Showing papers on "BCH code published in 2003"

Decoding a received BCH encoded signal

Abstract: A method or apparatus for decoding of a BCH encoded signal begins by determining whether the received BCH encoded signal includes error. The decoding process continues when the received BCH encoded signal includes error by determining whether the error is correctable. This may be done by determining a number of errors of the received BCH encoded signal, identifying bit locations of the received BCH encoded signal having the error; counting the number of bit locations of the received BCH encoded signal having the error, comparing the number of errors to the number of bit locations of the received BCH encoded signal having the error, when the number of bit locations of the received BCH encoded signal having the error equals the number of errors, ceasing the identifying of the bit locations of the received BCH encoded signal having the error, and correcting information contained in the bit locations of the received BCH encoded signal having the error when the identifying of the bit locations is ceased.

Fundamentals of Information Theory and Coding Design

Abstract: From the Publisher: "Without abandoning the theoretical foundations, Fundamentals of Information Theory and Coding Design presents working algorithms and implementation that can be used to design and create real systems. The emphasis is on the underlying concepts governing information theory and the mathematical basis for modern coding systems, but the authors also provide the practical details of important codes like Reed-Solomon, BCH, and Turbo codes. Also setting this book apart are discussions on the cascading of information channels and the additivity of information, the details of arithmetic coding, and the connection between coding of extensions and Markov modeling." Complete, balanced coverage, an outstanding format, and a wealth of examples and exercises make this an outstanding text for upper-level students in computer science, mathematics, and engineering, and a valuable reference for telecommunications engineers and coding theory researchers.

Parallel decoding of a BCH encoded signal

Abstract: A method or apparatus for error identification of a BCH encoded signal includes processing that begins by receiving a BCH encoded signal in a binary polynomial format to produce a received polynomial. The processing then continues by converting the received polynomial into a plurality of error identifying polynomials. The processing then continues by recursively processing the plurality of binary error identifying polynomials to produce a plurality of error identifying values. The processing then continues by processing the plurality of error identifying values to produce an error locator polynomial that represents error in the received polynomial. The processing then continues by evaluating the error locator polynomial to identify the bit location of the error in the BCH encoded signal. The processing then continues by correcting the BCH encoded signal based on the bit location of the error.

Codes, graphs, and schemes from nonlinear functions

Abstract: We consider functions on binary vector spaces which are far from linear functions in different senses. We compare three existing notions: almost perfect nonlinear functions, almost bent (AB) functions, and crooked (CR) functions. Such functions are of importance in cryptography because of their resistance to linear and differential attacks on certain cryptosystems. We give a new combinatorial characterization of AB functions in terms of the number of solutions to a certain system of equations, and a characterization of CF in terms of the Fourier transform. We also show how these functions can be used to construct several combinatorial structures; such as semi-biplanes, difference sets, distance regular graphs, symmetric association schemes, and uniformly packed (BCH and Preparata) codes.

Designing fault-secure parallel encoders for systematic linear error correcting codes

Abstract: We consider the open problem of designing fault-secure parallel encoders for various systematic linear ECC. The main idea relies on generating not only the check bits for error correction but also, separately and in parallel, the check bits for error detection. Then, the latter are compared against error detecting check bits which are regenerated from the error correcting check bits. The detailed design is presented for encoders for CRC codes. The complexity evaluation of FPGA implementations of encoders with various degrees of parallelism shows that their fault-secure versions compare favorably against their unprotected counterparts both with respect to complexity and the maximal frequency of operation. Future research will include the design of FS decoders for CRC codes as well as the generalization of the presented ideas to design of FS encoders and decoders for other systematic linear ECC like nonbinary BCH codes and Reed-Solomon codes.

System and method for coding a digital wrapper frame

Abstract: A system and method are provided for coding a frame in a packet communications system using a G.709 Digital Wrapper Frame format. The method comprises: accepting digital information; outer encoding the digital information with a Reed Solomon (RS) encoding scheme; interleaving the outer encoded information; inner encoding the interleaved information using a BCH encoding scheme; and, forming a G.709 Digital Wrapper frame including payload and parity bytes. More specifically, a standard DW superframe is formed with 122,368 bits of payload and 8192 bits of parity. The outer encoding process uses an RS(1023,1007) parent code. In one aspect, 15 groups of RS(781,765) and 1 group of RS(778,762) codewords are formed per superframe. The inner encoding process uses a BCH(2047,1959) parent code. In one aspect, 64 groups of BCH(2040,1952) codewords are formed per superframe.

Single-chip FEC codec using a concatenated BCH code for 10 Gb/s long-haul optical transmission systems

Abstract: This paper describes a 10 Gb/s throughput FEC (forward error correction) codec for long-haul optical transmission systems. The FEC codec uses concatenated BCH(2040, 1930) and BCH(3860, 3824), which provides an excellent 7.98 dB net coding gain at 1E/sup -12/ corrected bit error rate with 6.69% redundancy by applying three times iterative decoding. Furthermore, by applying a time-multiplexed decoder and skipping Chien search process for uncorrectable BCH(3860, 3824) codewords, the FEC codec achieves a low power consumption of 3.15 W and a low gate count of 2.40 Mgates using 0.15 /spl mu/m CMOS technology.

Analysis of mapping strategies for turbo-coded space-time block codes

Abstract: We propose a "turbo" coding scheme for the multiple-input multiple-output (MIMO) Rayleigh fading channel consisting of the serial concatenation of a block code as the outer code and different orthogonal space-time block codes (STBC) for more than two transmit antennas as the inner code. Here, we consider the orthogonal STBC as a mapping scheme in space and time. At the receiver, we apply iterative space-time detection and decoding. We analyze the impact of different mapping strategies on the information transfer of the soft-input-soft-output (SISO) space-time detector. Moreover, we analytically show that additional performance gains over Gray mapping can be obtained by different mapping strategies. Furthermore, we use extrinsic information transfer characteristics (EXIT-charts) in order to predict the performance and the behavior of the system.

Efficient method for fast decoding of BCH binary codes

Abstract: A method for decoding a received word, including calculating a syndrome of the received word as a plurality of binary element vectors, generating respective logarithms of the binary element vectors, and determining, in response to the logarithms, an indication of a position of an erroneous bit in the received word.

Communication system and method

Abstract: A communication system comprises a transmitting apparatus and receiving apparatus for transmitting/receiving a N-ary signal, based on a prime number exceeding 2, inclusive of an efficient error correction system. The transmitting apparatus includes a binary to N-ary converting unit for converting a binary transmit signal into an N-ary signal, as an information sequence, where the N of the N-ary number is a prime number exceeding 2, an encoding unit for generating a transmit sequence, comprised of BCH code on a Galois field with the number of elements being a prime number exceeding 2, and a multi-level modulating unit for multi-level modulating the transmit sequence and for transmitting the multi-level modulated transmit sequence to a receiving apparatus. The receiving apparatus includes a demodulating unit for demodulating the received signal from the transmitting apparatus and for producing a multi-leveled received sequence, with the multi-level of the received sequence being a prime number exceeding 2, a decoding unit for decoding the received sequence by a decoding method for a BCH code on a Galois field with the number of elements being a prime number exceeding 2, and a N-ary to binary conversion unit for producing a decoded binary signal from the N-ary signal as a decoded sequence.

An adaptive two-stage algorithm for ML and sub-ML decoding of binary linear block codes

Abstract: Two distinct codeword-searching procedures based on iterative bounded-distance decoding (BDD) are combined to form an adaptive two-stage maximum-likelihood (ML) decoder for binary linear block codes. During the first stage of the algorithm, a tight upper bound on an error likelihood metric ("discrepancy") is established iteratively for the ML codeword. First-stage processing requires sorting and storage. Adaptive switching to the second stage removes the sorting and storage requirements and allows to rule out redundant BDDs efficiently. Second-stage processing accounts for all codewords with discrepancy lower bound below the upper bound of the ML codeword and guarantees ML performance. In addition, the proposed two-stage algorithm is inherently tunable for controlled suboptimum operation. Under sub-ML operation, the overall scheme can be interpreted as a generalization of the Chase (1972) algorithm. Simulation studies for the (24,12,8) extended Golay and the (64,30,14) and (128,64,22) extended Bose-Chaudhuri-Hocquenghem (BCH) codes illustrate and support these theoretical developments.

Lee-metric decoding of BCH and Reed-Solomon codes

Abstract: A Lee-metric list-decoding algorithm for Reed–Solomon (RS) codes over GF( p) is presented. The algorithm is obtained by generalising the Guruswami–Sudan (Hamming metric) list-decoding algorithm for RS codes. The algorithm can be used to decode the Lee-metric BCH codes, and outperforms the known Lee-metric decoding algorithm for BCH codes.

Design space exploration of a hardware-software co-designed GF(2m) galois field processor for forward error correction and cryptography

Abstract: This paper describes a hardware-software co-design approach for flexible programmable Galois Field Processing for applications which require operations over GF(2m), such as RS and BCH codes, Elliptic Curve Cryptography and the AES Complexities of flexible implementations of different applications on a same computation architecture can be migrated to software during design time However, the underlying GF(2m) arithmetic architecture needs to be designed with software programmability (or reconfigurability) in mind We describe novel reconfigurable subword parallel GF(2m) arithmetic architectures designed with an associated instruction set architecture for different applications over GF(2m) and same applications with differing parameters Design space exploration is carried out with two simple parameters P and Q which can be changed at design time and will affect the performance of different applications and flexibility of the final implementation We show implementation results given for an FPGA prototype of the processor and programmed for RS and BCH coding, AES and elliptic curve cryptography with differing parameters Complexity figures and configuration overheads for subword parallel GF(2m) arithmetic architectures are also estimated and discussed

A key equation and the computation of error values for codes from order domains

Abstract: We study the computation of error values in the decoding of codes con- structed from order domains. Our approach is based on a sort of analog of the key equation for decoding Reed-Solomon and BCH codes. We identify a key equation for all codes from order domains which have finitely-generated value semigroups; the field of fractions of the order domain may have arbitrary transcendence degree, however. We provide a natural interpretation of the construction using the theory of Macaulay's inverse systems and duality. O'Sullivan's generalized Berlekamp-Massey- Sakata (BMS) decoding algorithm applies to the duals of suitable evaluation codes from these order domains. When the BMS algorithm does apply, we will show how it can be understood as a process for constructing a collection of solutions of our key equation.

Alternant and BCH codes over certain rings

Abstract: Alternant codes over arbitrary finite commutative local rings with identity are constructed in terms of parity-check matrices. The derivation is based on the factorization of xs - 1 over the unit group of an appropriate extension of the finite ring. An efficient decoding procedure which makes use of the modified Berlekamp-Massey algorithm to correct errors and erasures is presented. Furthermore, we address the construction of BCH codes over Zm under Lee metric.

Coding gain of basic FEC block-codes in the presence of ASE noise

Abstract: We investigate the performance of Hamming and BCH block codes for the case of ASE (amplified spontaneous emission) noise as the dominating impairment. Numerical results obtained by Monte-Carlo simulations show excellent agreement with theoretical considerations.

Enhanced robustness in image watermarking using turbo codes

Abstract: In this paper, we investigate the improvement achieved by turbo coding for robustness in still image watermarking. We use an error correcting scheme based on the concatenation of a BCH product code and a repetition code. The product code is iteratively decoded using the Chase-Pyndiah algorithm (turbo decoded). For this study, we set the watermarking distortion to around 38 dB and we consider different payloads that can correspond to different applications and services. We compare different coding strategies (i.e. repetition code only and a concatenation of product and repetition codes) in terms of robustness to different photometric attacks, in particular additive noise or lossy compression. Typically, for a payload of 121 bits, the robustness gain for a given message error probability is significant: first error appears at a JPEG quality factor of about 25% with the new coding scheme instead of about 50% when using only repetition codes.© (2003) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Affine invariant extended cyclic codes over Galois rings

Abstract: Recently, Blackford and Ray-Chaudhuri used transform domain techniques to permutation groups of cyclic codes over Galois rings. They used the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length 2/sup m/ over any subring of GR(4,m) to be affine invariant. Here, we use the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length p/sup m/ over any subring of GR(p/sup e/,m) to be affine invariant, for e=2 with arbitrary p and for p=2 with arbitrary e. These are used to find two new classes of affine invariant Bose-Chaudhuri-Hocquenghem (BCH) and generalized Reed-Muller (GRM) codes over Z/sub 2//sup e/ for arbitrary e and a class of affine invariant BCH codes over Z/sub p//sup 2/ for arbitrary prime p.

A forward error correction scheme for ultra long haul optical transmission systems based on low-density parity-check codes

Abstract: FEC scheme based on LDPC codes is presented in this paper. We show that LDPC codes provide a significant system performance improvement with respect to the state-of-the-art FEC schemes, such as turbo BCH and RS codes, employed in optical communications systems. The system performance is further improved by a code design that eliminates short cycles in a graph employed in iterative decoding. As opposed to additive white Gaussian noise (AWGN) model for optical fiber channel, which is used very often in the analysis of error controlling schemes, our model takes into account all major impairments in a long-haul optical transmission such as amplifier spontaneous emission (ASE) noise, pulse distortion due to fiber nonlinearities, chromatic dispersion, crosstalk, intersymbol-interference, etc.

Optical link optimization using embedded forward error correcting codes

Abstract: The design of a single-chip optical transceiver to optimize the performance of a short-distance optical datalink is proposed. The transceiver includes an embedded hybrid automatic repeat request (ARQ) controller capable of operation at several gigahertz clock rates. The hybrid ARQ controller uses a combination of packet retransmission protocols and forward error correction (FEC) to minimize bit errors and achieve a transmitter power coding gain of several dB. Conventional FEC codes such as Reed-Solomon codes cannot be used due to their excessive hardware cost and delays. A practical multilevel coding scheme is explored. The inner codes consist of small linear block codes with reasonable FEC capability, such as small BCH codes, which can be encoded and decoded with reasonable hardware cost and delay. The outer code for a complete packet consists of a long linear block code with excellent error detection ability, such as a cycle redundancy check code. Low-power pipelined on-chip FEC decoders with estimated throughputs of several hundred gigabits per second per square millimeter are proposed. Mathematical analysis indicates that substantial coding gains are possible, which can be used to increase the data rate or the distance span of the link. The proposed designs can be used in short-distance optical transceivers for 10-Gb ethernet, fiberchannel, and very short reach optical datalinks, and are scalable to future two-dimensional optical datalinks with Terabits of capacity.

Processor for performing arithmetic operations in composite operands

Abstract: The present invention relates to a processor and, more particularly, a truly flexible and versatile Configurable Galois Field Processor for applications which require operations over GF(2 ). Embodiments of the present invention have a unique configuration structure allows them to operate in either Single Instruction Multiple Data Mode (SIMD) (.i.e. p GF(2 ) operations) or Single Instruction Single Data Mode (SISD) (i.e. one GF(2 ) operation, where m <= p.q.). The choice of p and q is determined by the type of application intended and the level. of performance required. Accordingly, the same realised processor can be programmed for both forward error correction (Reed Solomon Codes, BCH codes) and Cryptography (Advanced Encryption Standard Lirid Elliptic Curve Cryptography) under various performance constraints. The properties of embodiments of processor can also be adapted for use in implementing DSP algorithms over Galois Fields (such as Transforms and adaptive filters). The dual mode design of the GF processor ensures a very high performance regardless of the type of applications.

Some categorical aspects of bch-algebras

Abstract: We show that the category BCH of BCH-algebras and BCH-homomorphisms is complete. We also show that it has coequalizers, kernel pairs, and an image factorization system. It is also proved that onto homomorphisms and coequalizers, and monomorphisms and one-to-one homomorphisms coincide, respectively, in BCH. It is shown that MBCI is a coreflexive subcategory of BCH. Regular homomorphisms have been defined and their properties are studied. An open problem has been posed.

An Optimized Multi-bits Blind Watermarking Scheme

Abstract: This paper presents a new multi-bits watermarking scheme in DCT domain based on a chaotic Direct Sequence Spread Spectrum (DSSS) communication system, which is combined with error correcting codes (ECC) and Human Visual System (HVS) model in spatial domain. To extract the hidden watermark from a possibly corrupted watermarked image without error, we model watermarking as a digital communication problem and apply BCH channel coding and shuffling. To ensure optimal adaptive DCT watermark, we also demonstrate how to optimally embed a watermark given the constraints imposed by the mask in the spatial domain. The robustness of the algorithm has been tested with StirMark 4.0. Without the original image during the decoding process, the algorithm allows for the recovery of 64 bits of information in a 256×256 graylevel image after a significant JPEG compression and other common signal processing attack.

Switching threshold and coding-rate optimisation for turbo convolutional and turbo bch coded adaptive modulation

Abstract: Decision feedback equaliser (DFE) aided turbo coded wideband adaptive quadrature amplitude modulation (AQAM) is capable of accommodating the temporal channel quality variation of fading channels. A procedure is suggested for determining the AQAM switching thresholds and the specific turbo coding rates capable of maintaining the target BER, while aiming for achieving a high effective bits per symbol (BPS) throughput. Subsequently, the performances of both turbo convolutional (TC) and turbo BCH (TBCH) coded fixed and adaptive modulation schemes are characterised and compared when communicating over the COST 207 typical urban (TU) wideband fading channel. It was found that at a given throughput the set of TBCH coded fixed modulation schemes considered outperforms the corresponding TC counterparts, when using a high-latency turbo and channel interleaver. However, both schemes have a similar performance, when applied in conjunction with adaptive modulation.

A Decoding Algorithm for Turbo Product Codes Using Optimality Test and Amplitude Clipping

Abstract: This paper presents an iterative soft-input/soft-output (SISO) decoder for product code using optimality test and amplitude clipping. A modified expression for computing the soft-output of SISO decoder is proposed. The correlation discrepancy is employed to provide an optimality test on the decision codeword. The optimality test is performed in row and column decoding to evaluate the reliability of row and column decision codewords. Based on the optimality test, the variable reliability factor is introduced for optimization of turbo decoding. A stopping criterion with very little performance degradation is also designed for turbo decoding of product codes by using the optimality test. Besides, the amplitude clipping is employed to improve the performance of turbo product code. Simulation results on the performance of the introduced SISO decoder are presented.

WLANs with extended communication ranges

Abstract: The aim of this paper is to quantify the performance gains in terms of transmission ranges for a broad class of the outer and inner channel coding schemes and deployed space-time coding applied to HIPERLAN2 (H2). As such, the H2 PHY layer represents a broad class of OFDM-based W-LAN technologies. This allows a direct translation of obtained results into performance gains for other standards, such as IEEE802.11a. The outer coding class used comprises a broad class of BCH codes. The inner channel code is either a convolutional or turbo code. The space-time codes deployed comprise higher order space-time trellis and space-time block codes. Assuming a given pathloss model, performance gains are quantified in terms of transmission ranges gained for each H2 PHY mode.

An efficient prime-length DFT algorithm over finite fields GF(2m)

Abstract: This letter describes a discrete Fourier transform (DFT) for prime transform lengths N ≥ 3, where the sample values are elements of finite (Galois) fields GF(2m). A regular and uniform system of additions and multiplications is presented which reduces the multiplicative complexity by at least one-quarter, compared with a brute-force implementation. The benefits of the algorithm are shown with respect to BCH decoding; in particular, the efficient computation of syndromes will be discussed. Copyright © 2003 AEI.

Errors and erasures decoding of BCH and Reed-Solomon codes for reduced M-ary orthogonal signaling

Abstract: The paper presents a comparison of communication systems using different signal constellation sizes and Reed-Solomon or Bose-Chaudhuri-Hocquengem codes with different rates so that the overall required bandwidth is the same for each system. In these comparisons, the channel symbol size is smaller than the code symbol size, so that a code symbol contains parts of multiple channel symbols. Thus, the normal assumption of independent code symbols does not apply. Instead, consideration must be taken to obtain the best arrangement of channel symbols in each code symbol. Analytical expressions are developed to compare the bit error probability performance of comparable systems, based on individual codewords using errors-only decoding and errors and erasures decoding with transmission over a Rayleigh fading channel.

Errors and Erasures Decoding of BCH and Reed-Solomon Codes for Reduced M-ary

Abstract: This paper presents a comparison of communication systems using different signal constellation sizes and Reed- Solomon or Bose-Chaudhuri-Hocquengem codes with different rates so that the overall required bandwidth is the same for each system. In these comparisons, the channel symbol size is smaller than the code symbol size, so that a code symbol contains parts of multiple channel symbols. Thus, the normal assumption of independent code symbols does not apply. Instead, consideration must be taken to obtain the best arrangement of channel symbols in each code symbol. Analytical expressions are developed to com- pare the bit error probability performance of comparable systems, based on individual codewords using errors-only decoding and errors and erasures decoding with transmission over a Rayleigh fading channel.