Showing papers on "BCH code published in 1992"

Method and apparatus for decoding code words protected wordwise by a non-binary BCH code from one or more symbol errors

Abstract: A method and apparatus for decoding code words which are error protected by a non-binary BCH-code against at least one symbol error. Upon reception of an input code word a syndrome information thereof is generated under control of code defining information. From the syndrome information, a key equation is set up and solved for generating an error locator polynomial sig (z) and an error evaluator polynomial w(z). On the basis of the polynomial sig (z) and its formal derivative polynomial sig' (z), Euclid's algorithm is employed to calculate two accessory polynomials b(z), c(z), such that b(z)sig(z)+c(z)sig'(z)=1. From these, a Lagrangian polynomial L(z) is calculated which for any inversed error location value has the associated error symbol value. The error data is then calculated by evaluating the error locator polynomial and the Lagrangian polynomial.

Reed-Muller codes and supersingular curves. I

Abstract: © Foundation Compositio Mathematica, 1992, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

The algebraic decoding of the (41, 21, 9) quadratic residue code

Abstract: A new algebraic approach for decoding the quadratic residue (QR) codes, in particular the (41, 21, 9) QR code, is presented. The key ideas behind this decoding technique are a systematic application of the Sylvester resultant method to the Newton identities associated with the syndromes to find the error-locator polynomial, and next a method for determining error locations by solving certain quadratic, cubic, and quartic equations over GF(2/sup m/) in a new way which uses Zech's logarithms for the arithmetic. The logarithms developed for Zech's logarithms save a substantial amount of computer memory by storing only a table of Zech's logarithms. These algorithms are suitable for implementation in a programmable microprocessor or special-purpose VLSI chip. It is expected that the algebraic methods developed can apply generally to other codes such as the BCH and Reed-Solomon codes. >

Studying the locator polynomials of minimum weight codewords of BCH codes

Abstract: Primitive binary cyclic codes of length n=2/sup m/ are considered. A BCH code with designed distance delta is denoted B(n, delta ). A BCH code is always a narrow-sense BCH code. A codeword is identified with its locator polynomial, whose coefficients are the symmetric functions of the locators. The definition of the code by its zeros-set involves some properties for the power sums of the locators. Moreover, the symmetric functions and the power sums of the locators are related to Newton's identities. An algebraic point of view is presented in order to prove or disprove the existence of words of a given weight in a code. The principal result is the true minimum distance of some BCH codes of length 255 and 511. which were not known. The minimum weight codewords of the codes B(n2/sup h/-1) are studied. It is proved that the set of the minimum weight codewords of the BCH code B(n,2/sup m-2/-1) equals the set of the minimum weight codewords of the punctured Reed-Muller code of length n and order 2, for any m. >

Error-Correcting Codes and Finite Fields

Abstract: Introduction Block codes, weight, and distance Linear codes Error processing for linear codes Hamming codes Appendix: Linear algebra Introduction and an example Euclid's algorithm Invertible and irreducible elements The construction of finite fields The structure of finite fields Roots of polynomials Primitive elements Appendix: Polynomials over a field BCH-codes as subcodes of Hamming codes BCH codes as polynomial codes Decoding BCH codes (1) the fundamental equation Decoding BCH codes: (2) an error processing algorithm Reed-Solomon codes and burst error correction Bounds on codes Classical Goppa codes Classical Goppa codes: error processing Introduction to algebraic curves Functions on algebraic curves A survey of the theory of algebraic curves Geometric Goppa codes An error processor for geometric Goppa codes.

A high-performance full-motion video compression chip set

Abstract: A seven-chip set which performs the functions associated with video and image compression algorithms, and CCITT H.261 in particular, has been designed, fabricated, and is fully functional. The major functions performed by the devices include motion estimation, DCT and IDCT, forward and inverse quantization, Huffman coding and decoding, BCH error correction, and loop filtering. The chips that perform the predictive and transform coding section of the algorithm operate with pixel rates up to 40 MHz. Array-based technologies of 1.5 and 1.0 mu m CMOS were used extensively to achieve a 28 man-month design time. Each die is less than 10 mm on a side. >

A burst-error-correcting algorithm for Reed-Solomon codes

Abstract: It is known that for a burst-error environment, the error-correcting capability of Reed-Solomon codes can be extended beyond the Singleton bound with a high degree of confidence. This is significant in that an (n, k) code with an arbitrarily small probability of a miscorrection can correct more than (n-k)/2 errors. A decoding algorithm for correcting a burst of length greater than (n-k)/2 is presented. >

Real-time binary BCH decoder

Abstract: A decoder is utilized for the correction of bit errors occurred in BCH (Bose-Chaudhuri-Hocquenghem) codes. The decoder first calculates the syndromes of a received word. The syndrome values form a first group of syndrome matrices whose determinant values are used for determining the weight of the error pattern of the received word. Subsequently, during each error trial testing, the bits constituting the received word are cyclically shifted and a predetermined bit is inverted to form a new word to see how the corresponding weight of the error pattern is changed thereby. If the weight is increased, the predetermined bit before being inverted is a correct one; otherwise if decreased the same is an erroneous one and thus correcting action is undertaken. The weight of the error pattern of a word is determined by the zeroness of the determinants of a plurality of matrices formed by the syndrome values thereof.

Cyclic concatenated codes with constacyclic outer codes

Abstract: The known construction of cyclic concatenated codes is based on the fact that the inner is a cyclic minimal code, the outer code is cyclic, and the lengths of the inner and outer codes are relatively prime. It is shown that if the outer code is a suitably chosen constacyclic code the overall concatenated code is always cyclic regardless of the length of the outer code. Moreover, it follows that any cyclic code of composite length is a direct sum of cyclic concatenated codes with inner cyclic minimal codes and outer constacyclic codes. This description of cyclic codes of composite length leads to the introduction of the concept of a poor-code length (PCL). All but low-rate codes of this length have a poor minimum distance. A PCL is directly related to the existence of irreducible binomials. In the binary case, the first few PCLs are 9, 25, 27, 45, 49, 75, 81 and 99. Arbitrarily long binary cyclic codes that are better than binary BCH codes of primitive length are constructed. >

Error correcting neural networks for channels with Gaussian noise

Abstract: The main features of error correcting codes and standard decoding techniques are reviewed. Feedforward neural networks for soft-decision decoding of block codes in channels with additive white Gaussian noise are presented. When the noise is not white, the authors deduce the optimal set of weights for the connections of the network. These weights are also approximately obtained by an error backpropagation algorithm. A practical realization for a BCH

On minimum distance bounds for abelian codes

Abstract: This paper is an exposition of two methods of formulating a lower bound for the minimum distance of a code which is an ideal in an abelian group ring. The first, a generalization of the cyclic BCH (Bose-Chaudhuri-Hoquenghem) bound, was proposed by Camion [2]. The second method, presented by Jensen [4], allows the application of the BCH bound or any of its improvements by viewing an abelian code as a direct sum of concatenations of cyclic codes. This second method avoids the mathematical analysis required for a direct generalization of a cyclic bound to the abelian case. It can produce a lower bound that improves the generalized BCH bound. We present simple algorithms for 1) deriving the generalized BCH bound for an abelian code 2) determining direct sum decompositions of an abelian code to concatenated codes and 3) deriving a bound on an abelian code, viewed as a direct sum of concatenated codes, by applying the cyclic BCH bound to the inner and outer code of each concatenation. Finally, we point out the applicability of these methods to codes that are not ideals in abelian group rings.

Frequency-hopped multiple access communications with coding and side information

Abstract: The authors consider frequency-hopped spread-spectrum multiple-access communications using M-ary modulation and error-correction coding. The major concerns are multiple-access interference and the network capacity in terms of the number of users that can transmit simultaneously for a given level of codeword error probability. Block coding is studied in detail. The authors first consider the use of Q-ary Reed-Solomon (RS) codes in combination with M-ary modulation with mismatched alphabets so that Q>M. It is shown that the network capacity is drastically reduced in comparison with the system with matched alphabets. As a remedy, the use of matched M-ary BCH codes is proposed as an alternative to mismatched RS codes. It is shown that when the number of users in the system is large, a BCH code outperforms an RS code with a comparable code rate and decoding complexity. The authors consider the use of a robust technique for generation of reliable side information based on a radio-threshold test. They analyze its performance in conjunction with MFSK and error-erasure correction decoding. It is shown that this nonideal ratio-threshold method can increase the network capacity in comparison with the system with perfect side information. >

Runlength limited binary error control codes

Abstract: A systematic procedure is presented for modifying standard binary error control codes (ECCs) to form runlength-limited ECCs without reducing rates or affecting minimum distances. This procedure is then optimised to find the tightest runlength bound given an EEC defined by its generator matrix. A simple example based on the BCH (15,5) ECC is presented to illustrate the procedure, which is nonetheless more generally applicable. The effects of the modification on spectral characteristics are also considered.

Voice transmission at a very low bit rate on a noisy channel: 800 bps vocoder with error protection to 1200 bps

Abstract: A codec was studied at 800 b/s source coding and total bit rate of 1200 b/s after error protection. After an LPC 10 analysis, the 800-b/s coder encodes several frames together in order to use the intercorrelation between successive frames. The coder chooses from eight different spectrum coding schemes the one which best corresponds to the speech on the superframe, viz. the energy is quantized by a VQ of the evolution of the power on the superframe and a scalar quantization of a reference energy. The redundancy of 3/2 (1200/800) is filled up with block codes. Comparison between, on the one hand, BCH(63,36) codes and, on the other hand, a combination of Golay codes and short BCH gives, after mathematical simulations and listening tests, an advantage to the latter protection scheme. The vocoder remains nearly unaltered up to 3% BER on the 1200 b/s channel and is intelligible up to 5% BER. >

Method and apparatus for implementing a triple error detection and double error correction code

Abstract: A method and apparatus for achieving the detection of triple errors and the correction of double errors in data stored in a memory or processed in a data processing system. The method and apparatus being based on a modification of a standard Bose Chauduri Hocquenghem (BCH) code that permits a reduction of the decoding circuitry needed to achieve the detection and correction of the errors.

Real-valued error control coding by using DCT

Abstract: The discrete cosine theorem (DCT)-based real-valued linear code is derived for the first time in the literature. A BCH-like subclass of DCT linear codes is also developed, for which fast decoding algorithms exist. It is shown that the DCT-based error control codes proposed by the authors can be viewed as a bridge to link the fields of source coding and channel coding.

On the spectra of the duals of binary BCH codes of designed distance delta =9

Abstract: An infinite series of duals of binary BCH codes with designed distance delta =9 and lengths l=2/sup n/ with n odd is found, such that their weights almost satisfy the Carlitz-Uchiyama bound. This solves negatively a problem formulated by F.J. MacWilliams and N.J.A. Sloane (1977). >

Synchronous fibre-optic code division multiple access networks with error control coding

Abstract: For a fibre-optic synchronous code divison multiple access (S/CDMA) network, the number of users able to simultaneously utilise the network is a major concern. A theoretical analysis of the S/CDMA network employing error control codes to enhance the reliability of the network and to increase the number of concurrent users is presented. The error probabilities for both the uncoded and coded systems are analysed. The example shows that by using a (63, 36) BCH code the influence of interference arising from other users can be greatly reducted and the number of active users can also be increased significantly.

Error correction circuit for BCH codewords

Abstract: An error correction circuit for detecting and correcting errors in a received BCH codeword comprises first and second syndrome generators for generating patterns of binary signals representative of first and second syndromes of the received BCH codeword. An error detector circuit utilizes the generated syndromes to determine whether the received codeword has no multi-bit errors. Single bit errors are corrected while multibit errors are only indicated. The single bit error is corrected by successively generating patterns of binary signals representative of the powers a k of a primitive number a where k=p-1, p-2, . . . 2, 1, 0 and where p is the number of bits in the received codeword. When the binary pattern representative of the first syndrome is identical to the binary pattern representative of a k for a particular value of k, the particular value k indicates the location of the single bit error in the received codeword.

Designing Efficient Maximum-Likelihood Soft-Decision Decoding Algorithms for Linear Block Codes Using Algorithm A*

Abstract: In this report we present a class of efficient maximum-likelihood soft-decision decoding algorithms for linear block codes. The approach used here is to convert the decoding problem into a search problem through a graph which is a trellis for an equivalent code of the transmitted code. Algorithm A*, which uses a priority-first search strategy, is employed to search through this graph. This search is guided by an evaluation function f defined to take advantage of the information provided by the received vector and the inherent properties of the transmitted code. This function f is used to drastically reduce the search space and to make the decoding efforts of this decoding algorithm adaptable to the noise level. For example, simulation results for the (128,64) binary extended BCH code indicate that for most real channels the proposed decoding algorithm is at least fifteen orders of magnitude more efficient in time and in space than that proposed by Wolf. Simulation results for the (104, 52) binary extended quadratic residue code are also given. These simulation results indicate that the use of Algorithm A* for decoding has resulted not only in an efficient soft-decision decoding algorithm for hitherto intractable linear block codes, but an algorithm which is in fact optimal as well.

Weight distribution of cosets of 2-error-correcting binary BCH codes of length 15, 63 and 255

Abstract: The weight distributions of cosets are not known for the class of binary 2-error-correcting BCH codes of length n=2/sup m/-1, m even (the nonuniformly packed case). By using the graph theoretical concepts of the combinatorial matrix of a code and an r-partition design introduced in previous works, the authors obtain these distributions of cosets in the three particular cases of length 15, 63, and 255. It is observed that in these three cases the number of distinct weight distributions is a constant equal to 8. >

A fast algorithm to determine the burst-correcting limit of cyclic or shortened cyclic codes

Abstract: A novel fast algorithm is developed for computing the burst-correcting limit of a cyclic or shortened cyclic code from the parity-check polynomial of the cyclic code. The algorithm is similar to the algorithm of H.J. Matt and J.L. Massey (1980) which, up to now, has been the most efficient method for determining the burst-correcting limit of a cyclic code, but is based on apolarity of binary forms instead of linear complexity. The running times of implementations in C of both algorithms on an IBM RISC System/6000 are compared for several binary cyclic codes of practical interest. A table of the burst-correcting limit of primitive binary BCH codes of length up to 1023 is included. >

Wavelet-coded image transmission over land mobile radio channels

Abstract: An image compression and channel coding scheme is proposed for efficient image transmission over mobile radio channels. The image-compression system is based on a wavelet representation coupled with the JPEG baseline coding algorithm standardized by CCITT/ISO. The wavelet representation forms a pyramid in which information is layered in terms of different priorities, which facilitates more efficient use of forward error correction coding (FEC). After error sensitivity analysis for different layers over a 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 picture quality degradation. Application of the new technique to the proposed Telecommunications Industry Association (TIA) North American digital cellular standard (IS-54) under different vehicle speeds is also investigated. >

Modelling of frequency hopped VHF channels in an urban environment and error correction on the channel

Abstract: Experimental and analytical results of applying Fritchman partitioned Markov chains to model the error sequences of mobile and stationary channels are obtained from real-time measurements. The Markov chain channel models were applied to the system using four modulation schemes, namely FSK, DPSK, QPSK and 8-ary PSK. Using error distributions obtained from both the experimental measurements and the channel models, BCH and Reed-Solomon error correcting codes of various block lengths and information rates are investigated. The effect of different interleaving depths on the block error rate is shown. >

Nonlinear, nonbinary cyclic group codes

Abstract: New cyclic group codes of length 2(exp m) - 1 over (m - j)-bit symbols are introduced. These codes can be systematically encoded and decoded algebraically. The code rates are very close to Reed-Solomon (RS) codes and are much better than Bose-Chaudhuri-Hocquenghem (BCH) codes (a former alternative). The binary (m - j)-tuples are identified with a subgroup of the binary m-tuples which represents the field GF(2 exp m). Encoding is systematic and involves a two-stage procedure consisting of the usual linear feedback register (using the division or check polynomial) and a small table lookup. For low rates, a second shift-register encoding operation may be invoked. Decoding uses the RS error-correcting procedures for the m-tuple codes for m = 4, 5, and 6.