The present study proposes a scheme in which variable-length orthogonal codes generated by combining inverse discrete Fourier transform matrices over a finite field multiplex user data into a multiplexed sequence and its sequence forms one or a plural number of codewords for Reed-Solomon coding. The proposed scheme realizes data multiplexing, error correction coding, and multi-rate transmitting at the same time. This study also shows a design example and its performance analysis of the proposed scheme.

Many studies have applied the three-dimensional discrete wavelet transform (3D DWT) to video coding. It is known that corruptions of the lowest frequency sub-band (LL) coefficients of 3D DWT severely affect the visual quality of video. Recently, we proposed an error resilient 3D DWT video coding method (the conventional method) that employs dispersive grouping and an error concealment (EC). The EC scheme of our conventional method adopts a replacement technique of the lost LL coefficients. In this paper, we propose a new 3D DWT video transmission method in order to enhance error resilience. The proposed method adopts an error correction scheme using invertible codes to protect LL coefficients. We use half-rate Reed-Solomon (RS) codes as invertible codes. Additionally, to improve performance by using the effect of interleave, we adopt a new configuration scheme at the RS encoding stage. The evaluation by computer simulation compares the performance of the proposed method with that of other EC methods, and indicates the advantage of the proposed method.

Secret sharing is a method of information protection for security. The information is divided into n shares and reconstructed from any k shares, but no knowledge of the information is revealed from k-1 shares. Physical layer security is a method of achieving favorable reception conditions at the destination terminal in wireless communications. In this study, we propose a security enhancement technique for wireless packet communications. The technique uses secret sharing and physical layer security to exchange a secret encryption key. The encryption key for packet information is set as the secret information in secret sharing, and the secret information is divided into n shares. Each share is located in the packet header. The base station transmits the packets to the destination terminal by using physical layer security based on precoded multi-antenna transmission. With this transmission scheme, the destination terminal can receive more than k shares without error and perfectly recover the secret information. In addition, an eavesdropper terminal can receive less than k-1 shares without error and recover no secret information. In this paper, we propose a protection technique using secret sharing based on systematic Reed-Solomon codes. The technique establishes an advantageous condition for the destination terminal to recover the secret information. The evaluation results by numerical analysis and computer simulation show the validity of the proposed technique.

To improve the BER performance of the conventional cooperative communication, this letter proposes an efficient method for the reliability, and it uses hierarchical modulation that has both the high priority (HP) layer and the low priority (LP) layer. To compensate more reliable transmission, the proposed method uses the error correction capability of Reed-Solomon (RS) codes additionally. The simulation results show that the proposed method can transmit data more reliably than the basic RS coded decode-and-forward (DF) method.

In this paper, a method is proposed for reconstruction of the parameters of a non-binary block encoder using an intercepted sequence of noisy coded data. The proposed method is a generalization of the Barbier's method for the reconstruction of binary block codes to the more problematic case of non-binary codes. It has been shown mathematically that considering some revisions in definitions, such a generalization is possible. The proposed method is able to estimate the code parameters such as the code length, the code dimension, number of bits per symbol, and the dual-code subspace, and also to synchronize the sequence. Since the Reed-Solomon code is the most important type of non-binary block codes, an additional method is proposed to reconstruct the generator polynomial in the case of Reed-Solomon codes. The proposed method is evaluated via computer simulations which verify its strength and effectiveness.

In this paper, we derive a simple formula to generate a wide-sense systematic generator matrix(we call it quasi-systematic) B for a Reed-Solomon code. This formula can be utilized to construct an efficient interpolation based erasure-only decoder with time complexity O(n2) and space complexity O(n). Specifically, the decoding algorithm requires 3kr + r2 - 2r field additions, kr + r2 + r field negations, 2kr + r2 - r + k field multiplications and kr + r field inversions. Compared to another interpolation based erasure-only decoding algorithm derived by D.J.J. Versfeld et al., our algorithm is much more efficient for high-rate Reed-Solomon codes.

A low-complexity Reed-Solomon (RS) decoder design based on the modified Euclidean (ME) algorithm proposed by Truong is presented in this paper. Low complexity is achieved by reformulating Truong's ME algorithm using the proposed polynomial manipulation scheme so that a more compact polynomial representation can be derived. Together with the developed folding scheme and simplified boundary cell, the resulting design effectively reduces the hardware complexity while meeting the throughput requirements of optical communication systems. Experimental results demonstrate that the developed RS(255, 239) decoder, implemented in the TSMC 0.18 µm process, can operate at up to 425 MHz and achieve a throughput rate of 3.4 Gbps with a total gate count of 11,759. Compared to related works, the proposed decoder has the lowest area requirement and the smallest area-time complexity.

This paper presents a high-speed, low-complexity VLSI architecture based on the modified Euclidean (ME) algorithm for Reed-Solomon decoders. The low-complexity feature of the proposed architecture is obtained by reformulating the error locator and error evaluator polynomials to remove redundant information in the ME algorithm proposed by Truong. This increases the hardware utilization of the processing elements used to solve the key equation and reduces hardware by 30.4%. The proposed architecture retains the high-speed feature of Truong's ME algorithm with a reduced latency, achieved by changing the initial settings of the design. Analytical results show that the proposed architecture has the smallest critical path delay, latency, and area-time complexity in comparison with similar studies. An example RS(255,239) decoder design, implemented using the TSMC 0.18 µm process, can reach a throughput rate of 3 Gbps at an operating frequency of 375 MHz and with a total gate count of 27,271.

This study presents a partition decoding algorithm for an (mN, mK) binary image of an (N, K) Reed Solomon code over GF(2m). A proposed partition decoding algorithm includes several steps. Firstly we compute m's segmental reliability values of a received subvector of length N and determine which one with the least segmental reliability value. A permutation is performed on a binary generator matrix of an RS code and a received vector, which are then partitioned into two submatrices and two subvectors. The first subvector of length N(m-1) associate with the first submatrix and the second subvector with the least segmental reliability value relates to the second submatrix. Secondly, an MLD algorithm based on the first submatrix is employed to decode the first subvector. Thirdly, an MLD algorithm based on a BCH generator matrix is employed to decode the second subvector. A codeword is finally outputted after performing the inverse permutation on a concatenation of code vectors decoded from these two decoding. The error coefficient and minimum Hamming distance of the code sequences generated in the first submatrix are fewer than those of a corresponding binary image. Simulation results show that at low and medium SNRs, the effect of error coefficient becomes more significant than that of minimum Hamming distance. Minimum Hamming distances and error coefficients of code sequences generated in the first submatrices and their corresponding binary images have been explored in this work. For (60,36,7)RS(b), (155,125,7)RS(b), (155,105,11)RS(b) and (889, 847,7))RS(b) being binary images of (15,9,7)RS, (31,25,7)RS, (31,21,11)RS and (127,121,7)RS codes respectively, with BPSK signaling over AWGN channels, the decoding performances of proposed partition decoding algorithm are a little poorer than those of MLD [10] by 1.0 to 1.4 dB at BER 10-5, but better than those of GMD decoding by [1] 0.8 to 1.1 dB. For SNR of 5 dB, proposed partition decoding algorithm only takes 50% to 60% amount of bit operations of an MLD [10]. Under a constraint of decoding complexity, proposed partition decoding algorithm may be a solution to decode binary images of long RS codes, which provides superior performance to GMD decoding with much lower complexity than an MLD.

Error control codes have extensively been applied to semiconductor memories using high density RAM chips with wide I/O data, e.g., with 8-bit or 16-bit I/O data. Recently, spotty byte errors called s-spotty byte errors are newly defined as t or fewer bits errors in a byte having length b bits, where 1 ≤ t ≤ b. This paper proposes another type of spotty byte errors, i.e., m-spotty byte errors, where more than t bits errors in a byte may occur due to hit by high energetic particles. For these errors, this paper presents generalized m-spotty byte error control codes with minimum m-spotty distance d.

Efficient real-time contents distribution services on the Internet are only possible by suppressing the influence of packet losses. One solution for UDP transmission is the use of Forward Error Correction (FEC) based on Reed-Solomon codes. However, a more efficient method is required since this causes the increase of network traffic and includes the weakness to burst packet losses. In this paper, we propose a data recovery method that generates redundant data with the combination of Reed-Solomon codes and convolution of neighboring blocks. We realize the small amount of redundancy and the high reliability in data transmission compared with using only Reed-Solomon codes in the environment that burst packet losses are occurred frequently. We implement proposal method into the network bridge and confirm its efficiency from the viewpoint of data reconstruction from burst packet losses.

This paper examines a coded Gaussian Minimum Shift Keying (GMSK) system which uses Reed-Solomon (RS) codes both in Additive White Gaussian Noise (AWGN) channels and Rayleigh fading channels. The performance of GMSK and RS code combinations is compared with the constraint that the transmitted signal bandwidth is constant. The coding gains were obtained using simulations and the best combination of GMSK and RS codes was found. The optimal code rates over AWGN and Rayleigh fading channels were also compared.

In this paper, an efficient architecture for an adaptive Reed-Solomon decoder is presented, where the block length n and the message length k can be varied from their minimum allowable values up to their selected values. This eliminates the need of inserting zeros before decoding shortened RS codes. And the error-correcting capability t can be changed adaptively to channel state at every codeword block. The decoder allows efficient decoding in both burst mode and continuous mode, and it permits 3-step pipelined processing based on the modified Euclid's algorithm. Each step in decoding is designed to be clocked by a separate clock. Thus, each step can be efficiently pipelined with no help of multiplexing. Also, it makes it possible to employ no additional buffer even when the decoder input and output clocks are different. The adaptive RS decoder over GF(28) having the error-correcting capability of upto 10 has been designed in VHDL, and successfully synthesized in an FPGA chip. It can be used in a wide range of applications because of its versatility.

One of the categories of decoding techniques for DFT codes in erasure channels is the class of iterative algorithms. Iterative algorithms can be considered as kind of alternating mapping methods using the given information in a repetitive way. In this paper, we propose a new iterative method for decoding DFT codes. It will be shown that the proposed method outperforms the well-known methods such as Wiley/Marvasti, and ADPW methods in the decoding of DFT codes in erasure channels.

We propose multibits/sequence-period optical code division multiple access (MS-OCDMA) systems with double optical hardlimiters (DHL) in the presence of APD noise, thermal noise, and channel interference. We apply Reed-Solomon (RS) codes to MS-OCDMA to further improve the error rate performance. We show that the MS-OCDMA receiver with DHL improves the bit error probability of MS-OCDMA systems when the received laser power is large. We also show that the performance of RS coded MS-OCDMA system is better than that of on-off keying OCDMA (OOK-OCDMA) system at the same bit rate and at the same chip rate, respectively.

Based on a modified step-by-step decoding procedure, a high-speed pipelined Reed-Solomon decoder is presented. The decoder requires only the delay time of three 2-input XOR gates for decoding each coded symbol. The decoder can be operated in a bit rate of Gbits/sec order and thus suitable for the very high speed data transmission systems.

In this paper, we introduce a lower bound for the generalized Hamming weights, which is applicable to arbitrary linear code, in terms of the notion of well-behaving. We also show that any [n,k] linear code C over a finite field F is the t-th rank MDS for t such that g(C)+1 t k where g(C) is easily calculated from the basis of Fn so chosen that whose first n-k elements generate C. Finally, we apply our result to Reed-Solomon, Reed-Muller and algebraic geometry codes on Cab, and determine g(C) for each code.

A simple near-orthogonal code is used as frequency-hopping patterns for the frequency-hopped multiple access systems. Extended RS code is used as channel coding to deplete the effects of hits from simultaneous users. Packet error probability and channel throughput for the system utilizing the near-orthogonal code are evaluated and compared to the corresponding values obtained from the system utilizing random patterns. Results show that the former can provide substantial improvement over the latter. In our illustrated examples, we also show that under the constraint of packet error probability PE 10-2, the maximum achievable number of users with most (n,k) RS codes of interest is less than the number of distinct codewords in the near-orthogonal code. Thus, the number of codewords of the near-orthogonal code is large enough to support the practical application.

Generalized minimum-distance (GMD) decoding is well-known as a soft decision decoding technique for such linear block codes as BCH and RS codes. The GMD decoding algorithm generates a set of candidate codewords and selects as a decoded codeword that candidate with the smallest reliable distance. In this paper, for a GMD decoder of RS and BCH codes, we present a new sufficient condition for the decoded codeword to be optimal, and we show that this sufficient condition is less stringent than the one presented by Taipale and Pursely.

The performance of coded multi-pulse pulse position modulation (MPPM) consisting of m slots and 2 pulses, denoted as (m, 2) MPPM, with imperfect slot synchronization is analyzed. Convolutional codes and Reed-Solomon (RS) codes are employed for (m, 2) MPPM, and the bit error probability of coded (m, 2) MPPM in the presence of the timing offset is derived. In each coded (m, 2) MPPM, we compare the performance of some different code rate systems. Moreover, we compare the performance of both systems at the same information bit rate. It is shown that in both coded systems, the performance of code rate-1/2 coded (m, 2) MPPM is the best when the timing offset is small. Wheji the timing offset is somewhat large, however, uncoded (m, 2) MPPM is shown to perform better than coded (m, 2) MPPM. Further, convolutional coded (m, 2) MPPM with the constraint length k7 is shown to perform better than RS coded (m, 2) MPPM for the same code rate.