Showing papers on "BCH code published in 1991"

Error-Correcting Output Codes: A General Method for Improving

Abstract: Multiclass learning problems involve finding a definition for an unknown function f(x) whose range is a discrete set containing k < 2 values (i.e., k "classes"). The definition is acquired by studying large collections of training examples of the form [xi, f(xi)]. Existing approaches to this problem include (a) direct application of multiclass algorithms such as the decision-tree algorithms ID3 and CART, (b) application of binary concept learning algorithms to learn individual binary functions for each of the k classes, and (c) application of binary concept learning algorithms with distributed output codes such as those employed by Sejnowski and Rosenberg in the NETtalk system. This paper compares these three approaches to a new technique in which BCH error-correcting codes are employed as a distributed output representation. We show that these output representations improve the performance of ID3 on the NETtalk task and of back propagation on an isolated-letter speech-recognition task. These results demonstrate that error-correcting output codes provide a general-purpose method for improving the performance of inductive learning programs on multiclass problems.

VLSI design of inverse-free Berlekamp-Massey algorithm

Abstract: The Berlekamp-Massey iterative algorithm for decoding BCH codes is modified to eliminate the calculation of inverses. This new algorithm is useful in the practical application of multiple-error-correcting BCH or RS codes. A VLSI architecture is developed for this algorithm.< >

Bit-level soft-decision decoding of Reed-Solomon codes

Abstract: A Reed-Solomon decoder that makes use of bit-level soft-decision information is presented. A Reed-Solomon generator matrix that possesses a certain inherent structure in GF(2) is derived. This structure allows the code to be represented as a union of cosets, each coset being an interleaver of several binary BCH codes. Such partition into cosets provides a clue for efficient bit-level soft-decision decoding. Two decoding algorithms are derived. In the development of the first algorithm a memoryless channel is assumed, making the value of this algorithm more conceptual than practical. The second algorithm, which is obtained as a modification of the first, does account for channel memory and thus accommodates a bursty channel. Both decoding algorithms are, in many cases, orders of magnitude more efficient than conventional techniques. >

Error-correcting WOM-codes

Abstract: A problem raised by R.L. Rivest and A. Shamir (1982), namely, constructing write-once-memory (WOM) codes capable of error correction, is considered. The authors call a (n,m,t)-WOM code a scheme that allows t successive writings of m arbitrary bits (i.e., one message among 2/sup m/) on a WOM of size n. WOM codes have been studied from an information-theoretic viewpoint by J.K. Wolf et al. (1984) and constructed using classical coding theory by G.D. Cohen et al. (1986, 1987) (for example, with parameters, (23,11,3), (2/sup m-1/,m,2/sup m-2/+2/sup m-4/+1)). The authors adapt those methods in order to solve the problem raised by Rivest. Large classes of easily decodable single-error-correcting WOM codes are obtained. >

A Fast Matrix Decoding Algorithm for Rank-Error-Correcting Codes

Abstract: The so-called term-rank and rank metrics and appropriate codes were introduced and investigated in [1 –7]. These metrics and codes can be used for correcting array errors in a set of parallel channels, for scrambling in channels with burst errors, as basic codes in McEliece public key cryptosystem [8], etc. For codes with maximal rank distance (MRD codes) there exists a fast decoding algorithm based on Euclid's Division Algorithm in some non-commutative ring [6]. In this paper a new construction of MRD codes is given and a new fast matrix decoding algorithm is proposed which generalizes Peterson's algorithm [9] for BCH codes.

Decoding cyclic and BCH codes up to actual minimum distance using nonrecurrent syndrome dependence relations

Abstract: The decoding capabilities of algebraic algorithms, mainly the Berlekamp-Massey algorithm, the Euclidean algorithm, and the authors' (1989) generalizations of these algorithms, are basically constrained by the minimum distance bounds of the codes. The authors introduce a more general procedure which breaks away from this restriction and which can determine the, error locations from nonrecurrent dependence relations among the syndromes. It can decode many cyclic and BCH codes up to their actual minimum distance and is seen to be a generalization of the procedure introduced by W.W. Peterson and E.J. Weldon (1972). >

Programmable systolic BCH decoder

Abstract: A programmable decoder that provides both error and erasure decoding for all Reed-Solomon, primitive BCH, non-primitive BCH, and binary BCH codes of any rate over any field is disclosed. The user can specify decoding parameters including the code block-length, the code-generator polynomial, and the field-generator polynomial. The basic architecture, less the small overhead for programmability, is also recommended for fixed-code applications. The decoding processor of the decoder includes systolic arrays implementing a syndrome calculator, a key equation solver, a Chien search, a recursive extender, and an inverse transform. The number of cells required for each of the five functions is on order of the error correction capability t. The systolic arrays can be fabricated on a single VLSI microchip that is itself systolic. Each of the individual systolic arrays can extended by arraying microchips together, so that any desired error correction capability can be attained by using multiple systolic microchips with a single controller.

Decoding binary 2-D cyclic codes by the 2-D Berlekamp-Massey algorithm

Abstract: A method of decoding two-dimensional (2-D) cyclic codes by applying the 2-D Berlekamp-Massey algorithm is proposed. To explain this decoding method, the author introduces a subclass of 2-D cyclic codes, which are called 2-D BCH codes due to their similarity with BCH codes. It is shown that there are some short 2-D cyclic codes with a better cost parameter value. The merit of the approach is verified by showing several simple examples of 2-D cyclic codes. >

Minimum distance bounds for cyclic codes and Deligne's theorem

Abstract: At the present time, there are very good methods to obtain bounds for the minimum distance of BCH codes and their duals. On the other hand, there are few other bounds suitable for general cyclic codes. Therefore, research Problem 9.9 of MacWilliams and Sloane (1977), The Theory of Error-Correcting Codes, asks if the bound of Deligne (1974) for exponential sums in several variables or the bound of Lang and Weil (1954), can be used to obtain bounds on the minimum distance of codes. This question is answered in the affirmative by showing how Deligne's theorem can be made to yield a lower bound on the minimum distance of certain classes of cyclic codes. In the process, an infinite family of binary cyclic codes is presented for which the bound on minimum distance so derived is as tight as possible. In addition, an infinite family of polynomials of degree 3 in 2 variables over a field of characteristic 2, for which Deligne's bound is tight, is exhibited. Finally, a bound is presented for the minimum distance of the duals of the binary subfield subcodes of generalized Reed-Muller codes as well as for the corresponding cyclic codes. It is noted that these codes contain examples of the best binary cyclic codes. >

Foundations of coding : theory and applications of error-correcting codes, with an introduction to cryptography and information theory

Abstract: CODING AND INFORMATION THEORY Coding and Decoding Huffman Codes Data Compression and Entropy Reliable Communication Through Unreliable Channels ERROR--CORRECTING CODES Binary Linear Codes Groups and Standard Arrays Linear Algebra Linear Codes Reed--Muller Codes: Weak Codes with Easy Decoding Cyclic Codes Polynomials and Finite Fields BCH Codes: Strong Codes Correcting Multiple Errors Fast Decoding of BCH Codes Convolutional Codes CRYPTOGRAPHY Cryptography Appendices Bibliography List of Symbols Index

Finding BCH error locator polynomials in one step

Abstract: Syndrome polynomials produced in a BCH decoder span an ideal in a multivariate polynomial ring Because the error locators satisfy every member of the ideal, polynomial ideal theory is used to find another, more easily solvable system of equations having the same roots One such equation is the (univariate) error locator polynomial The Letter shows how to write this polynomial in one mathematical step from the syndrome polynomials

The (d,k) subcode of a linear block code

Abstract: A simple technique employing linear block codes to construct (d,k) error-correcting block codes is considered. This scheme allows asymptotically reliable transmission at rate R over a BSC channel with capacity C/sub BSC/ provided R >

The generation of rimitive olynomials in GF(q) with independent roots and their applications for ower residue codes, VLSI testing and finite field multipliers using normal basis

Abstract: The interest of this research is in finding rimitive olynomials with linearly independent roots over the Galois field of q elements, GF(q). Existing methods are sufficient only to generate a single olynomial. Here they need to be enumerated in order to apply further selections in view of the applications. The olynomials are generated through a search algorithm extensively runed using some known results and newly derived corollaries. Tables of these olynomials are given over fields up to GF(19) for the first time. The common background for the applications is in forming a normal basis from the linearly independent roots. Three applications are discussed, each including new results: the transformation of ower residue codes to quasi-cyclic codes; the VLSI implementation of multiplication and inverse operations over Galois fields and the acceleration of BCH error correcting code decoding; and the superior aliasing robabilities for the digital testing of integrated circuits

The Estimating On Performance To Interleaved BCH Codes Applied To The Mobile Communication Channel

Abstract: In this paper, the Markov model with three sates is used for the probability model to the mobile communication channel with long burst error. The error distribution probability P(n,m) for the mobile channel is calculated in terms of the relation between the parameters of the model and the error statistical parameters of the model and the error statistical parameters, and from this, the performance of error-correcting interleaved BCH codes are estimated, which can be applied to the mobile communication channel. As the first time, the method is proposed that the performance of interleaved BCH codes applied to the mobile channel is estimated by use of the parameters of BSC channel and the simulation of te performance of a few interleaved BCH codes is made in comparison with the result of the estimation.

On computing the weight spectrum of cyclic codes

Abstract: Two deterministic algorithms of computing the weight spectra of binary cyclic codes are presented. These algorithms have the lowest known complexity for cyclic codes. For BCH codes of lengths 63 and 127, several first coefficients of the weight spectrum in number sufficient to evaluate the bounded distance decoding error probability are computed. >

Closed solution of Berlekamp's algorithm for fast decoding of BCH codes

Abstract: A decision tree solution is presented for the most complicated step in decoding binary BCH codes, namely, the computation of an error location polynomial over GF(2/sup m/) from the syndrome vector of received data. The author runs S. Lin's (1970) iterative version of the Berlekamp-Massey algorithm symbolically, keeping the results at each level in the form of branches of a binary decision tree. A decoder can then be constructed that uses the derived formulas to evaluate a decision variable at each level. Complete traversal of the tree using the decision variables leads to the correct polynomial coefficients for the received vector. The decoder can be implemented in a very straightforward way with a simple processor or program that performs extension field arithmetic, or it can be realized entirely in hardware using lookup tables for multiplications, inverses, and exponents, and exclusive OR operations for addition. >

Outage control in digital cellular systems

Abstract: The use of convolutional, BCH, and Reed-Solomon codes to improve the spectral efficiency of digital cellular mobile radio systems is investigated. Optimal code rates are obtained as a function of the allowable outage, under the assumption of ideal interleaving. The outage predictions themselves are obtained by computer simulation. Of the codes considered, the block codes are more effective than convolutional codes for improving spectral efficiency. One major problem observed with error-correction coding is its ineffectiveness for channels with very slow fading. >

On Blahut's decoding algorithms for two-dimensional BCH codes

Abstract: It is shown that decoding of cyclic codes in the DFT domain is equivalent to an appropriate deconvolution problem. A two-dimensional (2-D) generalization of Blahut's (1979) one-dimensional (1-D) linear complexity theorem is obtained and utilized to determine the error-correcting capability of 2-D BCH codes, as afforded by code's defining array of zeros, with regard to correction of burst-errors. The 2-D linear complexity theorem is further utilized to present a new approach for decoding of cyclic codes, in general, and 2-D BCH codes in particular. An alternative exposition of Blahut's decoding algorithms, in the DFT domain, for random and burst error correction in 2-D BCH codes is given from a deconvolution viewpoint. Some modifications for efficient implementation of Blahut's decoding algorithms for random and burst error correction are suggested and improved decoding algorithms are presented. It is shown that the improved decoding algorithm requires at most half the number of passes through the Berlekamp-Massey algorithm compared to the Blahut's decoding algorithm. It is shown that Blahut's decoding algorithms have optimal error-correcting capability and improved decoding algorithms have less computational complexity. A comparative study of various time- and spectral-domain implementations of 2-D BCH decoding algorithms is also given.

On the undetected error probability of triple-error-correcting BCH codes

Abstract: The probability of undetected error P/sub u/( epsilon ) for the primitive triple-error-correcting BCH codes of blocklength 2/sup m/-1 used solely for error detection on a binary symmetric channel with crossover probability epsilon >

Error-rate performance of block-coded M-ary modulation and hard decision decoding systems on rayleigh fading channel

Abstract: Error rates of block-coded MPSK and/or MQAM systems with hard-decision decoding are evaluated theoretically on the Rayleigh fading channel characterizing the land mobile channel. BCH codes are employed as the block codes and analysis is carried out both for nondiversity reception and for dual MRC diversity reception. It is shown that on the Rayleigh fading channel, in contrast to the AWGN channel, BCH-coded MPSK systems can achieve significant coding gain compared to uncoded BPSK even in the hard-decision decoding case without changing spectral efficiency. It is shown also that diversity reception is needed for sufficient large coding gain in higher spectral efficiency. Moreover, it is suggested that the code length longer than a threshold is necessary for large coding gain under the given number of multilevel modulation and given spectral efficiency.

Parallel decoding of binary BCH codes

Abstract: A parallel decoding procedure for the BCH codes is introduced, which is particularly useful for decoding BCH codes with small error-correcting capability. The high regularity inherent in the scheme enable it to be easily implemented with VLSI circuits.

Neural Networks as Error Correcting Systems in Digital Communications

Abstract: Feed forward neural network are very powerful devices for soft-decision decoding of linear block codes. A practical realization for a (7,4) BCH code is presented. The relationship between maximum likelihood decoding, winner-takes-all neural networks and neural networks with a sigmoidal response function is established. It is shown how learning with an error back propagation algorithm can drastically improved the performance of the decoding network in the case of an additive gaussian channel with memory.

Channel modeling and perforumance estimation of block codes with periodic and changeable interfeaving degree in the digital mobile channel

Abstract: In this paper, a simple partitioned Markov model with three states for the digital mobile channel is presented based on a field test curve. The expression of the model and its transition probability matrix P are obtained and error distribution probability P(n,m) is calculated In terms of the matrix P, From P(nm), the performance for a few of Interleaved block podes, which are BCH codes and Golay code is estimated by use of the BSC model and the simulation results is given. At the meantime, additional time-delay caused by the periodic interleaving is obtained. A few of group of the changeable Interleaving degree scheme is presented and the comparison between the periodic and changeable Interleaving degree scheme is obtained by means of the analysis to the error characterization in the digital mobile channel.

A new algorithm for correcting single burst errors with Reed-Solomon codes

Abstract: The author presents a fast algorithm which can correct a single burst error length up to n-k-1 for a RS (n,k) code. Several methods which can reduce the probability of misdecoding if the length of the burst is longer than ((n-k)/2) are discussed. An RS (Reed-Solomon) decoder design is proposed which performs a regular error correction and the proposed burst error correction simultaneously. This method can also be used for BCH codes and for RS codes over Galois fields of characteristics other than two. >

Channel coding/diversity reception on packet mobile radio

Abstract: The capacity of BCH coded π/4-shift QPSK packet mobile radio with postdetection selection diversity reception is investigated by computer simulation taking into account fast Rayleigh fading, shadow fading, and propagation loss. Results presented for 80-bit information packets show that overall capacity can be maximised by the use of high rate codes together with diversity reception.

On the problem of finding zero-concurring codewords

Abstract: Zero-concurring codewords disclose a certain structure of the code that may be used for efficient soft-decision decoding and for designing DC-free codes. Methods for constructing sets of zero-concurring codewords are presented for several families of codes. For the general case an algorithm solution of the problem is offered. A table of results obtained using the proposed techniques is supplied for all the primitive narrow-sense binary BCH codes of length up to 127. >

Codage pour modulation d'amplitude en quadrature à grand nombre d'états avec partition à plusieurs niveaux

Abstract: Les auteurs presentent une nouvelle technique de codage pour la modulation d’amplitude en quadrature (MAQ) a grand nombre d’etats (MAQ-64 et MAQ-256) avec partition a plusieurs niveaux. Ils effectuent une partition de la constellation en quatre classes reperees par deux bits qui sont codes chacun par un meme code BCH. En ce qui concerne la redondance, la simplicite du decodage et les performances obtenues, ce systeme, employe avec un code BCH correcteur d’erreurs doubles, donne de meilleurs resultats que les systemes actuellement connus (utilisant le code de Reed-Solomon (64,62), des codes de Lee-Nakamura ou de Hamming). Le probleme de l’ambiguite de phase est aisement resolu par codage differentiel prealable des deux bits codes et par l’emploi d’un code BCH transparent.