Extended Hamming Codes Research Articles

Implementation of Coding Theory with The Extended Hamming Code in Steganography

Implementation of Coding Theory with The Extended Hamming Code in Steganography

Jacobi polynomials and harmonic weight enumerators of the first-order Reed–Muller codes and the extended Hamming codes

Jacobi polynomials and harmonic weight enumerators of the first-order Reed–Muller codes and the extended Hamming codes

RSIS: A Secure and Reliable Secret Image Sharing System Based on Extended Hamming Codes in Industrial Internet of Things

In the industrial Internet of Things (IoT), large amounts of image data are collected, stored, and transmitted by low-power cooperative sensors and heterogeneous devices, such as monitoring pictures taken by surveillance cameras in intelligent transportation. Thus, the increasing image data would be outsourced and shared between different distributed servers. Recently, some secret image sharing with authentication (SISA) schemes were proposed, which can provide image sharing in the distributed IoT system. However, if the IoT data are tampered with by malicious attackers, the error cannot be corrected, which causes unreliability and data loss. Therefore, in order to solve the above issues, we propose a secure and reliable secret image-sharing system based on extended Hamming codes (RSIS) in IoT. In RSIS, a novel distributed IoT architecture-based secret sharing is proposed. A secret image is shared into a series of stego-images. These stego-images can be distributed among multiple edge servers in IoT, which is not easy to attract attentions of attackers. In RSIS, check bits generated by authentication bits and hamming code are embedded into the stego-images to identify the tampered places with a high probability. Theoretical analysis and experiments prove that the proposed scheme achieves a high authentication capability, and the bit errors of data can be accurately detected and corrected. Generally, the scheme is effective and practical.

Design of Extended Hamming Code Technique Encryption for Audio Signals by Double Code Error Prediction

Noise can scramble a message that is sent. This is true for both voicemails and digital communications transmitted to and from computer systems. During transmission, mistakes tend to happen. Computer memory is the most commonplace to use Hamming code error correction. With extra parity/redundancy bits added to Hamming code, single-bit errors may be detected and corrected. Short-distance data transmissions often make use of Hamming coding. The redundancy bits are interspersed and evacuated subsequently when scaling it for longer data lengths. The new hamming code approach may be quickly and easily adapted to any situation. As a result, it's ideal for sending large data bitstreams since the overhead bits per data bit ratio is much lower. The investigation in this article is extended Hamming codes for product codes. The proposal particularly emphasises on how well it functions with low error rate, which is critical for multimedia wireless applications. It provides a foundation and a comprehensive set of methods for quantitatively evaluating this performance without the need of time-consuming simulations. It provides fresh theoretical findings on the well-known approximation, where the bit error rate roughly equal to the frame error rate times the minimal distance to the codeword length ratio. Moreover, the analytical method is applied to actual design considerations such as shorter and punctured codes along with the payload and redundancy bits calculation. Using the extended identity equation on the dual codes, decoding can be done at the first instance. The achievement of 43.48% redundancy bits is obtained during the testing process which is a huge proportion reduced in this research work.

Weighted Posets and Digraphs Admitting the Extended Hamming Code to be a Perfect Code

Recently, Etzion et al. introduced metrics on ${\mathbb{F}}_{2}^{n}$ based on directed graphs on $n$ vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs, which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between the weighted poset metrics and the directed graph-based metrics. In the next, we classify weighted posets on a set with eight elements and directed graphs on eight vertices, which admit the extended Hamming code $\tilde {\mathcal {H }}_{3}$ to be a two-perfect code. We also construct some families of such structures for any $k \geq 3$ , which can be viewed as generalizations of some results presented by Etzion et al. and Hyun and Kim. Those families enable us to construct packing or covering codes of radius 2 under certain maps.

An MCS with Iterative Multi-dimensional Hamming Product Codes

Product codes are preferred in high data rate wireless communication systems to achieve good error performance. The combination of Block Product Code (BPC), Multi-Input Multi-Output (MIMO) antenna systems with Space–Time Block Code (STBC), and Orthogonal Frequency Division Multiplexing (OFDM) yields better error recovery. In this paper, two novel iterative hard decision decoding methods for multi-dimensional Hamming BPC are presented. The first method uses a pattern based approach to enhance detection and correction of double errors in two dimensional Extended Hamming Code (2DEHC); the second method is a perturbation based decoder for three dimensional Normal Hamming Code (3DNHC). The base data rate used in this study is 100 Mbps. From the simulation results, which show both the methods to achieve significant coding gain, we derive a Modulation and Coding Scheme (MCS). The MCS shows that the 3D decoder readily satisfies the requirements for heterogeneous network systems as it spans a wide range in EbNo and Effective Bits per Symbol (EBS) with fine steps. Such an MCS is also relevant and suitable for digital video broadcasting and advanced satellite communication technologies.

Improving Error Correction Codes for Multiple-Cell Upsets in Space Applications

Currently, faults suffered by SRAM memory systems have increased due to the aggressive CMOS integration density. Thus, the probability of occurrence of single-cell upsets (SCUs) or multiple-cell upsets (MCUs) augments. One of the main causes of MCUs in space applications is cosmic radiation. A common solution is the use of error correction codes (ECCs). Nevertheless, when using ECCs in space applications, they must achieve a good balance between error coverage and redundancy, and their encoding/decoding circuits must be efficient in terms of area, power, and delay. Different codes have been proposed to tolerate MCUs. For instance, Matrix codes use Hamming codes and parity checks in a bi-dimensional layout to correct and detect some patterns of MCUs. Recently presented, column–line–code (CLC) has been designed to tolerate MCUs in space applications. CLC is a modified Matrix code, based on extended Hamming codes and parity checks. Nevertheless, a common property of these codes is the high redundancy introduced. In this paper, we present a series of new low-redundant ECCs able to correct MCUs with reduced area, power, and delay overheads. Also, these new codes maintain, or even improve, memory error coverage with respect to Matrix and CLC codes.

Cascaded Channel Model, Analysis, and Hybrid Decoding for Spin-Torque Transfer Magnetic Random Access Memory

Spin-torque transfer magnetic random access memory (STT-MRAM) is a promising non-volatile memory technology widely considered to replace dynamic random access memory (DRAM). However, there still exist critical technical challenges to be tackled. For example, process variation and thermal fluctuation may lead to both write errors and read errors, severely affecting the reliability of the memory array. In this paper, we first propose a novel cascaded channel model for STT-MRAM that facilitates fast error rate simulations and more importantly the theoretical design and analysis of memory sensing and channel coding schemes. We analyze the raw bit error rate and probabilities of dominant error events, and derive the maximum likelihood decision criterion and the log-likelihood ratio of the cascaded channel. Based on these works, we further propose a two-stage hybrid decoding algorithm for extended Hamming codes for STT-MRAM. Simulation results demonstrate the effectiveness of the proposed hybrid decoding algorithm for improving the reliability of STT-MRAM in the presence of both write errors and read errors and supporting its potential applications, such as replacing DRAM.

Accelerating Extended Hamming Code Decoders on Graphic Processing Units for High Speed Communication

Accelerating Extended Hamming Code Decoders on Graphic Processing Units for High Speed Communication

New families of completely regular codes and their corresponding distance regular coset graphs

In this paper three new infinite families of linear binary completely regular codes are constructed. They have covering radius ? = 3 and 4, and are halves of binary Hamming and binary extended Hamming codes of length n = 2 m ?1 and 2 m , where m is even. There are also shown some combinatorial (binomial) identities which are new, to our knowledge.These completely regular codes induce, in the usual way, i.e., as coset graphs, three infinite families of distance-regular graphs of diameter three and four. This description of such graphs is new.

Stopping Set Distributions of Some Reed–Muller Codes

Stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> be a binary [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ] linear code with parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> , where the rows of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> may be dependent. A stopping set <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> with parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> is a subset of column indices of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> such that the restriction of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> does not contain a row of weight one. The stopping set distribution { <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ti</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> )} <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> =0 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> enumerates the number of stopping sets with size <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> with parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> . Note that stopping sets and stopping set distribution are related to the parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> . Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> be the parity-check matrix of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> which is formed by all the nonzero codewords of its dual code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> . A parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> is called BEC-optimal if <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ti</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> )= <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ti</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> ), <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> =0,1,..., <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> has the smallest number of rows. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes, and the extended Hamming codes, which are some Reed-Muller codes or their shortening or puncturing versions.

Higher weights for codes over rings

A generalized definition of higher weights for codes over finite chain rings and principal ideal rings and bounds on the minimum higher weights in this setting are given. Using this we generalize the definition for higher MDS and MDR codes. Computationally, the higher weight enumerator of lifted Hamming and Simplex codes over $${\mathbb{Z}_4}$$, the minimum higher weights for the lifted code of the binary [8,4,4] self-dual extended Hamming code, the lifted code of the ternary [12,6,6] self-dual Golay code and the lifted code of the binary [24,12,8] self-dual Golay code are given. Joint weight enumerators are used to produce MacWilliams relations for specific higher weight enumerators.

Design of Memories with Concurrent Error Detection and Correction by Nonlinear SEC-DED Codes

In this paper we propose memory protection architectures based on nonlinear single-error-correcting, double-error-detecting (SEC-DED) codes. Linear SEC-DED codes widely used for design of reliable memories cannot detect and can miscorrect lots of errors with large Hamming weights. This may be a serious disadvantage for many modern technologies when error distributions are hard to estimate and multi-bit errors are highly probable. The proposed protection architectures have fewer undetectable errors and fewer errors that are miscorrected by all codewords than architectures based on linear codes with the same dimension at the cost of a small increase in the latency penalty, the area overhead and the power consumption. The nonlinear SEC-DED codes are generalized from the existing perfect nonlinear codes (Vasil'ev codes, Probl Kibern 8:375---378, 1962; Phelps codes, SIAM J Algebr Discrete Methods 4:398---403, 1983; and the codes based on one switching constructions, Etzion and Vardy, IEEE Trans Inf Theory 40:754---763, 1994). We present the error correcting algorithms, investigate and compare the error detection and correction capabilities of the proposed nonlinear SEC-DED codes to linear extended Hamming codes and show that replacing linear extended Hamming codes by the proposed nonlinear SEC-DED codes results in a drastic improvement in the reliability of the memory systems in the case of repeating errors or high multi-bit error rate. The proposed approach can be applied to RAM, ROM, FLASH and disk memories.

Hypercube 1-factorizations from extended Hamming codes

Abstract Based on the extended Hamming code, for n = 2 r we construct a proper edge-coloring of the n-dimensional hypercube Qn with n colors (i.e. 1-factorization) such that every edge has exactly one parallel edge from each color class except its own. Two edges are said to be parallel if they are opposite on a common 4-cycle. We study properties of this coloring and applications for some Turan-type problems.

Error-block codes and poset metrics

Let $P = (${$1, 2,\ldots,n$}$,$≤$)$ be a poset, let $V_1, V_2,\ldots, V_n$ be a family of finite-dimensional spaces over a finite field $\mathbb F_q$ and let <br> $ V = V_1 \oplus V_2 \oplus\ldots \oplus V_n.$ <br> In this paper we endow $V$ with a poset metric such that the $P$-weight is constant on the non-null vectors of a component $V_i$, extending both the poset metric introduced by Brualdi et al. and the metric for linear error-block codes introduced by Feng et al.. We classify all poset block structures which admit the extended binary Hamming code $[8; 4; 4]$ to be a one-perfect poset block code, and present poset block structures that turn other extended Hamming codes and the extended Golay code $[24; 12; 8]$ into perfect codes. We also give a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we obtain the group of linear isometries of the error-block metric spaces.

Design of High-Rate Serially Concatenated Codes with Low Error Floor

In this paper we look at the serial concatenation of short linear block codes with a rate-1 recursive convolutional encoder, with a goal of designing high-rate codes with low error floor. We observe that under turbo-style decoding the error floor of the concatenated codes with extended Hamming codes is due to detectable errors in many cases. An interleaver design addressing this is proposed in this paper and its effectiveness is verified numerically. We next examine the use of extended BCH codes of larger minimum distance, resulting in an improved weight spectrum of the overall code. Reduced complexity list decoding is used to decode the BCH codes in order to obtain low decoding complexity for a negligible loss in performance.

Heuristic construction of high-rate linear block codes

We propose a new heuristic construction technique that generates all the even codes of length greater than 8 with full rank property for Hamming distance-4. The codes generated by the proposed method include the extended Hamming codes and the Reed Muller codes of distance-4. The way of obtaining the codes with higher minimum distance using the proposed method is also described.

High-rate serially concatenated coding with extended hamming codes

This letter shows that concatenation of extended Hamming codes with rate-1 recursive convolutional encoder and iterative decoding achieves near Shannon limit performance for very high rate coding. These codes can be decoded efficiently, and have large adaptability in code lengths and code rates.

On the Performance and Implementation Issues of Interleaved Single Parity Check Turbo Product Codes

In this paper, both performance and complexity aspects of two-dimensional single parity check turbo product codes (I-SPC-TPC) are investigated. Based on the proposed I-SPC-TPC coding scheme, a parallel decoding structure is developed to increase the decoding throughput with minor performance degradation compared with the serial structure. For both decoding architectures, a new helical interleaver is constructed to further improve the coding gain. In terms of decoding algorithm, the extremely simple Sign-Min decoding is alternatively derived with only three additions needed to compute each bit's extrinsic information. For performance evaluation, (16, 14, 2)2 single parity check turbo product code with code rate 0.766 over AWGN channel using QPSK modulation is considered. The simulation results using Sign-Min decoding show that it can achieve bit-error-rate of 10?5 at signal-to-noise ratio of 3.8 dB with 8 iterations. Compared to the same rate and codeword length turbo product code composed of extended Hamming codes, the considered scheme can achieve similar performance with much less complexity. Important implementation issues such as the finite precision analysis, efficient sorting circuit design and interleaver memory management are also presented.

Extended Hamming Product Codes Analytical Performance Evaluation for Low Error Rate Applications

We study product codes based on extended Hamming codes. We focus on their performance at low error rates, which are important for wireless multimedia applications. We present the basis and a complete set of techniques which allows one to analytically evaluate this performance without resorting to extremely long simulations. We present new theoretical results concerning the popular approximation where the bit error rate is nearly equal to the frame error rate times the ratio of the minimum distance to the codeword length. We prove that: 1) binary codes with a transitive automorphism group satisfy this approximation with equality; and 2) extended Hamming product codes belong to this class. Closed-form expressions for their dominant multiplicity values are derived. Analytical curves are plotted, discussed, and validated by comparison with iterative decoding. This analytical approach is then extended to both shortened and punctured codes, which are important for practical design. The first case is solved by applying the extended MacWilliams identity to the dual codes. For punctured codes, we present a new analytical approach for estimating their average performance using a "random" puncturer.