constructed using the discrete Walsh transform. They are a class of double-error correcting codes defined on the field of real numbers. An iterative decoding algorithm for Walsh transform codes is developed and implemented. The error correcting performance of Walsh transform codes over an AWGN channel is evaluated. Selected Walsh transform code parameters are compared to those of the well-known BCH and RS codes.
01 Jan 1999
TL;DR: The areas of system theory and coding theory are more or less equally “young”, having their origins around the fifties, and it is pointed out that what the Berlekamp-Massey algorithm solves is “a version of the widely conceived engineering black box problem”.
Abstract: The areas of system theory and coding theory are more or less equally “young”, having their origins around the fifties. Throughout their history, there have been observations on the existence of connections between the two areas. One of the first of these observations was concerned with the Berlekamp-Massey algorithm, derived in [5, 22] for the purpose of decoding BCH/Reed-Solomon codes. Indeed, following upon Massey’s exposition in [22], Sain pointed out in [26] that what the Berlekamp-Massey algorithm solves is “a version of the widely conceived engineering black box problem”.
TL;DR: This theorem is able to give several classes of BCH codes of composite length whose minimum distance does not exceed the BCH bound and can be used to determine the true minimum distance of some other cyclic codes with composite-length.
Abstract: We derive a theorem which generalizes Theorem 3 in Chapter 9 of the book "The Theory of Error-Correcting Codes" by F.J. MacWilliams and N.J.A. Sloane (North-Holland, 1977). By this theorem, we are able to give several classes of BCH codes of composite length whose minimum distance does not exceed the BCH bound. Moreover, we show that this theorem can also be used to determine the true minimum distance of some other cyclic codes with composite-length.
05 Dec 1999
TL;DR: This paper proposes an optimized transmission system for still images that comprises a VQ source encoder with a compression rate of 9.7 dB and uses a new joint source-channel turbo decoding algorithm of the product code to improve channel decoding algorithms.
Abstract: In this paper we propose an optimized transmission system for still images. This system comprises a VQ source encoder with a compression rate of 9. The source data is encoded using a BCH product code and mapped on a 16-QAM constellation. The codebook and channel encoder have been optimized to minimize distortion in presence of transmission errors. We use a new joint source-channel turbo decoding algorithm of the product code. As compared to classical channel decoding algorithms (soft decoding), we obtain an improvement of 0.7 dB in terms of coding gain for a BER of 10/sup -5/. The PSNR of the received image exhibits an improvement of 11 dB as compared to conventional turbo decoding at low SNRs.
Patent•
14 Sep 1999TL;DR: In this paper, a randomizer circuit that combines the randomizer sequence with all the symbols of ECC code words that are encoded using a BCH code over GF(2 m ) is presented.
Abstract: A system that produces one or more non-repeating randomizer sequences of up to 2 m −1 or more m-bit symbols includes a randomizer circuit that is set up in accordance with a polynomial with primitive elements of GF(2 m ) as coefficients. The system combines the randomizer sequence with all the symbols of ECC code words that are encoded using a BCH code over GF(2 m ) to produce a randomized code word. The particular primitive elements used and/or an initial state of one or more registers in the system specifies the particular sequence produced by the system. The initial state of each of the one or more registers is a selected one of the 2 m −1 elements of GF(2 m ), and thus, 2 m −1 different sequences may be produced by selecting a different initial state for a given one of the registers. If the coefficients are also selected from, for example, a set of “p” possible values, the system produces p*(2 m −1) different sequences. The system may thus be used to encrypt the ECC code word by associating the code word with a particular selected initial state and/or coefficient. The coefficients may be selected to produce randomizer sequences that are predetermined minimum distances away from both the ECC code words.
TL;DR: In this article, a quantum error-correcting code based on classical BCH codes is presented. And the quantum erasure channel model is discussed, and the parameters of quantum BCH code are provided.
Abstract: After a brief introduction to both quantum computation and quantum error correction, we show how to construct quantum error-correcting codes based on classical BCH codes. With these codes, decoding can exploit additional information about the position of errors. This error model - the quantum erasure channel - is discussed. Finally, parameters of quantum BCH codes are provided.
TL;DR: A twisted squaring construction for the Preparata and Goethals codes is given, based on the double- and triple-error-correcting extended primitive BCH codes, respectively.
Abstract: The trellis complexity of the Preparata and Goethals codes is examined. It is shown that at least for a given set of permutations these codes are rectangular. Upper bounds on the state complexity profiles of the Preparata and Goethals codes are given. The upper bounds on the state complexity of the Preparata and Goethals codes are determined by the dimension/length profiles (DLP) of the extended primitive double- and triple-error-correcting BCH codes, respectively. A twisted squaring construction for the Preparata and Goethals codes is given, based on the double- and triple-error-correcting extended primitive BCH codes, respectively.