Some asymptotically optimal burst-correcting codes and their relation to single-error-correcting Reed-Solomon codes

A class of asymptotically optimal burst-correcting codes that are closely related to the Fire codes is defined. The codes are quasi-cyclic as defined by Townsend and Weldon. However, decoding can be accomplished with a very simple algorithm similar to that used for cyclic burst-correcting codes. It is shown that these codes are equivalent to certain Reed-Solomon codes. From this it follows that such Reed-Solomon codes can be easily encoded and decoded without any computations in an extension field.     

A class of high-speed decodable burst-correcting codes (Corresp.)

A class of high-speed decodable burst-correcting codes is presented. This class of codes is obtained by modifying burst-correcting convolutional codes into block codes and does not require any cyclic shifts in the decoding process. With the appropriate choices of parameters, the codes can approximate minimum-redundancy codes. The high-speed decodability is expected to make these codes suitable for application to computer systems.     

A note on Gilbert burst-correcting codes

The codes discussed here, due to E. N. Gilbert, are capable of detecting and correcting certain bursts, i.e., errors which occur in clusters. In particular, it is shown that Gilbert's burst-correcting cedes have capabilities beyond those previously recognized. D. A. Huffman's graph-code formulation is used, which greatly facilitates the appreciation of the results. A cyclic code formulation is also given, and certain Gilbert codes are seen to be Fire codes.     

Burst-trapping techniques for a compound channel

An adaptive decoding technique called burst trapping is presented to correct both random and burst errors. Two decoding algorithms are used, one for random errors, and the other for bursts. The former is based on a conventional correction technique, the latter utilizes an encoding procedure for which each information digit appears twice in the data stream, first unchanged, and second combined with (addition modulo2) a check digit of a widely separated later block. Whenever the number of errors within a code block are detected to be too large to correct with the random-error-correcting algorithm, the burst-correcting algorithm corrects these errors by recovering the information from later blocks where it appears in combination with check digits. It is shown that the scheme requires very limited guard space and has limited error propagation. Furthermore, the storage requirement is even smaller than the guard space. This is the only known coding system that has this desirable feature. Results of simulation of such codes over telephone channels indicate that the performance of such codes, when compared with interleaved block codes, offers better results at significantly lower cost.     

A family of efficient burst-correcting array codes

A family of binary burst correcting array codes that are defined as follows is discussed: consider an n₁×n n₂ array with n₁=4u+ν+2 and n₂=6u+2ν+5, u⩾1, ν⩾0, ν≠1 where each row and column has even parity. The bits are read diagonally starting from the upper-left corner. The columns are viewed cyclically, i.e. the array is a cylinder. If one diagonal has been read out, one proceeds with the second diagonal preceding it. It is proven that the codes of this type can correct any burst of length up to n₁. The burst-correcting efficiency of this family tends to 4/5 as u→∞. As a comparison, the burst-correcting efficiency of other families of array codes tends to 2/3; the same is true for Fire codes. A simple decoding algorithm for the codes is also presented     

Implementation of burst-correcting convolutional codes

A general procedure is formulated for decoding any convolutional code with decoding delayNblocks that corrects all bursts confined toror fewer consecutive blocks followed by a guard space of at leastN-1consecutive error-free blocks. It is shown that all such codes can be converted to a form called "doubly systematic" which simplifies the decoding circuitry. The decoding procedure can then be implemented with a circuit of the same order of complexity as a parity-checking circuit for a block-linear code. A block diagram of a complete decoder is given for an optimal burst-correcting code. It is further shown that error propagation after a decoding mistake is always terminated by the occurrence of a double guard space of error-free blocks.     

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.     

基于改进2-DGRS码的QC-LDPC码高效构造

利用GRS(generalized reed-solomon)码的生成多项式提出了基于改进的2-D GRS(two-dimensional GRS)码设计和构造QC-LDPC(quasi-cyclic low density parity-check)码的方法,使所构造的码具有较好的译码性能。同时在码的构造过程中,考虑到了准双对角线结构和合适的度分布。不同码率的LDPC码用于和新设计的QC-LDPC码进行测试和比较。实验结果表明,所提出的码构造方法可加快LDPC码校验矩阵的构造,同时基于所提出方法构造的QC-LDPC码可提高译码性能,并降低编码复杂度。     

DC-free error-control block codes

DC-free codes and error-control (EC) codes are widely used in digital transmission and storage systems. To improve system performance in terms of code rate, bit-error rate (BER), and low-frequency suppression, and to provide a flexible tradeoff between these parameters, this paper introduces a new class of codes with both dc-control and EC capability. The new codes integrate dc-free encoding and EC encoding, and are decoded by first applying standard EC decoding techniques prior to dc-free decoding, thereby avoiding the drawbacks that arise when dc-free decoding precedes EC decoding. The dc-free code property is introduced into standard EC codes through multimode coding techniques, at the cost of minor loss in BER performance on the additive white Gaussian noise channel, and some increase in implementation complexity, particularly at the encoder. This paper demonstrates that a wide variety of EC block codes can be integrated into this dc-free coding structure, including binary cyclic codes, binary primitive BCH codes, Reed-Solomon codes, Reed-Muller codes, and some capacity-approaching EC block codes, such as low-density parity-check codes and product codes with iterative decoding. Performance of the new dc-free EC block codes is presented.     

状态网格图的LDPC码在OFDM系统中的应用

介绍了LDPC码的编译码技术,提出了一种新颖的2状态网格图译码算法,研究了该码在OFDM系统中的性能,对不同的译码算法进行了比较.仿真结果表明,LDPC码在OFDM基带传输系统中用2状态网格图对其译码能够更好的对错误码进行纠错,提高码字性能,信息传输速率会大大提高.     

Rotationally invariant nonlinear trellis codes for two-dimensionalmodulation

A general parity-check equation is presented that defines rotationally invariant trellis codes of rate k/(k+1) for two-dimensional signal sets. This parity-check equation is used to find rate k/(k+1) codes for 4PSK, 8PSK, 16PSK, and QAM signal sets by systematic code searches. The MPSK codes exhibit smaller free Euclidean distances than nonrotationally invariant linear codes with the same number of states. However, since the nonlinear codes have a smaller number of nearest neighbors, their performance at moderate signal to noise ratios is close to that of the best linear codes. The rotationally invariant QAM codes with 8, 32, 64, and 256 states achieve the same free Euclidean distance as the best linear codes. Transparency of user information under phase rotations is accomplished either by conventional differential encoding and decoding, or by integrating this function directly into the code trellis     

On type-B1 burst-error-correcting convolutional codes

Two new classes of type-B1 burst-error-correcting convolutional codes are introduced. One of them requires a shorter length of guard space and a smaller number of shift register stages than optimum type-B2 codes used for type-B1 burst correction. Another class of codes improves the required number of shift register stages considerably when the correctable burst length is very large. In addition, these codes require a very short length of additional guard space to restore the decoder to correct operation after a decoding failure. Both classes of codes are derived in a straightforward manner and their implementations are also very simple. Thus, we can avoid type-B2 code procedures to correct type-B1 bursts. The codes derived here result in the more efficient and simply implemented type-B1 burst-correcting convolutional codes.     

Data structure of Huffman codes and its application to efficient encoding and decoding (Corresp.)

The data structure of Huffman codes and its application to efficient encoding and decoding of Huffman codes are studied in detail. The tree structure is presented by a two-dimensional array which can be applied for the decoding of Huffman codes as a state transition table of the finite-state decoding automaton. Inversion produces a one-dimensional state transition table of the semiautonomous finite-state sequential machine which can be used as a Huffman encoder with a push-down stack. The encoding and decoding procedures are simple and efficient. It is not only possible to implement by simple hardware but is also applicable to software implementation.     

量子突发纠错乘积码的构造

量子突发纠错码是以CSS量子码的纠错原理和构造技术为基础,在量子计算和量子通信中有着十分重要的作用。首次利用GF(q)上的任意线性码C1=〖JB([〗n,k1,d1q和满足对偶包含关系的BCH码C2=〖JB([〗n,k2,d2q,来构造乘积码C1C2和(C1C2)⊥,当满足n2>2k1k2时,在CSS构造的基础上便可构造参数为[[n2,n2－n]]的量子突发纠错乘积码,并给出其突发纠错能力。     

一种伪随机LDPC码构造方法与性能研究

低密度校验(Low-Density Parity-Check)码作为迄今为止性能最好的纠错码之一, 目前已经被许多数字通信标准广泛采用。伪随机低密度校验码(Pseudo-Random)是 LDPC 码的一个子类,已被应用于空间通信和无线网络技术。本文给出了一种基于有限域的伪随机 LDPC 码构造方法,并采用理论分析和仿真结果分析相结合的方法,对伪随机 LDPC 码的构造和编译码方法进行了研究,并给出了实现中适合的译码算法及量化方案。     

M-ary phase coding for the noncoherent AWGN channel

This work considers coded M-ary phase-shift keying (MPSK) schemes with noncoherent detection. A class of block codes called module-phase codes is described. The algebraic framework used for describing these codes relies on elements from module theory which are discussed along with a method for constructing such codes for noncoherent detection. It is shown that differential encoding may be viewed as a specific code from a particular class of module-phase codes. Two classes of codes that achieve significant coding gain with respect to coherent detection of uncoded MPSK are presented. In the first class of module-phase codes, the coding gain is achieved at the expense of bandwidth expansion. In the second class, however, the coding gain is achieved at the expense of signal constellation expansion without expanding bandwidth. Finally, an integrated demodulation/decoding technique based on a modification of information set decoding is presented. It Is shown that this reduced-complexity, suboptimal decoding strategy performs nearly as well as maximum-likelihood decoding     

On Diamond codes

A Diamond code is an error-correcting code obtained from two component codes. As in a product code, any symbol in a word of a Diamond code is checked by both component codes. However, the “code directions” for the component codes have been selected to minimize the memory that is required between successive decoding stages for the component codes. Diamond codes combine the error correcting power of a product code with the reduced memory requirements of the cross interleaved Reed-Solomon code (CIRC), applied in the compact disk system. We discuss encoding, decoding, and minimum distance properties of Diamond codes. Variations on the Diamond code construction are proposed that result in codes that are suited for use in rewritable block-oriented applications     

Reliability-based code-search algorithms for maximum-likelihooddecoding of block codes

Efficient code-search maximum-likelihood decoding algorithms, based on reliability information, are presented for binary Linear block codes. The codewords examined are obtained via encoding. The information set utilized for encoding comprises the positions of those columns of a generator matrix G of the code which, for a given received sequence, constitute the most reliable basis for the column space of G. Substantially reduced computational complexity of decoding is achieved by exploiting the ordering of the positions within this information set. The search procedures do not require memory; the codeword to be examined is constructed from the previously examined codeword according to a fixed rule. Consequently, the search algorithms are applicable to codes of relatively large size. They are also conveniently modifiable to achieve efficient nearly optimum decoding of particularly large codes     

Concatenated Reed-Muller codes for unequal error protection

We present a new coding scheme that combines the advantages of a product-like concatenation of Reed-Muller codes with so-called iterative “turbo” decoding and provides powerful unequal error protection abilities. It is shown that various levels of error protection can be realized using a sophisticated encoding scheme for Reed-Muller codes. A discussion of this code construction, the resulting distance profile between the different levels and the iterative decoding scheme is given. The results are very promising and impressively confirm the unequal error protection capabilities of the presented coding scheme     

A fast LDPC encoder/decoder for small/medium codes

Although Low-Density Parity-Check (LDPC) codes perform admirably for large block sizes — being mostly resilient to low levels of channel SNR and errors in channel equalization — real time operation and low computational effort require small and medium sized codes, which tend to be affected by these two factors. For these small to medium codes, a method for designing efficient regular codes is presented and a new technique for reducing the dependency of correct channel equalization, without much change in the inner workings or architecture of existing LDPC decoders is proposed. This goal is achieved by an improved intrinsic Log-Likelihood Ratio (LLR) estimator in the LDPC decoder — the ILE-Decoder, which only uses LDPC decoder-side information gathered during standard LDPC decoding. This information is used to improve the channel parameters estimation, thus improving the reliability of the code correction, while reducing the number of required iterations for a successful decoding. Methods for fast encoding and decoding of LDPC codes are presented, highlighting the importance of assuring low encoding/decoding latency with maintaining high throughput. The assumptions and rules that govern the estimation process via subcarrier corrected-bit accounting are presented, and the Bayesian inference estimation process is detailed. This scheme is suitable for application to multicarrier communications, such as OFDM. Simulation results in a PLC-like environment that confirm the good performance of the proposed LDPC coder/decoder are presented.