Binary Golay Code Research Articles

5-DESIGN AND COUNTER 5-DESIGN IN THE BINARY GOLAY CODE OF LENGTH 24

5-DESIGN AND COUNTER 5-DESIGN IN THE BINARY GOLAY CODE OF LENGTH 24

Extended binary Golay codes by a group algebra

Extended binary Golay codes by a group algebra

On the Generalized Concatenated Construction for the Nordstrom–Robinson Code and the Binary Golay Code

On the Generalized Concatenated Construction for the Nordstrom–Robinson Code and the Binary Golay Code

(Re)constructing Code Loops

The Moufang loop named for Richard Parker is a central extension of the extended binary Golay code. It the prototypical example of a general class of nonassociative structures known today as code loops, which have been studied from a number of different algebraic and combinatorial perspectives. This expository article aims to highlight an experimental approach to computing in code loops, by a combination of a small amount of precomputed information and making use of the rich identities that code loops’ twisted cocycles satisfy. As a byproduct we demonstrate that one can reconstruct the multiplication in Parker’s loop from a mere fragment of its twisted cocycle. We also give relatively large subspaces of the Golay code over which Parker’s loop splits as a direct product.

Reversible Data Hiding in Encrypted DICOM Images Using Cyclic Binary Golay (23, 12) Code

In this paper, a novel reversible data hiding method for encrypted images (RDHEI) is proposed. An efficient coding scheme based on cyclic binary Golay (23, 12) code is designed to embed additional data into the least significant bits (LSBs) of the encrypted image. The most significant bits (MSBs) are used to ensure the reversibility of the embedding process. The proposed scheme is lossless, and based on the receiver’s privileges, allows recovery of marked data, original data and embedded data. Furthermore, the scheme can be used with any type of data, however it is best suited to 16-bit DICOM images of monochrome photometric interpretation. A modification to the standard DICOM network model was also introduced, to point out an example application of the proposed RDHEI method, i.e. an anonymized data storage outsourcing. A computer-based analysis has been carried out and simulation results are shown at the end of this paper.

Розширені бінарні коди Голея за груповою алгеброю однієї групи

Extended binary Golay codes are examples of extreme binary self-dual codes of Type II (linear binary self-dual codes with Hamming distance between arbitrary codewords which are multiples of 4 that has the highest possible minimum Hamming distance among such codes with a fixed dimension of codeword space and their length). These codes have been studied for a long time and many different constructions have been established to build these codes. In addition, extended binary Golay codes are easy to obtain from binary Golay codes and vice versa. The latter are perfect codes and together with binary codes they give us all possible parameters of nontrivial binary perfect codes.In the paper the construction of linear binary codes, in particular of binary Golay codes extended by the group algebra $\mathbb{F}_2G$ of finite group $G=(C_6 \times C_2)~\rtimes~C_2$ of order $n=24$ over the field of two elements $\mathbb{F}_2$ has been considered. Extended binary Golay code is defined as any binary linear code, for which the length of the codewords is 24, the dimension of the subspase of the codewords is 12 and the minimum Hamming distance of the code is 8, that is, any [24,12,8]-code. Considering these codes, we apply the elements of the presentation theory, in particular regular presentations $v\to \sigma (v)$ of algebra $\mathbb{F}_2G$. For the element $v$ we define the binary code $C(v)$ as the subspace of $\mathbb{F}_2^{n}$ generated by the rows of the matrix $\sigma (v)$. The criterion of self-dual codes $C(v)$ for an arbitrary finite group $G$ of order 24 was used and easily verified necessary conditions for binary code $C(v)$ for elements $v$ of group algebra $\mathbb{F}_2G$ of the group $G=(C_6 \times C_2)~\rtimes~C_2$ to be self-dual was found. As a result of numerical calculations which involves verifying the found necessary conditions, we get the number of elements $v\in\mathbb{F}_2G$ that $C(v)$ is self-dual. Quantitative results for comparison with the number of the same elements when $v=v^*$ are presented. Previously, in this form, extended binary Golay codes were found only for elements $v$ that $v=v^*$. As a result of calculations we obtained all 27~648 elements $v$ of group algebra $\mathbb{F}_2G$ that $C(v)$ is extended binary Golay code.

Reversible Biosignal Steganography Approach for Authenticating Biosignals Using Extended Binary Golay Code.

We present a reversible biosignal steganography method to authenticate the source of biosignal in this paper. Cloud is being a popular platform for storing a large volume of biosignals such as an electrocardiogram (ECG), electroencephalogram (EEG), and photoplethysmogram (PPG). However, outsourcing biosignals to the cloud may introduce authenticity issues. For instance, patient data can be altered, or fake patient data can be inserted by the dishonest cloud service provider or attacker for giving benefits to business organizations such as insurance service providers. Steganography approaches can be used to hide data source's identification data before outsourcing to the cloud for maintaining authenticity. Existing biosignal steganography approaches fail to reconstruct original biosignal after applying a reverse data hiding technique. In other words, current biosignal steganography approaches are irreversible. Reversible biosignal steganography method is required for protecting biosignal data from deterioration and efficient use by its stakeholders. In this work, we develop a reversible biosignal steganography approach using the Extended Binary Golay Code based error correction method. Our proposed method embeds secret authentication message as an error within different types of biosignals such as ECG, PPG, and EEG. Extended Binary Golay Code based error correction method is used to extract the secret message, and reconstruct original biosignal. We conduct a set of experiments for evaluating the performance of our proposed method.

New Two-Stage Automorphism Group Decoders for Cyclic Codes

Recently, error correcting codes in the erasure channel have drawn great attention for various applications such as distributed storage systems and wireless sensor networks, but many of their decoding algorithms are not practical because they have higher decoding complexity and longer delay. Thus, the automorphism group decoder (AGD) for cyclic codes in the erasure channel was introduced, which has good erasure decoding performance with low decoding complexity. In this paper, we propose new two-stage AGDs (TS-AGDs) for cyclic codes in the erasure channel by modifying the parity-check matrix and introducing the preprocessing stage to the AGD scheme. The proposed TS-AGD is analyzed for binary extended Golay and BCH codes. Also, TS-AGD can be used in the error channel using ordered statistics. Through numerical analysis, it is shown that the proposed decoding algorithm has good erasure decoding performance with lower decoding complexity than the conventional AGD. For some cyclic codes, it is shown that the proposed TS-AGD achieves the performance nearly identical to the maximum likelihood (ML) decoder in the erasure channel and the ordered statistics decoder (OSD) in the error channel.

Extended maximal self-orthogonal codes

Self-dual and maximal self-orthogonal codes over [Formula: see text], where [Formula: see text] is even or [Formula: see text](mod 4), are extensively studied in this paper. We prove that every maximal self-orthogonal code can be extended to a self-dual code as in the case of binary Golay code. Using these results, we show that a self-dual code can also be constructed by gluing theory even if the sum of the lengths of the gluing components is odd. Furthermore, the (Hamming) weight enumerator [Formula: see text] of a self-dual code [Formula: see text] is given in terms of a maximal self-orthogonal code [Formula: see text], where [Formula: see text] is obtained by the extension of [Formula: see text].

A Novel and Fast Hardware Implementation for Golay Code Encoder

The Error Correction Code (ECC) is utilized to reduce the probability of error in digital systems. The binary Golay code is an ECC that can correct any combination of three or fewer random errors over a block of 23 digits. This code can be extended by appending a parity check bit to each codeword. There are several algorithms for constructing of Golay code, but more of them are not comfortable for hardware implementation. In this paper, an efficient hardware architecture is presented for the encoder of both binary Golay code and extended Golay code based on CRC. The proposed Golay code encoder is constructed of three units, which are designed carefully: data path, control unit and conversion unit. The proposed architecture is implemented on FPGA using Xilinx ISE 14.2. The implementation results demonstrate that low latency, high throughput, low area and less complexity are the advantages of this architecture compared to previous architectures. Thus, this hardware module can be used for high-speed digital systems.

The sextuply shortened binary Golay code is optimal

The maximum size of unrestricted binary three-error-correcting codes has been known up to the length of the binary Golay code, with two exceptions. Specifically, denoting the maximum size of an unrestricted binary code of length n and minimum distance d by A(n, d), it has been known that $$64 \le A(18,8) \le 68$$ and $$128 \le A(19,8) \le 131$$ . In the current computer-aided study, it is shown that $$A(18,8)=64$$ and $$A(19,8)=128$$ , so an optimal code is obtained even after shortening the extended binary Golay code six times.

A Review of design of Binary Golay Code and Extended Binary Golay Code for error correction

A Review of design of Binary Golay Code and Extended Binary Golay Code for error correction

Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces

Alves, Panek and Firer (Error-block codes and poset metrics, Adv. Math. Commun., 2 (2008), 95-111) classified all poset block structures which turn the [8,4,4] extended binary Hamming code into a 1-perfect poset block code. However, the proof needs corrections that are supplied in this paper. We provide a counterexample to show that the extended binary Golay code is not 1-perfect for the proposed poset block structures. All poset block structures turning the extended binary and ternary Golay codes into 1-perfect codes are classified.

On codes over 𝓡 k, m and constructions for new binary self-dual codes

Abstract In this work, we study codes over the ring 𝓡 k, m = 𝔽2[u, v]/ 〈uk , vm , uv − vu〉, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from 𝓡 k, m to F 2 k m $\mathbb{F}_2^{km}$ together with a Lee weight. After proving the MacWilliams identities for codes over 𝓡 k, m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over 𝓡 k, m . In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72, 36, 12] and 105 new Type II binary self-dual codes of parameter [72, 36, 12].

Efficient Hardware Implementation of Encoder and Decoder for Golay Code

This brief lays out cyclic redundancy check-based encoding scheme and presents an efficient implementation of the encoding algorithm in field programmable gate array (FPGA) prototype for both the binary Golay code ( $G_{\mathrm {\mathbf {23}}}$ ) and extended binary Golay code ( $G_{\mathrm {\mathbf {24}}}$ ). High speed with low-latency architecture has been designed and implemented in Virtex-4 FPGA for Golay encoder without incorporating linear feedback shift register. This brief also presents an optimized and low-complexity decoding architecture for extended binary Golay code (24, 12, 8) based on an incomplete maximum likelihood decoding scheme. The proposed architecture for decoder occupies less area and has lower latency than some of the recent work published in this area. The encoder module runs at 238.575 MHz, while the proposed architecture for decoder has an operating clock frequency of 195.028 MHz. The proposed hardware modules may be a good candidate for forward error correction in communication link, which demands a high-speed system.

A Multivariate Weight Enumerator for Tail-biting Trellis Pseudocodewords

Tail-biting-trellis representations of codes allow for iterative decoding algorithms, which are limited in effectiveness by the presence of pseudocodewords. We introduce a multivariate weight enumerator that keeps track of these pseudocodewords. This enumerator is invariant under many linear transformations, often enabling us to compute it exactly. The extended binary Golay code has a particularly nice tail-biting-trellis and a famous unsolved question is to determine its minimal AWGN pseudodistance. The new enumerator provides an inroad to this problem.

A New Construction for the Extended Binary Golay Code

We give a new construction of the extended binary Golay code. The construction is carried out by taking the Gray image of a self-dual linear code over the ring R=F2+ uF2+ vF2+ uvF2 of length 6 and size 2 12 . Writing a typical generating matrix of the form (I3jA), with A being a 3� 3 matrix over R, and finding some dependencies among the entries of A, we are able to set a general form for the generating matrices of self-dual codes of length 6. Using some special properties of elements of R, we end up with a family of generating matrices all of which give us the extended binary Golay code. We also prove the minimum distance property analytically.

High-Speed Decoding of the Binary Golay Code

Recently, some table-lookup decoding algorithms (TLDAs) have been used to correct the binary Golay code. Thispaper proposes an efficient high-speed TLDA called message-syndrome decoding algorithm (MSDA) by using themessage syndrome to correct the binary systematic Golay code. The proposed MSDA is based on the novelmessage-syndrome lookup table (MSLT). The MSLT merely consists of 12 candidate syndromes and its memory sizeis significantly smaller than other existing lookup tables. Computer software simulation results show that the decodingspeed of the proposed MSDA is superior to other proposed TLDAs.

Combinatorics associated with inflections and bitangents of plane quartics

After a preliminary survey and a description of some small Steiner systems from the standpoint of the theory of invariants of binary forms, we construct a binary Golay code (of length 24) using ideas from J. Grassmann's thesis of 1875. One of our tools is a pair of disjoint Fano planes. Another application of such pairs and properties of plane quartics is a construction of a new block design on 28 objects. This block design is a part of a dissection of the set of 288 Aronhold sevens. The dissection distributes the Aronhold sevens into 8 disjoint block designs of this type.

Generalization of all similar constructions of extended binary Golay code

Generalization of all similar constructions of extended binary Golay code