Zigzag Codes Research Articles

A study on the big data scientific research model and the key mechanism based on blockchain

Abstract In an era of open data sharing, the scientific research field puts forward an urgent need for the value of big data. However, big data still form “data islands,” which seriously affects the level of scientific research and the progress of scientific research. In this regard, this article proposes the research and realization of the big data scientific research model and key mechanism based on blockchain. This article uses the K-means algorithm to cluster scientific research data and reasonably utilizes the decentralization, smart contracts, and non-tampering characteristics of the blockchain to design a distributed data model based on the blockchain. This article proposes that a BIZi network is formed based on a blockchain Interplanetary File System (IPFS) and Zigzag code (blockchain, IPF Sand Zigzag code, BIZi for short) to achieve reliable data connection and through a set of data access control mechanisms and data service customization mechanism to effectively provide data requirements for scientific research. Finally, IPFS network transmission speed performance can better meet the needs of scientific research. The larger the number of file blocks, the higher the fault tolerance rate of the scheme and the better the storage efficiency. In a completely open data-sharing scenario, the fault tolerance rate of Byzantine nodes is extremely high to ensure the stability of the blockchain. The current optimal consensus algorithm fault tolerance rate reaches 49%.

Optimal Rebuilding of Multiple Erasures in MDS Codes

MDS array codes are widely used in storage systems due to their computationally efficient encoding and decoding procedures. An MDS code with $r$ redundancy nodes can correct any $r$ node erasures by accessing all the remaining information in the surviving nodes. However, in practice, $e$ erasures is a more likely failure event, for $1\le e<r$. Hence, a natural question is how much information do we need to access in order to rebuild $e$ storage nodes? We define the rebuilding ratio as the fraction of remaining information accessed during the rebuilding of $e$ erasures. In our previous work we constructed MDS codes, called zigzag codes, that achieve the optimal rebuilding ratio of $1/r$ for the rebuilding of any systematic node when $e=1$, however, all the information needs to be accessed for the rebuilding of the parity node erasure. The (normalized) repair bandwidth is defined as the fraction of information transmitted from the remaining nodes during the rebuilding process. For codes that are not necessarily MDS, Dimakis et al. proposed the regenerating codes framework where any $r$ erasures can be corrected by accessing some of the remaining information, and any $e=1$ erasure can be rebuilt from some subsets of surviving nodes with optimal repair bandwidth. In this work, we study 3 questions on rebuilding of codes: (i) We show a fundamental trade-off between the storage size of the node and the repair bandwidth similar to the regenerating codes framework, and show that zigzag codes achieve the optimal rebuilding ratio of $e/r$ for MDS codes, for any $1\le e\le r$. (ii) We construct systematic codes that achieve optimal rebuilding ratio of $1/r$, for any systematic or parity node erasure. (iii) We present error correction algorithms for zigzag codes, and in particular demonstrate how these codes can be corrected beyond their minimum Hamming distances.

Optimal Exact Repair Strategy for the Parity Nodes of the $(k+2,k)$ Zigzag Code

In this paper, we reinterpret the (k+2, k) zigzag code in coding matrix and then propose an optimal exact repair strategy for its parity nodes, whose repair disk I/O approaches a lower bound derived in this paper.

A New Construction of (k+2,k) Minimal Storage Regenerating Code Over &lt;b&gt;F&lt;/b&gt;&lt;sub&gt;3&lt;/sub&gt; with Optimal Access Property for All Nodes

The optimal access property is highly desirable for distributed storage systems. In this letter, based on the $(k+2,k)$ zigzag code, we propose a new $(k+2,k)$ minimal storage regenerating code. The new code inherits the key properties of the zigzag code, such as the optimal access property for all systematic nodes and small alphabet F3. Furthermore, the new code possesses the optimal access property for the two parity nodes, at the expense of doubled storage per node of the original zigzag code. While possessing the same properties (optimal access property for all nodes and small alphabet F3), the new code requires only half of the storage per node compared with that of the modified zigzag code.

Performance of parallel concatenated zigzag codes for different segment length

General turbo codes have an excellent bit-error-rate (BER) performance at low signal-to-noise ratios (SNR) in additive white Gaussian noise (AWGN) channels. However, the standard turbo decoding algorithm is highly complex. To reduce its decoding computational complexity, zigzag codes are currently used. A zigzag code is a family of error-correcting codes similar to the single parity check codes. It is an important issue, how the segment length called J and the interleaver can affect concatenated zigzag code performance. In this study, the segment size and interleaver structure are investigated in order to obtain a high bit-error-rate performance for the concatenated zigzag code with four encoders and three random interleavers. Realized computer simulation results are compared with/without interleaver for various values of J over a noisy channel.

FPGA Implementation of a Modified Turbo Encoder

convolutional codes (TCC) are excellent error correcting codes, however TCC decoding based on A- Posteriori Probability (APP) algorithm is computationally complex and the complexity is not significantly reduced even if puncturing mechanism is used. To overcome the above disadvantage turbo codes need to be concatenated with other coding techniques such that the decoding complexity is significantly reduced and at the same time the Signal to Noise Ratio (SNR) can be as close to the Shannon limit as possible. In this paper one such modification is described, whereby convolutional coding as well as block coding technique of Zig-Zag codes will be used. First the simulation results of the encoder using MATLAB are presented and then the FPGA results using Artix-7 board will be shown.

Zigzag-Coded Modulation for High-Speed Fiber Optical Channels

We propose new coded modulation schemes for ultra-high-speed optical transmission (e.g., 100 Gb/s and above) using component concatenated zigzag codes in combination with high-order digital modulation and coherent detection. In particular, we propose single-level coded modulation and multi-level coded modulation using concatenated zigzag codes. For single-level coded modulation, only one concatenated zigzag code is employed to encode the information bits, and Gray mapping is employed to map the coded bits to channel symbols. For multi-level coded modulation, the mapping bits are divided into several layers, and one component concatenated zigzag code is employed at each layer; the layers are decoded successively, and the decoded layers are used to assist the decoding of the subsequent layers. Furthermore, we propose a rate allocation scheme for the component concatenated zigzag codes and a parameter selection scheme for multi-level coded modulation. Simulation results demonstrate that single-level and multi-level coded systems with zigzag codes show up to 0.5 dB performance gain over structured low-density parity check codes.

Performance Comparison between Turbo Code and Rate-Compatible LDPC Code for Evolved UTRA Downlink OFDM Radio Access

This paper compares the turbo code and rate-compatible low-density parity-check (LDPC) codes based on the block error rate (BLER) performance and decoding complexity in order to clarify which channel coding scheme is most appropriate for the channel coding scheme in the OFDM based Evolved UTRA (E-UTRA) downlink. Simulation results and the decoding complexity analysis show that although the Rate-Compatible/Quasi-Cyclic (RC/QC)-LDPC code employing an offset layered belief propagation (BP) method can reduce the computational complexity by approximately 30% for the channel coding rate of R ≥ 1/2, the required average received signal energy per bit-to-noise power spectrum density ratio (Eb/N0) is degraded by approximately 0.2-0.3dB for R=1/3, 1/2 and 3/4 compared to that for the turbo code. Moreover, the decoding complexity level of the RC/QC-LDPC code with the δ-min algorithm is almost the same or higher than that for the turbo code with a slight degradation in the required received Eb/N0. Although the decoding complexity level of the ZigZag code is lower than that of the turbo code, the code brings about a distinct loss in the required average received Eb/N0 of approximately 0.4dB. Finally, the turbo Single Parity Check (SPC) code improves the BLER performance compared to the ZigZag code, i.e., achieves almost the same BLER performance as that for the turbo code, at the cost of a two-fold increase in the decoding complexity. As a result, we conclude that the turbo code with a contention free interleaver is more promising than the LDPC codes for prioritizing the achievable performance over complexity and as the channel coding scheme for the shared data channel in the E-UTRA.

Joint linear interleaver design for concatenated zigzag codes

The design of a class of well-structured low-density parity-check (LDPC) codes, namely linear interleaver based concatenated zigzag (LICZ) codes, is investigated. With summary distances as the design metric, short LICZ codes with large minimum distances can be constructed. Moreover, an efficient cycle-based method is proposed to compute the minimum distances of LICZ codes. Simulation results show that LICZ codes outperform both CZ codes with random interleavers and LDPC codes by the progressive edge growth algorithm.

On the Performance of Short Forward Error-Correcting Codes

This letter investigates the performance of short forward error-correcting (FEC) codes. Reed-Solomon (RS) codes and concatenated zigzag codes are chosen as representatives of classical algebraic codes and modern simple iteratively decodable codes, respectively. Additionally, random binary linear codes are used as a baseline reference. Our main results (demonstrated by simulations and ensemble distance spectrum analysis) are as follows: 1) Short RS codes are as good as random binary linear codes; 2) Carefully designed short low-density parity-check (LDPC) codes are almost as good as random binary linear codes; 3) Low complexity belief propagation decoders incur considerable performance loss at short coding lengths. Thus, future work could focus on developing low-complexity (near) optimal decoders for RS codes and/or LDPC codes.

On the asymptotic input-output weight distributions of some accumulate-based codes

In this letter, we show how to compute the asymptotic growth rate of input-output weight enumerator (AGR-IOWE) for some accumulate-based codes by using the sharp tools already developed. Numerical results on the AGR-IOWE for irregular repeat-accumulate (IRA) codes, systematic regular RA (SRA) codes, and concatenated zigzag codes are reported. It is observed that the SRA code has the same AGR-IOWE as a comparable concatenated zigzag code. For both SRA and concatenated zigzag codes, if keeping the code rate fixed, the increase of the grouping factor for the component punctured accumulate code may result in better asymptotic performance under maximum-likelihood decoding, but often worse performance under iterative sum-product decoding.

A data link control protocol for broadband wireless networks with adaptive coding rate

AbstractOne of the challenges of broadband wireless multimedia networks is to design a wireless data link control (W‐DLC) protocol which provides reliable data delivery over the noisy wireless link in an efficient manner. This paper presents our proposal of such a W‐DLC protocol, based on a new coding scheme called multi‐dimensional concatenated zigzag code. Both real‐time and non‐real‐time traffic can be efficiently supported by adopting different error control mechanisms. A distinct feature of the protocol is that the coding rate can be adapted to a wide range of channel conditions. Hence, our protocol utilizes the scarce wireless channel resources more efficiently. Copyright © 2005 John Wiley &amp; Sons, Ltd.

On Concatenated Zigzag Codes and Their Decoding Schemes

In this letter, we show that a concatenated zigzag code can be viewed as a low-density parity-check (LDPC) code. Based on the bipartite graph representation for such a parallel-concatenated code, various sum-product based decoding algorithms are introduced and compared. The results show that the improved versions of sum-product algorithm exhibit better convergence rate while maintaining the essential parallel form.

New near shannon limit codes: Zigzag codes and concatenated zigzag codes

This paper introduces a family of error-correcting codes called zigzag codes. A zigzag code is described by a highly structured zigzag graph. Due to the structural properties of the graph, very low-complexity soft-in, soft-out decoding rules can be implemented. We present a decoding rule, based on the Max-Log-APP(MLA) formulation, which requires a total of only 20 addition-equivalent-operations per information bit per iteration. Simulation of a rate-1/2 concatenated zigzag code with four constituent encoders with interleaver length 65536 yields a bit error rate (BER) and of 10−5 at 0.9 dB and 1.4 dB away from the Shannon limit by optimal (APP) and low-cost sub-optimal (MLA) decoders, respectively.

Zigzag codes and concatenated zigzag codes

This paper introduces a family of error-correcting codes called zigzag codes. A zigzag code is described by a highly structured zigzag graph. Due to the structural properties of the graph, very low-complexity soft-in/soft-out decoding rules can be implemented. We present a decoding rule, based on the Max-Log-APP (MLA) formulation, which requires a total of only 20 addition-equivalent operations per information bit, per iteration. Simulation of a rate-1/2 concatenated zigzag code with four constituent encoders with interleaver length 65 536, yields a bit error rate (BER) of 10/sup -5/ at 0.9 dB and 1.3 dB away from the Shannon limit by optimal (APP) and low-cost suboptimal (MLA) decoders, respectively. A union bound analysis of the bit error probability of the zigzag code is presented. It is shown that the union bounds for these codes can be generated very efficiently. It is also illustrated that, for a fixed interleaver size, the concatenated code has increased code potential as the number of constituent encoders increases. Finally, the analysis shows that zigzag codes with four or more constituent encoders have lower error floors than comparable turbo codes with two constituent encoders.

Adaptive type II hybrid ARQ scheme using zigzagcode

The authors propose a new hybrid type II ARQ scheme based on the simple multidimensional concatenated zigzag code. Owing to the fact that the error-correcting capability of zigzag codes is far better than that of conventional convolutional codes, the throughput of our scheme is better than existing hybrid ARQ schemes.

A more efficient notion of zigzag stability

In [6] a definition of zigzag stability for zigzag submonoids is given. We give here a new definition of zigzag stability, which is simpler and very close to the usual definition of stability for ordinary submonoids. We give a short proof that zigzag stability is decidable in the rational case. Moreover this notion of zigzag stability enables one to obtain an algorithm to decide whether a rational language is a zigzag code. Despite being in exponential time, the complexity of this algorithm is better than that of every other known algorithm [3], [8], [6].

Sur les codes zigzag et leur décidabilité

This paper deals with zigzag factorizations and zigzag codes. The language of “zigzag” over a regular language is represented by constructing a special family of two-way automata. Decidability of zigzag codes, previously shown for the finite languages, is proved here for all regular languages by the analysis of the set of “crossing sequences” produced by a two-way automation in the family. We also obtain that it is decidable whether or not a two-way automation of a certain type is non-ambiguous.