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Journal articles on the topic 'Graphe de code'

Author: Grafiati

Published: 15 February 2025

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1

Azkarate, Igor, Mikel Ayani, Juan Carlos Mugarza, and Luka Eciolaza. "Petri Net-Based Semi-Compiled Code Generation for Programmable Logic Controllers." Applied Sciences 11, no. 15 (August 3, 2021): 7161. http://dx.doi.org/10.3390/app11157161.

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Industrial discrete event dynamic systems (DEDSs) are commonly modeled by means of Petri nets (PNs). PNs have the capability to model behaviors such as concurrency, synchronization, and resource sharing, compared to a step transition function chart or GRAphe Fonctionnel de Commande Etape Transition (GRAFCET) which is a particular case of a PN. However, there is not an effective systematic way to implement a PN in a programmable logic controller (PLC), and so the implementation of such a controller outside a PLC in some external software that will communicate with the PLC is very common. There have been some attempts to implement PNs within a PLC, but they are dependent on how the logic of places and transitions is programmed for each application. This work proposes a novel application-independent and platform-independent PN implementation methodology. This methodology is a systematic way to implement a PN controller within industrial PLCs. A great portion of the code will be validated automatically prior to PLC implementation. Net structure and marking evolution will be checked on the basis of PN model structural analysis, and only net interpretation will be manually coded and error-prone. Thus, this methodology represents a systematic and semi-compiled PN implementation method. A use case supported by a digital twin (DT) is shown where the automated solution required by a manufacturing system is carried out and executed in two different devices for portability testing, and the scan cycle periods are compared for both approaches.

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MÜLLER, T., and J. S. SERENI. "Identifying and Locating–Dominating Codes in (Random) Geometric Networks." Combinatorics, Probability and Computing 18, no. 6 (August 11, 2009): 925–52. http://dx.doi.org/10.1017/s0963548309990344.

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We model a problem about networks built from wireless devices using identifying and locating–dominating codes in unit disk graphs. It is known that minimizing the size of an identifying code is -complete even for bipartite graphs. First, we improve this result by showing that the problem remains -complete for bipartite planar unit disk graphs. Then, we address the question of the existence of an identifying code for random unit disk graphs. We derive the probability that there exists an identifying code as a function of the radius of the disks, and we find that for all interesting ranges of r this probability is bounded away from one. The results obtained are in sharp contrast to those concerning random graphs in the Erdős–Rényi model. Another well-studied class of codes is that of locating–dominating codes, which are less demanding than identifying codes. A locating–dominating code always exists, but minimizing its size is still -complete in general. We extend this result to our setting by showing that this question remains -complete for arbitrary planar unit disk graphs. Finally, we study the minimum size of such a code in random unit disk graphs, and we prove that with probability tending to one, it is of size (n/r)2/3+o(1) if r ≤ /2−ϵ is chosen such that nr2 → ∞, and of size n1+o(1) if nr2 ≪ lnn.

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Hudry, Olivier, Junnila Ville, and Antoine Lobstein. "On Iiro Honkala’s Contributions to Identifying Codes." Fundamenta Informaticae 191, no. 3-4 (July 22, 2024): 165–96. http://dx.doi.org/10.3233/fi-242178.

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A set C of vertices in a graph G = (V, E) is an identifying code if it is dominating and any two vertices of V are dominated by distinct sets of codewords. This paper presents a survey of Iiro Honkala’s contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes.

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Saenpholphat, Varaporn, and Ping Zhang. "Conditional resolvability in graphs: a survey." International Journal of Mathematics and Mathematical Sciences 2004, no. 38 (2004): 1997–2017. http://dx.doi.org/10.1155/s0161171204311403.

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For an ordered setW={w1,w2,…,wk}of vertices and a vertexvin a connected graphG, the code ofvwith respect toWis thek-vectorcW(v)=(d(v,w1),d(v,w2),…,d(v,wk)), whered(x,y)represents the distance between the verticesxandy. The setWis a resolving set forGif distinct vertices ofGhave distinct codes with respect toW. The minimum cardinality of a resolving set forGis its dimensiondim(G). Many resolving parameters are formed by extending resolving sets to different subjects in graph theory, such as the partition of the vertex set, decomposition and coloring in graphs, or by combining resolving property with another graph-theoretic property such as being connected, independent, or acyclic. In this paper, we survey results and open questions on the resolving parameters defined by imposing an additional constraint on resolving sets, resolving partitions, or resolving decompositions in graphs.

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José, Marco, and Gabriel Zamudio. "Symmetrical Properties of Graph Representations of Genetic Codes: From Genotype to Phenotype." Symmetry 10, no. 9 (September 8, 2018): 388. http://dx.doi.org/10.3390/sym10090388.

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It has long been claimed that the mitochondrial genetic code possesses more symmetries than the Standard Genetic Code (SGC). To test this claim, the symmetrical structure of the SGC is compared with noncanonical genetic codes. We analyzed the symmetries of the graphs of codons and their respective phenotypic graph representation spanned by the RNY (R purines, Y pyrimidines, and N any of them) code, two RNA Extended codes, the SGC, as well as three different mitochondrial genetic codes from yeast, invertebrates, and vertebrates. The symmetry groups of the SGC and their corresponding phenotypic graphs of amino acids expose the evolvability of the SGC. Indeed, the analyzed mitochondrial genetic codes are more symmetrical than the SGC.

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Tang, C. S., and Tyng Liu. "The Degree Code—A New Mechanism Identifier." Journal of Mechanical Design 115, no. 3 (September 1, 1993): 627–30. http://dx.doi.org/10.1115/1.2919236.

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An important step in the structural synthesis of mechanisms requires the identification of isomorphism between the graphs which represents the mechanism topology. Previously used methods for identifying graph isomorphism either yield incorrect results for some cases or their algorithms are computationally inefficient for this application. This paper describes a new isomorphism identification method which is well suited for the automated structural synthesis of mechanisms. This method uses a new and compact mathematical representation for a graph, called the Degree Code, to identify graph isomorphism. Isomorphic graphs have identical Degree Codes; nonisomorphic graphs have distinct Degree Codes. Therefore, by examining the Degree Codes of the graphs, graph isomorphism is easily and correctly identified. This Degree Code algorithm is simpler and more efficient than other methods for identifying isomorphism correctly. In addition, the Degree Code can serve as an effective nomenclature and storage system for graphs or mechanisms. Although this identification scheme was developed specifically for the structural synthesis of mechanisms, it can be applied to any area where graph isomorphism is a critical issue.

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7

Leslie, Martin. "Hypermap-homology quantum codes." International Journal of Quantum Information 12, no. 01 (February 2014): 1430001. http://dx.doi.org/10.1142/s0219749914300010.

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We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m2, 2, m] code as compared to the toric code which is a [2m2, 2, m] code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

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8

Schlingemann, D. "Stabilizer codes can be realized as graph codes." Quantum Information and Computation 2, no. 4 (June 2002): 307–23. http://dx.doi.org/10.26421/qic2.4-4.

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We establish the connection between a recent new construction technique for quantum error correcting codes, based on graphs, and the so-called stabilizer codes: Each stabilizer code can be realized as a graph code and vice versa.

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Hwang, Yongsoo, and Jun Heo. "On the relation between a graph code and a graph state." Quantum Information and Computation 16, no. 3&4 (March 2016): 237–50. http://dx.doi.org/10.26421/qic16.3-4-3.

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A graph state and a graph code respectively are defined based on a mathematical simple graph. In this work, we examine a relation between a graph state and a graph code both obtained from the same graph, and show that a graph state is a superposition of logical qubits of the related graph code. By using the relation, we first discuss that a local complementation which has been used for a graph state can be useful for searching locally equivalent stabilizer codes, and second provide a method to find a stabilizer group of a graph code.

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Al-Kadhimi, Aymen M., Ammar E. Abdulkareem, and Charalampos C. Tsimenidis. "Performance Enhancement of LDPC Codes Based on Protograph Construction in 5G-NR Standard." Tikrit Journal of Engineering Sciences 30, no. 4 (November 1, 2023): 1–10. http://dx.doi.org/10.25130/tjes.30.4.1.

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To meet the high throughput demands, the 3rd Generation Partnership Project has specified the low-density parity check (LDPC) codes in the fifth generation-new radio 5G-NR standard with rate and length compatibility and scalability. This paper presents an extensive performance evaluation and enhancement of LPDC using the protograph-based construction defined in the 5G-NR standard. Firstly, the protograph-LDPC with layered offset min-sum (OMS) decoding, polar with successive cancellation list (SCL), and block turbo code are implemented and compared. Puncturing and shortening are applied to maintain block length at 1024 and code rate at 1/2 for all codes for comparison fairness. The results showed that P-LDPC outperforms its counterparts in terms of bit/ frame error rate (BER/ FER) behavior for given signal-to-noise ratios. Then, different P-LDPC settings were realized to study the effects of base graph selection (Graph1 or Graph2), code rate change (1/3 - 2/3), and block lengths increase (260 – 4160 bits). The simulation outcomes proved that BER performed better for lower coding rates or higher block lengths. Furthermore, P-LDPC behavior was examined over a Rayleigh flat-fading channel to achieve a 12.5 dB coding gain at 0.001 BER compared with uncoded transmission.

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Manickam, Machasri, and Kalyani Desikan. "Eigenvalue Interlacing of Bipartite Graphs and Construction of Expander Code using Vertex-split of a Bipartite Graph." European Journal of Pure and Applied Mathematics 17, no. 2 (April 30, 2024): 772–89. http://dx.doi.org/10.29020/nybg.ejpam.v17i2.5057.

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The second largest eigenvalue of a graph is an important algebraic parameter which is related with the expansion, connectivity and randomness properties of a graph. Expanders are highly connected sparse graphs. In coding theory, Expander codes are Error Correcting codes made up of bipartite expander graphs. In this paper, first we prove the interlacing of the eigenvalues of the adjacency matrix of the bipartite graph with the eigenvalues of the bipartite quotient matrices of the corresponding graph matrices. Then we obtain bounds for the second largest and second smallest eigenvalues. Since the graph is bipartite, the results for Laplacian will also hold for Signless Laplacian matrix. We then introduce a new method called vertex-split of a bipartite graph to construct asymptotically good expander codes with expansion factor D/2 < alpha < D and epsilon < 1/2 and prove a condition for the vertex-split of a bipartite graph to be k-connected with respect to the second largest eigenvalue: Further, we prove that the vertex-split of G is a bipartite expander. Finally, we construct an asymptotically good expander code whose factor graph is a graph obtained by the vertex-split of a bipartite graph.

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Fimmel, Elena, Christian J. Michel, and Lutz Strüngmann. "n -Nucleotide circular codes in graph theory." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374, no. 2063 (March 13, 2016): 20150058. http://dx.doi.org/10.1098/rsta.2015.0058.

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The circular code theory proposes that genes are constituted of two trinucleotide codes: the classical genetic code with 61 trinucleotides for coding the 20 amino acids (except the three stop codons { TAA , TAG , TGA }) and a circular code based on 20 trinucleotides for retrieving, maintaining and synchronizing the reading frame. It relies on two main results: the identification of a maximal C 3 self-complementary trinucleotide circular code X in genes of bacteria, eukaryotes, plasmids and viruses (Michel 2015 J. Theor. Biol. 380, 156–177. ( doi:10.1016/j.jtbi.2015.04.009 ); Arquès & Michel 1996 J. Theor. Biol. 182, 45–58. ( doi:10.1006/jtbi.1996.0142 )) and the finding of X circular code motifs in tRNAs and rRNAs, in particular in the ribosome decoding centre (Michel 2012 Comput. Biol. Chem. 37, 24–37. ( doi:10.1016/j.compbiolchem.2011.10.002 ); El Soufi & Michel 2014 Comput. Biol. Chem. 52, 9–17. ( doi:10.1016/j.compbiolchem.2014.08.001 )). The univerally conserved nucleotides A1492 and A1493 and the conserved nucleotide G530 are included in X circular code motifs. Recently, dinucleotide circular codes were also investigated (Michel & Pirillo 2013 ISRN Biomath. 2013, 538631. ( doi:10.1155/2013/538631 ); Fimmel et al. 2015 J. Theor. Biol. 386, 159–165. ( doi:10.1016/j.jtbi.2015.08.034 )). As the genetic motifs of different lengths are ubiquitous in genes and genomes, we introduce a new approach based on graph theory to study in full generality n -nucleotide circular codes X , i.e. of length 2 (dinucleotide), 3 (trinucleotide), 4 (tetranucleotide), etc. Indeed, we prove that an n -nucleotide code X is circular if and only if the corresponding graph is acyclic. Moreover, the maximal length of a path in corresponds to the window of nucleotides in a sequence for detecting the correct reading frame. Finally, the graph theory of tournaments is applied to the study of dinucleotide circular codes. It has full equivalence between the combinatorics theory (Michel & Pirillo 2013 ISRN Biomath. 2013, 538631. ( doi:10.1155/2013/538631 )) and the group theory (Fimmel et al. 2015 J. Theor. Biol. 386, 159–165. ( doi:10.1016/j.jtbi.2015.08.034 )) of dinucleotide circular codes while its mathematical approach is simpler.

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Wenni, Mariza. "BILANGAN KROMATIK LOKASI DARI GRAF P m P n ; K m P n ; DAN K , m K n." Jurnal Matematika UNAND 2, no. 1 (March 10, 2013): 14. http://dx.doi.org/10.25077/jmu.2.1.14-22.2013.

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Let G and H be two connected graphs. Let c be a vertex k-coloring of aconnected graph G and let = fCg be a partition of V (G) into the resultingcolor classes. For each v 2 V (G), the color code of v is dened to be k-vector: c1; C2; :::; Ck(v) =(d(v; C1); d(v; C2); :::; d(v; Ck)), where d(v; Ci) = minfd(v; x) j x 2 Cg, 1 i k. Ifdistinct vertices have distinct color codes with respect to , then c is called a locatingcoloring of G. The locating chromatic number of G is the smallest natural number ksuch that there are locating coloring with k colors in G. The Cartesian product of graphG and H is a graph with vertex set V (G) V (H), where two vertices (a; b) and (a)are adjacent whenever a = a0and bb02 E(H), or aa0i2 E(G) and b = b, denotedby GH. In this paper, we will study about the locating chromatic numbers of thecartesian product of two paths, the cartesian product of paths and complete graphs, andthe cartesian product of two complete graphs.

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Darmoun, Saber, Abdelmoumen Khalid, and Ben-Azza Hussain. "Heights of error-correcting codes." Gulf Journal of Mathematics 18, no. 2 (December 1, 2024): 29–46. https://doi.org/10.56947/gjom.v18i2.2397.

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In this work, we investigate the evaluation of odd polynomials P defined on a finite field on the class of error-correcting codes C. We exploit the correspondence between codes and Tanner graphs. Thus, we formally define P(C), a polynomial code. Then the new notion of height of a code emerges, whose properties are studied. We extended the lower bound of Tanner on the minimum distance of a code to the case of a polynomial code, by using spectral graph theory. Computer algebra software enable us to give numerical results to illustrate the theory of polynomial codes for various classes of error-correcting codes.

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Ma, Xuanlong, Ruiqin Fu, Xuefei Lu, Mengxia Guo, and Zhiqin Zhao. "Perfect codes in power graphs of finite groups." Open Mathematics 15, no. 1 (December 9, 2017): 1440–49. http://dx.doi.org/10.1515/math-2017-0123.

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Abstract The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.

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Sarkar, Rahul, and Theodore J. Yoder. "A graph-based formalism for surface codes and twists." Quantum 8 (July 18, 2024): 1416. http://dx.doi.org/10.22331/q-2024-07-18-1416.

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Twist defects in surface codes can be used to encode more logical qubits, improve the code rate, and implement logical gates. In this work we provide a rigorous formalism for constructing surface codes with twists generalizing the well-defined homological formalism introduced by Kitaev for describing CSS surface codes. In particular, we associate a surface code to any graph G embedded on any 2D-manifold, in such a way that (1) qubits are associated to the vertices of the graph, (2) stabilizers are associated to faces, (3) twist defects are associated to odd-degree vertices. In this way, we are able to reproduce the variety of surface codes, with and without twists, in the literature and produce new examples. We also calculate and bound various code properties such as the rate and distance in terms of topological graph properties such as genus, systole, and face-width.

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Cappelletti, Luca, Tommaso Fontana, Elena Casiraghi, Vida Ravanmehr, Tiffany J. Callahan, Carlos Cano, Marcin P. Joachimiak, et al. "GRAPE for fast and scalable graph processing and random-walk-based embedding." Nature Computational Science 3, no. 6 (June 26, 2023): 552–68. http://dx.doi.org/10.1038/s43588-023-00465-8.

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AbstractGraph representation learning methods opened new avenues for addressing complex, real-world problems represented by graphs. However, many graphs used in these applications comprise millions of nodes and billions of edges and are beyond the capabilities of current methods and software implementations. We present GRAPE (Graph Representation Learning, Prediction and Evaluation), a software resource for graph processing and embedding that is able to scale with big graphs by using specialized and smart data structures, algorithms, and a fast parallel implementation of random-walk-based methods. Compared with state-of-the-art software resources, GRAPE shows an improvement of orders of magnitude in empirical space and time complexity, as well as competitive edge- and node-label prediction performance. GRAPE comprises approximately 1.7 million well-documented lines of Python and Rust code and provides 69 node-embedding methods, 25 inference models, a collection of efficient graph-processing utilities, and over 80,000 graphs from the literature and other sources. Standardized interfaces allow a seamless integration of third-party libraries, while ready-to-use and modular pipelines permit an easy-to-use evaluation of graph-representation-learning methods, therefore also positioning GRAPE as a software resource that performs a fair comparison between methods and libraries for graph processing and embedding.

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Mudaber, M. H., N. H. Sarmin, and I. Gambo. "Subset Perfect Codes of Finite Commutative Rings Over Induced Subgraphs of Unit Graphs." Malaysian Journal of Mathematical Sciences 16, no. 4 (December 23, 2022): 783–91. http://dx.doi.org/10.47836/mjms.16.4.10.

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The induced subgraph of a unit graph with vertex set as the non unit elements of a ring R is a graph obtained by deleting all unit elements of R. In a graph 􀀀, a subset of the vertex set is called a perfect code if the balls with radius 1 centred on the subset are pairwise disjoint and their unions yield the whole vertex set. In this paper, we determine the perfect codes of induced subgraphs of the unit graphs associated with some finite commutative rings R with unity that has a vertex set as non unit elements of R. Moreover, we classify the commutative rings in which their associated induced subgraphs of unit graphs admit the trivial and non-trivial perfect codes. We also characterize the commutative rings based on the induced subgraph of unit graphs that do not admit the perfect codes. Furthermore, we prove that the complement induced subgraph of unit graph admit only the trivial subring perfect code.

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Bliadze, A., and V. Gurevich. "Equivalent to the Probability of Error with Block Error Correction Codes." Telecom IT 7, no. 3 (December 31, 2019): 1–6. http://dx.doi.org/10.31854/2307-1303-2019-7-3-1-6.

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The probabilities of bit and code errors are compared when using non-redundant and noise-resistant block codes in a digital radio system for transmitting information. The analysis results are accompanied by graphs that allow you to select the bit rate of the noise-tolerant code necessary to ensure the specified probability of code errors.

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Shin, Jae Kyun, and S. Krishnamurty. "Development of a Standard Code for Colored Graphs and Its Application to Kinematic Chains." Journal of Mechanical Design 116, no. 1 (March 1, 1994): 189–96. http://dx.doi.org/10.1115/1.2919345.

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The development of an efficient solution procedure for the detection of isomorphism and canonical numbering of vertices of colored graphs is introduced. This computer-based algorithm for colored graphs is formed by extending the standard code approach developed earlier for the canonical numbering of simple noncolored graphs, which fully utilizes the capabilities of symmetry analysis of such noncolored graphs. Its application to various kinematic chains and mechanisms is investigated with the aid of examples. The method never failed to produce unique codes, and is also found to be robust and efficient. Using this method, every kinematic chain and mechanism, as well as path generators and function generators, will have their own unique codes and a corresponding canonical numbering of their respective links. Thus, based on its efficiency and applicability, this method can be used as a universal standard code for identifying isomorphisms, as well as for enumerating nonisomorphic kinematic chains and mechanisms.

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Aouf, H. "Locating Chromatic Number of Middle Graph of Path, Cycle, Star, Wheel, Gear and Helm Graphs." Journal of Combinatorial Mathematics and Combinatorial Computing 119, no. 1 (March 31, 2024): 335–45. http://dx.doi.org/10.61091/jcmcc119-32.

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Let \(c\) be a proper \(k\)-coloring of a connected graph \(G\) and \(\pi=\{S_{1},S_{2},\ldots,S_{k}\}\) be an ordered partition of the vertex set \(V(G)\) into the resulting color classes, where \(S_{i}\) is the set of all vertices that receive the color \(i\). For a vertex \(v\) of \(G\), the color code \(c_{\pi}(v)\) of \(v\) with respect to \(\pi\) is the ordered \(k\)-tuple \(c_{\pi}(v)=(d(v,S_{1}),d(v,S_{2}),\ldots,d(v,S_{k}))\), where \(d(v,S_{i})=min\{d(v,u):\textit{ } u\in S_{i}\}\) for \(1\leqslant i \leqslant k\). If all distinct vertices of \(G\) have different color codes, then \(c\) is called a locating coloring of \(G\). The locating chromatic number is the minimum number of colors needed in a locating coloring. In this paper, we determine the locating-chromatic number for the middle graphs of Path, Cycle, Wheel, Star, Gear and Helm graphs.

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Wang, Yun Yun, and Li Zhen Jiang. "Measures of 7-qubit Stabilizer Codes for Graph States." Advanced Materials Research 1049-1050 (October 2014): 1844–47. http://dx.doi.org/10.4028/www.scientific.net/amr.1049-1050.1844.

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Most quantum error-correcting codes constructed are stabilizer codes which are potentially more efficient and significant.Under the concept of graph states theory, we focused on 7-qubit graph states characteristics of stabilizer codes, by calculating and analyzing the entanglement properties of graphical codes. For the instance of code ((7,1,3)) with 16 inequitable graphs, the figure of entanglement properties can be measured.And we analyzed 7-qubit graph characteristics of stabilizers codes and calculated the entanglement measures by iterative algorithm.

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Joshi, Sandeep S., and Kishor F. Pawar. "ENERGY OF SOME GRAPHS OF PRIME GRAPH OF A RING." Jnanabha 50, no. 01 (2020): 14–19. http://dx.doi.org/10.58250/jnanabha.2020.50101.

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Let R be a commutative ring and PG(R) is a graph whose vertices are all the elements of ring R and two vertices are adjacent if their product is zero. In this article, we study the energy of 1-Quasitotal and 2-Quasitotal Prime Graph of a Ring Zp and also find the energy of PG 1 (Zp ) and PG 2 (Zp ), p prime. A General SCILAB Software code for our calculation is also presented

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Pilongo, Jupiter, Leonard Mijares Paleta, and Philip Lester P. Benjamin. "Vertex-weighted $(k_{1},k_{2})$ $E$-torsion Graph of Quasi Self-dual Codes." European Journal of Pure and Applied Mathematics 17, no. 2 (April 30, 2024): 1369–84. http://dx.doi.org/10.29020/nybg.ejpam.v17i2.4867.

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In this paper, we have introduced a graph $G_{EC}$ generated by type-$(k_{1},k_{2})$ $E$-codes which is $(k_{1},k_{2})$ $E$-torsion graph. The binary code words of the torsion code of $C$ are the set of vertices, and the edges are defined using the construction of $E$-codes. Also, we characterized the graph obtained when $k_{1}=0$ and $k_{2}=0$ and calculated the degrees of every vertex and the number of edges of $G_{EC}$. Moreover, we presented necessary and sufficient conditions for a vertex to be in the center of a graph given the property of the code word corresponding to the vertex. Finally, we represent every quasi-self dual codes of short length by defining the vertex-weighted $(k_{1},k_{2})$ $E$-torsion graph, where the weight of every vertex is the weight of the code word corresponding to the vertex.

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Seneviratne, P. "Codes from multipartite graphs and minimal permutation decoding sets." Discrete Mathematics, Algorithms and Applications 07, no. 04 (December 2015): 1550060. http://dx.doi.org/10.1142/s1793830915500603.

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Permutation decoding method developed by MacWilliams and described in [Permutation decoding of systematic codes, Bell Syst. Tech. J. 43 (1964) 485–505] is a decoding technique that uses a subset of the automorphism group of the code called a PD-set. The complexity of the permutation decoding algorithm depends on the size of the PD-set and finding a minimal PD-set for an error correcting code is a hard problem. In this paper we examine binary codes from the complete-multipartite graph [Formula: see text] and find PD-sets for all values of [Formula: see text] and [Formula: see text]. Further we show that these PD-sets are minimal when [Formula: see text] is odd and [Formula: see text].

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Semri, Ahmed, and Hillal Touati. "Optimal Identifying Codes in Oriented Paths and Circuits." International Journal of Mathematical Models and Methods in Applied Sciences 15 (March 26, 2021): 9–14. http://dx.doi.org/10.46300/9101.2021.15.2.

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Identifying codes in graphs are related to the classical notion of dominating sets [1]. Since there first introduction in 1998 [2], they have been widely studied and extended to several application, such as: detection of faulty processor in multiprocessor systems, locating danger or threats in sensor networks. Let G=(V,E) an unoriented connected graph. The minimum identifying code in graphs is the smallest subset of vertices C, such that every vertex in V have a unique set of neighbors in C. In our work, we focus on finding minimum cardinality of an identifying code in oriented paths and circuits

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Hsu, Cheng-Ho, and Kin-Tak Lam. "Topological code of graphs." Journal of the Franklin Institute 329, no. 1 (January 1992): 99–109. http://dx.doi.org/10.1016/0016-0032(92)90100-u.

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28

Gold, Robert. "Control flow graphs and code coverage." International Journal of Applied Mathematics and Computer Science 20, no. 4 (December 1, 2010): 739–49. http://dx.doi.org/10.2478/v10006-010-0056-9.

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Control flow graphs and code coverageThe control flow of programs can be represented by directed graphs. In this paper we provide a uniform and detailed formal basis for control flow graphs combining known definitions and results with new aspects. Two graph reductions are defined using only syntactical information about the graphs, but no semantical information about the represented programs. We prove some properties of reduced graphs and also about the paths in reduced graphs. Based on graphs, we define statement coverage and branch coverage such that coverage notions correspond to node coverage, and edge coverage, respectively.

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Lv, Yijie, Jiguang He, Weikai Xu, and Lin Wang. "Design of Low-Density Parity-Check Code Pair for Joint Source-Channel Coding Systems Based on Graph Theory." Entropy 25, no. 8 (August 10, 2023): 1189. http://dx.doi.org/10.3390/e25081189.

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In this article, a graph-theoretic method (taking advantage of constraints among sets associated with the corresponding parity-check matrices) is applied for the construction of a double low-density parity-check (D-LDPC) code (also known as LDPC code pair) in a joint source-channel coding (JSCC) system. Specifically, we pre-set the girth of the parity-check matrix for the LDPC code pair when jointly designing the two LDPC codes, which are constructed by following the set constraints. The constructed parity-check matrices for channel codes comprise an identity submatrix and an additional submatrix, whose column weights can be pre-set to be any positive integer numbers. Simulation results illustrate that the constructed D-LDPC codes exhibit significant performance improvement and enhanced flexible frame length (i.e., adaptability under various channel conditions) compared with the benchmark code pair.

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Frolov, A. A., and V. V. Zyablov. "Bounds on the minimum code distance for nonbinary codes based on bipartite graphs." Problems of Information Transmission 47, no. 4 (December 2011): 327–41. http://dx.doi.org/10.1134/s0032946011040028.

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31

Mudaber, Mohammad Hassan, Nor Haniza Sarmin, and Ibrahim Gambo. "Non-Trivial Subring Perfect Codes in Unit Graph of Boolean Rings." Malaysian Journal of Fundamental and Applied Sciences 18, no. 3 (August 4, 2022): 374–82. http://dx.doi.org/10.11113/mjfas.v18n3.2503.

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The aim of this paper is to investigate the non-trivial subring perfect codes in a unit graph associated with the Boolean rings. We prove a subring perfect code of size , where , in the unit graphs associated with the finite Boolean rings . Moreover, we give a necessary and sufficient condition for a subring of an infinite Boolean ring to be a perfect code of size infinity in the unit graph.

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Lavi, Inbal, Shai Avidan, Yoram Singer, and Yacov Hel-Or. "Proximity Preserving Binary Code Using Signed Graph-Cut." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 4535–44. http://dx.doi.org/10.1609/aaai.v34i04.5882.

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We introduce a binary embedding framework, called Proximity Preserving Code (PPC), which learns similarity and dissimilarity between data points to create a compact and affinity-preserving binary code. This code can be used to apply fast and memory-efficient approximation to nearest-neighbor searches. Our framework is flexible, enabling different proximity definitions between data points. In contrast to previous methods that extract binary codes based on unsigned graph partitioning, our system models the attractive and repulsive forces in the data by incorporating positive and negative graph weights. The proposed framework is shown to boil down to finding the minimal cut of a signed graph, a problem known to be NP-hard. We offer an efficient approximation and achieve superior results by constructing the code bit after bit. We show that the proposed approximation is superior to the commonly used spectral methods with respect to both accuracy and complexity. Thus, it is useful for many other problems that can be translated into signed graph cut.

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Ling, Xiang, Lingfei Wu, Saizhuo Wang, Gaoning Pan, Tengfei Ma, Fangli Xu, Alex X. Liu, Chunming Wu, and Shouling Ji. "Deep Graph Matching and Searching for Semantic Code Retrieval." ACM Transactions on Knowledge Discovery from Data 15, no. 5 (June 26, 2021): 1–21. http://dx.doi.org/10.1145/3447571.

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Code retrieval is to find the code snippet from a large corpus of source code repositories that highly matches the query of natural language description. Recent work mainly uses natural language processing techniques to process both query texts (i.e., human natural language) and code snippets (i.e., machine programming language), however, neglecting the deep structured features of query texts and source codes, both of which contain rich semantic information. In this article, we propose an end-to-end deep graph matching and searching (DGMS) model based on graph neural networks for the task of semantic code retrieval. To this end, we first represent both natural language query texts and programming language code snippets with the unified graph-structured data, and then use the proposed graph matching and searching model to retrieve the best matching code snippet. In particular, DGMS not only captures more structural information for individual query texts or code snippets, but also learns the fine-grained similarity between them by cross-attention based semantic matching operations. We evaluate the proposed DGMS model on two public code retrieval datasets with two representative programming languages (i.e., Java and Python). Experiment results demonstrate that DGMS significantly outperforms state-of-the-art baseline models by a large margin on both datasets. Moreover, our extensive ablation studies systematically investigate and illustrate the impact of each part of DGMS.

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Li, Chi-Kwong, and Ingrid Nelson. "Perfect codes on the towers of Hanoi graph." Bulletin of the Australian Mathematical Society 57, no. 3 (June 1998): 367–76. http://dx.doi.org/10.1017/s0004972700031774.

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We characterise all the perfect k-error correcting codes that can be defined on the graph associated with the Towers of Hanoi puzzle. In particular, a short proof for the existence of 1-error correcting code on such a graph is given.

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Slimani, Djamel, and Abdellah Kaddai. "An improved method for counting 6-cycles in low-density parity-check codes." Serbian Journal of Electrical Engineering 20, no. 1 (2023): 83–91. http://dx.doi.org/10.2298/sjee2301083s.

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Since their rediscovery in the early 1990s, low-density parity-check (LDPC) codes have become the most popular error-correcting codes owing to their excellent performance. An LDPC code is a linear block code that has a sparse parity-check matrix. Cycles in this matrix, particularly short cycles, degrade the performance of such a code. Hence, several methods for counting short cycles in LDPC codes have been proposed, such as Fan?s method to detect 4-cycles, 6- cycles, 8-cycles, and 10-cycles. Unfortunately, this method fails to count all 6- cycles, i.e., ignores numerous 6-cycles, in some given parity-check matrices. In this paper, an improvement of this algorithm is presented that detects all 6-cycles in LDPC codes, as well as in general bipartite graphs. Simulations confirm that the improved method offers the exact number of 6-cycles, and it succeeds in detecting those ignored by Fan?s method.

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Bao, Xingkai, and Jing Li. "Adaptive network coded cooperation (ANCC) for wireless relay networks: matching code-on-graph with network-on-graph." IEEE Transactions on Wireless Communications 7, no. 2 (February 2008): 574–83. http://dx.doi.org/10.1109/twc.2008.060439.

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37

Pareja, Aldo, Giacomo Domeniconi, Jie Chen, Tengfei Ma, Toyotaro Suzumura, Hiroki Kanezashi, Tim Kaler, Tao Schardl, and Charles Leiserson. "EvolveGCN: Evolving Graph Convolutional Networks for Dynamic Graphs." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 5363–70. http://dx.doi.org/10.1609/aaai.v34i04.5984.

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Graph representation learning resurges as a trending research subject owing to the widespread use of deep learning for Euclidean data, which inspire various creative designs of neural networks in the non-Euclidean domain, particularly graphs. With the success of these graph neural networks (GNN) in the static setting, we approach further practical scenarios where the graph dynamically evolves. Existing approaches typically resort to node embeddings and use a recurrent neural network (RNN, broadly speaking) to regulate the embeddings and learn the temporal dynamics. These methods require the knowledge of a node in the full time span (including both training and testing) and are less applicable to the frequent change of the node set. In some extreme scenarios, the node sets at different time steps may completely differ. To resolve this challenge, we propose EvolveGCN, which adapts the graph convolutional network (GCN) model along the temporal dimension without resorting to node embeddings. The proposed approach captures the dynamism of the graph sequence through using an RNN to evolve the GCN parameters. Two architectures are considered for the parameter evolution. We evaluate the proposed approach on tasks including link prediction, edge classification, and node classification. The experimental results indicate a generally higher performance of EvolveGCN compared with related approaches. The code is available at https://github.com/IBM/EvolveGCN.

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Raza, Mohd Arif, Adel N. Alahmadi, Widyan Basaffar, David G. Glynn, Manish K. Gupta, James W. P. Hirschfeld, Abdul Nadim Khan, Hatoon Shoaib, and Patrick Solé. "The Quantum States of a Graph." Mathematics 11, no. 10 (May 16, 2023): 2310. http://dx.doi.org/10.3390/math11102310.

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Quantum codes are crucial building blocks of quantum computers. With a self-dual quantum code is attached, canonically, a unique stabilised quantum state. Improving on a previous publication, we show how to determine the coefficients on the basis of kets in these states. Two important ingredients of the proof are algebraic graph theory and quadratic forms. The Arf invariant, in particular, plays a significant role.

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Steinert, Patrick, Stefan Wagenpfeil, Paul Mc Kevitt, Ingo Frommholz, and Matthias Hemmje. "Parallelization Strategies for Graph-Code-Based Similarity Search." Big Data and Cognitive Computing 7, no. 2 (April 6, 2023): 70. http://dx.doi.org/10.3390/bdcc7020070.

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The volume of multimedia assets in collections is growing exponentially, and the retrieval of information is becoming more complex. The indexing and retrieval of multimedia content is generally implemented by employing feature graphs. Feature graphs contain semantic information on multimedia assets. Machine learning can produce detailed semantic information on multimedia assets, reflected in a high volume of nodes and edges in the feature graphs. While increasing the effectiveness of the information retrieval results, the high level of detail and also the growing collections increase the processing time. Addressing this problem, Multimedia Feature Graphs (MMFGs) and Graph Codes (GCs) have been proven to be fast and effective structures for information retrieval. However, the huge volume of data requires more processing time. As Graph Code algorithms were designed to be parallelizable, different paths of parallelization can be employed to prove or evaluate the scalability options of Graph Code processing. These include horizontal and vertical scaling with the use of Graphic Processing Units (GPUs), Multicore Central Processing Units (CPUs), and distributed computing. In this paper, we show how different parallelization strategies based on Graph Codes can be combined to provide a significant improvement in efficiency. Our modeling work shows excellent scalability with a theoretical speedup of 16,711 on a top-of-the-line Nvidia H100 GPU with 16,896 cores. Our experiments with a mediocre GPU show that a speedup of 225 can be achieved and give credence to the theoretical speedup. Thus, Graph Codes provide fast and effective multimedia indexing and retrieval, even in billion-scale use cases.

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40

Albuquerque, C. D., R. Palazzo Jr., and E. B. Silva. "New classes of TQC associated with self-dual, quasi self-dual and denser tessellations." Quantum Information and Computation 10, no. 11&12 (November 2010): 956–70. http://dx.doi.org/10.26421/qic10.11-12-6.

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In this paper we present six classes of topological quantum codes (TQC) on compact surfaces with genus $g\ge 2$. These codes are derived from self-dual, quasi self-dual and denser tessellations associated with embeddings of self-dual complete graphs and complete bipartite graphs on the corresponding compact surfaces. The majority of the new classes has the self-dual tessellations as their algebraic and geometric supporting mathematical structures. Every code achieves minimum distance 3 and its encoding rate is such that $\frac{k}{n} \rightarrow 1$ as $n \rightarrow \infty$, except for the one case where $\frac{k}{n} \rightarrow \frac{1}{3}$ as $n \rightarrow \infty$.

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41

Kahl, Wolfram. "Towards “mouldable code” via nested code graph transformation." Journal of Logical and Algebraic Methods in Programming 83, no. 2 (March 2014): 225–34. http://dx.doi.org/10.1016/j.jlap.2014.02.010.

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42

Nie, Peng, Huanqin Wu, and Zhanchuan Cai. "Towards Automatic ICD Coding via Label Graph Generation." Mathematics 12, no. 15 (August 1, 2024): 2398. http://dx.doi.org/10.3390/math12152398.

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Automatic International Classification of Disease (ICD) coding, a system for assigning proper codes to a given clinical text, has received increasing attention. Previous studies have focused on formulating the ICD coding task as a multi-label prediction approach, exploring the relationship between clinical texts and ICD codes, parent codes and child codes, and siblings. However, the large search space of ICD codes makes it difficult to localize target labels. Moreover, there exists a great unbalanced distribution of ICD codes at different levels. In this work, we propose LabGraph, which transfers ICD coding into a graph generation problem. Specifically, we present adversarial domain adaptation training algorithms, graph reinforcement algorithms, and adversarial perturbation regularization. Then, we present a discriminator for label graphs that calculates the reward for each ICD code in the generator label graph. LabGraph surpasses existing state-of-the-art approaches on core assessment measures such as micro-F1, micro-AUC, and P@K, leading to the formation of a new state-of-the-art study.

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43

Baspin, Nouédyn, and Anirudh Krishna. "Connectivity constrains quantum codes." Quantum 6 (May 13, 2022): 711. http://dx.doi.org/10.22331/q-2022-05-13-711.

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Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D-dimensional hyperbolic space.

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44

Perezhogin, Aleksey Lvovich, and Igor Sergeevich Bykov. "Overview of constructions and properties of Gray codes." Mathematical Problems of Cybernetics, no. 20 (2022): 41–60. http://dx.doi.org/10.20948/mvk-2022-41.

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Cyclic enumeration of binary words of length n, where each pair of adjacent words differs in exactly one index, is called n-dimensional Gray code. Gray code determines hamiltonian cycle in boolean n-cube. In this paper we give a review of constructions and classifications of Gray codes. Constructions are divided into three main groups: recursive, toric and stream. We also give some of Gray code properties, as an example of applications of these constructions. In particular, we consider spectrum of edge directions, graphs of 2-subwords in transition sequence, local uniformity and others. Also some unresolved problems are given.

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45

Choo, Kelly, and Gary MacGillivray. "Gray code numbers for graphs." Ars Mathematica Contemporanea 4, no. 1 (March 28, 2011): 125–39. http://dx.doi.org/10.26493/1855-3974.196.0df.

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Raspaud, André, and Li-Da Tong. "The minimum identifying code graphs." Discrete Applied Mathematics 160, no. 9 (June 2012): 1385–89. http://dx.doi.org/10.1016/j.dam.2012.01.015.

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Klavžar, Sandi, Uroš Milutinović, and Ciril Petr. "1-perfect codes in Sierpiński graphs." Bulletin of the Australian Mathematical Society 66, no. 3 (December 2002): 369–84. http://dx.doi.org/10.1017/s0004972700040235.

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Sierpiński graphs S (n, κ) generalise the Tower of Hanoi graphs—the graph S (n, 3) is isomorphic to the graph Hn of the Tower of Hanoi with n disks. A 1-perfect code (or an efficient dominating set) in a graph G is a vertex subset of G with the property that the closed neighbourhoods of its elements form a partition of V (G). It is proved that the graphs S (n, κ) possess unique 1-perfect codes, thus extending a previously known result for Hn. An efficient decoding algorithm is also presented. The present approach, in particular the proposed (de)coding, is intrinsically different from the approach to Hn.

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48

Høholdt, Tom, and Jørn Justesen. "On the sizes of expander graphs and minimum distances of graph codes." Discrete Mathematics 325 (June 2014): 38–46. http://dx.doi.org/10.1016/j.disc.2014.02.005.

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49

Sun, Jiang Hong. "Kinematic Chain Isomorphism Identification Based on Loop-Code." Applied Mechanics and Materials 44-47 (December 2010): 3874–78. http://dx.doi.org/10.4028/www.scientific.net/amm.44-47.3874.

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A new method approach is presented to solve isomorphism identification of kinematic chain topology graphs. Kinematic chain topology graphs are depicted with loop-code based on the characteristic of kinematic chain topology graphs. The number of binary links in the limbs of topology graphs is arranged in group according to links with multi points of connection．Although the order of limbs in the groups and the order of links with multi points of connection outside the groups can change, the planar message of topology graphs can not change. Thereby forming loop-code.This representation is straightforward and not affected when drawing modes and labeling ways change in topology graphs．

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50

Durcek, Viktor, Michal Kuba, and Milan Dado. "Investigation of random-structure regular LDPC codes construction based on progressive edge-growth and algorithms for removal of short cycles." Eastern-European Journal of Enterprise Technologies 4, no. 9(112) (August 31, 2021): 46–53. http://dx.doi.org/10.15587/1729-4061.2021.225852.

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This paper investigates the construction of random-structure LDPC (low-density parity-check) codes using Progressive Edge-Growth (PEG) algorithm and two proposed algorithms for removing short cycles (CB1 and CB2 algorithm; CB stands for Cycle Break). Progressive Edge-Growth is an algorithm for computer-based design of random-structure LDPC codes, the role of which is to generate a Tanner graph (a bipartite graph, which represents a parity-check matrix of an error-correcting channel code) with as few short cycles as possible. Short cycles, especially the shortest ones with a length of 4 edges, in Tanner graphs of LDPC codes can degrade the performance of their decoding algorithm, because after certain number of decoding iterations, the information sent through its edges is no longer independent. The main contribution of this paper is the unique approach to the process of removing short cycles in the form of CB2 algorithm, which erases edges from the code's parity-check matrix without decreasing the minimum Hamming distance of the code. The two cycle-removing algorithms can be used to improve the error-correcting performance of PEG-generated (or any other) LDPC codes and achieved results are provided. All these algorithms were used to create a PEG LDPC code which rivals the best-known PEG-generated LDPC code with similar parameters provided by one of the founders of LDPC codes. The methods for generating the mentioned error-correcting codes are described along with simulations which compare the error-correcting performance of the original codes generated by the PEG algorithm, the PEG codes processed by either CB1 or CB2 algorithm and also external PEG code published by one of the founders of LDPC codes

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