Parallel error correcting codes

We introduce the concept of "parallel error correcting" codes, the error correcting codes for parallel channels. Here, a parallel channel is a set of channels such that the additive error over a finite field occurs in one of its members at time T if the same error occurs in all members at the same time. The set of codewords of a parallel error correcting code has to be a product set, if the messages transmitted are from independent information sources. We present a simple construction of optimal parallel error correcting codes based on ordinary optimal error correcting codes and a construction of optimal linear parallel codes for independent sources based on optimal ordinary linear error correcting codes. The decoding algorithms for these codes are provided as well     

On ternary error correcting line codes

The authors present Markov diagrams and tables with the capacities in bits/symbol for input restricted ternary channels with various restrictions on maximum runlengths, digital sum variation, and transitions between extreme signal levels. They derive Gilbert-type lower bounds on the minimum Hamming and Euclidean distances achievable with ternary line codes of rates lower than the capacity of the corresponding input restricted channel. They present some single-symbol-error-correcting ternary line codes, found by computer search methods     

High-throughput error correcting space-time block codes

This letter investigates using an error correction code (ECC) to construct the space-time block code (STBC). Block ECCs over several Galois fields are considered. The resulting STBCs have significantly higher throughput and better performance than orthogonal STBCs, at the cost of increased decoding complexity.     

Reed-Solomon codes for correcting phased error bursts

A code structure is introduced that represents a Reed-Solomon (RS) code in two-dimensional format. Based on this structure, a novel approach to multiple error burst correction using RS codes is proposed. For a model of phased error bursts, where each burst can affect one of the columns in a two-dimensional transmitted word, it is shown that the bursts can be corrected using a known multisequence shift-register synthesis algorithm. It is further shown that the resulting codes posses nearly optimal burst correction capability, under certain probability of decoding failure. Finally, low-complexity systematic encoding and syndrome computation algorithms for these codes are discussed. The proposed scheme may also find use in decoding of different coding schemes based on RS codes, such as product or concatenated codes.     

Construction of error correcting codes by interpolation

A generalized and unified method of interpolation and transformation is used to generate all known maximal distance codes and important subfield subcodes. Some powerful tools for the analysis and synthesis of maximal distance codes are presented, as well as a generalization of the Mattson-Solomon polynomial and Lagrange and Fourier transforms to more general functions. In certain cases new codes can be obtained by differentiating a kernel function. Some further generalizations of Srivastava codes are constructed. A general method of decoding is given which can be used for complete decoding of ali coset leaders.     

DC- and Nyquist-free error correcting convolutional codes

A class of DC- and Nyquist-free codes with error correcting capability is proposed. These codes are minimum-bandwidth codes because they have spectral null at Nyquist frequency. They also have a good error correcting capability because they are constructed from binary convolutional codes     

Block error correcting codes using finite-field wavelet transforms

This paper extends the popular wavelet framework for signal representation to error control coding. The primary goal of the paper is to use cyclic finite-field wavelets and filter banks to study arbitrary-rate L-circulant codes. It is shown that the wavelet representation leads to an efficient implementation of the block code encoder and the syndrome generator. A formulation is then given for constructing maximum-distance separable (MDS) wavelet codes using frequency-domain constraints. This paper also studies the possibility of finding a wavelet code whose tail-biting trellis is efficient for soft-decision decoding. The wavelet method may provide an easy way to look for such codes.     

The burst error correcting power of m-sequence codes

Linear maximum length sequence codes are shown to be asymptotically efficient burst error correcting codes. These codes are essentially single burst correctors and for binary alphabets have the following parameters : block length, n = 2^m ? 1 ; number of information digits, k = m; and burst error correcting capability, b≥2^m?1 ? m, m ≥ 2 ; where m is an integer. A generalization to multilevel alphabets is also presented.     

纠错码性能仿真中的误码率估计

在纠错码性能仿真中,一般是给定信噪比后,由实测的错误信息比特数与发达信息比特总数相比来估计误码率的,这种方法在大码长或者极低误码率情况下。可信程度不高。本文提出用仿真数据计算后验概率分布参数的方法来直接估计纠错误码率及其置信区间,给出了误码率的上界,由少量测试数据仍然可以获得比较可靠的误码率估计,由于这种后验估计考虑了信道特性,所以适用于任意广义信道模型。     

N-modular redundancy using the theory of error correcting codes

Using redundancy is basic to fault-tolerant computing. N-modular redundancy (NMR) is in some ways analogous to the use of a repetition code where an information symbol is replicated as parity symbols in a codeword. Linear error-correcting codes (ECC) use linear combinations of information symbols as parity symbols to generate syndromes for error patterns. In this paper, ECC theory has been applied to derive redundant circuits that tolerate faults in both the modules and checkers. Circuits using comparators for diagnosis are derived with a non-graph-theoretic approach. Coding theoretic principles are applied directly to NMR, so that extensive diagnosis of the fault-tolerant system is achieved.     

Single error correcting and multiple unidirectional error detecting cyclic an arithmetic codes

We present cyclic AN arithmetic codes capable of single error correction and multiple unidirectional error detection. These codes can be used throughout a fault-tolerant computer, and they eliminate the need for encoding/decoding circuits and code translation circuits. We use criteria for the determination of the unidirectional error detection capability for a given AN code, and we present a new error correction/detection scheme.     

Application of convolutional error correcting codes in ultrawideband M-ary PPM signaling

The interference issues related to ultrawideband (UWB) radio pose tight restrictions on the maximum data rate of UWB radio telecommunication systems. A possible solution is to reduce the required signal to interference ratio (SIR) that gives satisfactory performance to the UWB system. In this letter, we propose coded M-ary UWB radio communication systems. Two classes of convolutional codes, namely, low-rate superorthogonal codes and high-rate punctured codes are considered for this purpose. Simulation results on the bit error rate of the proposed system indicates that the system is capable to work in lower SIR's and therefore supports higher data transmission rates in a real interference environment compared to the previously proposed UWB communication systems.     

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

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.     

LDPC码的高性能信道纠错码理论

自从上世纪60年代信道编码理论提出至今,就如何构建一个模拟通信信道容量问题成为了各国学者努力研究的主要课题,并且形成了通信能力体现的一个主要问题。半个世纪以来,各国科学通过构造以两条主线为基础提出的香农条件算法。提出的LDPC码的优异性能吸引人们不断探讨它在各个领域的应用：在宽带接入网中的应用方面,基于二元LDPC的多电平编码计算方法,通过仿真实验证明了该方法在信道传输过程中的高性能算法。在进行算法记录模型方面,用磁记录中的高吞吐率和繁杂LDPC译码方案结合进行。     

Some classes of burst asymmetric or undirectional error correcting codes

Classes of systematic codes correcting burst asymmetric or unidirectional errors are proposed. These codes have less check bits than ordinary burst error correcting codes. Decoding algorithms for the proposed codes are also presented. Encoding and decoding of the codes is very easy.<>     

Compressed error and erasure correcting codes via rank‐metric codes in random network coding

The error control of random network coding has recently received a lot of attention because its solution can increase robustness and reliability of data transmission. To achieve this, additional overhead is needed for error correction. In this paper, we design a compressed error and erasure correcting scheme to decrease the additional overhead of error correction. This scheme reduces the computation overhead dramatically by employing an efficient algorithm to detect and delete linearly dependent received packets in the destination node. It also simplifies the hardware operations when the scheme reduces the received matrix Y to form E_k(Y ) instead of E(Y ) in the decoding process. If at most r original packets get combined in k packets of one batch, the payload of one packet can increase from M ? k to M ? O(rlog _qk) for the application of compressed code, where M is the packet length. In particular, the decoding complexity of compressed code is O(rm) operations in an extension field , which does not enhance the overall decoding complexity of the system. Finally, we also compare our scheme's performance with existing works. The numerical results and analyses illustrate the security and performance of our scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

[[2n,n]] Additive quantum error correcting codes

The self-orthogonal condition is analyzed with respect to symplectic inner product for the binary code that generated by [B1 I B2 B3],where Bi are the binary n ×n matrices,I is an identity matrix.By the use of the binary codes that generated by [B1 I B2 B2B1^T],asymptotic good[[2n ,n ]]additive quantum codes are obtained.     

STM-64 linearly repeatered optical transmission experiment usingforward error correcting codes

STM-64 (10 Gbit/s) nonregenerative transmission experiments using forward error correcting codes have been conducted for the first time. Bit errors caused by dispersion in the transmission line and optical noise of Er-doped fibre amplifiers were successfully corrected by using (18880, 18865) and (2370, 2358) shortened Hamming codes