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Error-correcting codes from graphs

Vladimir D. Tonchev
2002 Discrete Mathematics  
A relation between the linear binary codes derived from graphs and a class of quantum error-correcting codes is also discussed.  ...  Some interesting codes are obtainable from graphs with high degree of symmetry, such as strongly regular graphs.  ...  Since 13 = 14 × 13 × 7 2 × 49 ; The class of quantum-error-correcting codes of length n obtained from binary codes from A via Theorem 3.1 contains codes that can correct [(d − 1)=2] errors where d¿0:  ... 
doi:10.1016/s0012-365x(02)00513-7 fatcat:nt63565kpbgrrg6v3xjmrphauq

Error-correcting codes from k-resolving sets

Robert Bailey, González Ismael Yero
2018 Discussiones Mathematicae Graph Theory  
We demonstrate a construction of error-correcting codes from graphs by means of k-resolving sets, and present a decoding algorithm which makes use of covering designs.  ...  Along the way, we determine the k-metric dimension of grid graphs (i.e. Cartesian products of paths). MSC 2010: 05C12, 94B25 (primary); 05B40, 94B35 (secondary).  ...  Introduction Error-correcting codes Error-correcting codes are applied to the accurate transmission and storage of data.  ... 
doi:10.7151/dmgt.2087 fatcat:subevll35rgnfegy7cskuhlubq

Reconstructing Extended Perfect Binary One-Error-Correcting Codes From Their Minimum Distance Graphs

Ivan Yu. Mogilnykh, Patric R. J. Ostergard, Olli Pottonen, Faina I. Solov'eva
2009 IEEE Transactions on Information Theory  
A constructive proof for reconstructibility of an extended perfect binary one-error-correcting code from its minimum distance graph is presented.  ...  Consequently, inequivalent such codes have nonisomorphic minimum distance graphs.  ...  RECONSTRUCTING 1-PERFECT CODES We will handle the problem of reconstructing a 1-perfect code from its minimum distance graph by reducing it to the problem of reconstructing an extended 1-perfect code from  ... 
doi:10.1109/tit.2009.2018338 fatcat:gm5hqvesz5bbnizsmefgubov44

On Hardness of Approximation of Parameterized Set Cover and Label Cover: Threshold Graphs from Error Correcting Codes [article]

Karthik C. S., Inbal Livni-Navon
2020 arXiv   pre-print
Our reduction follows along the exact same lines, except that we generate the threshold graphs specified by Lin simply using the basic properties of the error correcting code C.  ...  In this paper, we prove a more scalable version of his result: given any error correcting code C over alphabet [q], rate ρ, and relative distance δ, we use C to create a reduction from the (k,k+1)-SetCover  ...  For every error correcting code C ⊆ Σ ℓ of relative distance δ we have, Definition 3.1 (Threshold Graphs from Error Correcting Codes).  ... 
arXiv:2009.02778v1 fatcat:kmtuu5bbp5filb7k4gragi2na4

Generating Burst-Error Correcting Codes from Orthogonal Latin Square Codes -- A Graph Theoretic Approach

Rudrajit Datta, Nur A. Touba
2011 2011 IEEE International Symposium on Defect and Fault Tolerance in VLSI and Nanotechnology Systems  
The method discussed in this paper models it as a graph coloring problem where the goal is to resolve conflicts in the existing OLS code in order for it to correct burst-errors.  ...  The final OLS code after reordering and/or reorganizing would be capable of correcting burst-errors of specific length in addition to its original error correction capabilities.  ...  Note that OLS coding does not need to generate a syndrome, but can "correct" errors directly from majority voting. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = t As an example, a single error correcting OLS code  ... 
doi:10.1109/dft.2011.3 dblp:conf/dft/DattaT11a fatcat:mxe7p36xazdd3isnugzgx6ysle

Linear-time encodable and decodable error-correcting codes

Daniel A. Spielman
1995 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing - STOC '95  
We present a new class of asymptotically good, linear error-correcting codes. These codes can be both encoded and decoded in linear time.  ...  We present both randomized and explicit constructions of these codes.  ...  From error reduction to error correction Our error-correcting codes are constructed from codes that we call error-reduction codes.  ... 
doi:10.1145/225058.225165 dblp:conf/stoc/Spielman95 fatcat:gdyukhp7yfdfrdbbpxl7fx7b4e

Linear-time encodable and decodable error-correcting codes

D.A. Spielman
1996 IEEE Transactions on Information Theory  
We present a new class of asymptotically good, linear error-correcting codes. These codes can be both encoded and decoded in linear time.  ...  We present both randomized and explicit constructions of these codes.  ...  From error reduction to error correction Our error-correcting codes are constructed from codes that we call error-reduction codes.  ... 
doi:10.1109/18.556668 fatcat:cqswhlwehngfhfjlcnoup3qah4

Graph-Theoretic Approach to Quantum Error Correction [article]

Robert Vandermolen, Duncan Wright
2022 arXiv   pre-print
These codes arise from an original graph-theoretic representation of sets of quantum errors.  ...  In this new framework, we represent the algebraic conditions for error correction in terms of edge avoidance between graphs providing a visual representation of the interplay between errors and error correcting  ...  Then any stabilizer code for S is an error-correcting code which will correct any error from E if and only if every conjugate error E ∈ E 2 satisfies either E ∈ S or E ∈ S ⊥ .  ... 
arXiv:2110.08414v2 fatcat:aeynp2dljbbf3knckd3ljydrjy

On the Number of Errors Correctable with Codes on Graphs

Alexander Barg, Arya Mazumdar
2011 IEEE Transactions on Information Theory  
We show that these codes can correct a linearly growing number of errors under simple iterative decoding algorithms.  ...  We study ensembles of codes on graphs (generalized low-density parity-check, or LDPC codes) constructed from random graphs and fixed local constrained codes, and their extension to codes on hypergraphs  ...  Error correction with codes on graphs has been studied along two lines, namely, by computing the average number of errors correctable with some decoding algorithm by codes from a certain ensemble of graph  ... 
doi:10.1109/tit.2010.2094812 fatcat:xkqjpgpllfgzvdzelq57syyloq

Quantum codes from classical graphical models [article]

Joschka Roffe and Stefan Zohren and Dominic Horsman and Nicholas Chancellor
2018 arXiv   pre-print
More importantly, this allows for a collaborative interplay where one can design new quantum error correction codes derived from classical codes.  ...  We introduce a new graphical framework for designing quantum error correction codes based on classical principles.  ...  For correction codes, the task of decoding involves deducing the most likely error from a set of parity check measurements.  ... 
arXiv:1804.07653v1 fatcat:e6cfvhmibfemlkc5xmpell5utu

Analysis of Iterated Hard Decision Decoding of Product Codes with Reed-Solomon Component Codes

Jorn Justesen, Tom Hoholdt
2007 2007 IEEE Information Theory Workshop  
The row codes correct all error patterns of weight at most T 1 , the column codes correct T 2 errors. Initially we let T 1 = T 2 = T .  ...  GRAPH CODES For a more general graph, consider the bipartite graph derived from a projective plane [5] .  ... 
doi:10.1109/itw.2007.4313069 fatcat:rkmtvxihrjhtbkcvz7cgvmztce

On the number of errors correctable with codes on graphs

Alexander Barg, Arya Mazumdar
2009 2009 IEEE International Symposium on Information Theory  
We estimate the number of errors corrected by two different ensembles of codes on graphs (generalized LDPC codes), namely codes on regular bipartite graphs and their extension to hypergraphs.  ...  Error correction with codes on graphs has been studied along two lines, namely, by computing the average number of errors correctable with some decoding algorithm by codes from a certain ensemble of graph  ...  Our results imply that codes on bipartite graphs and hypergraphs constructed from local codes with small distance can correct a positive proportion of errors under iterative decoding. II.  ... 
doi:10.1109/isit.2009.5206079 dblp:conf/isit/BargM09 fatcat:pm4mmtqlbjeyjk5quyqhosgz3q

Graph based linear error correcting codes [article]

Monika Polak, Eustrat Zhupa
2016 arXiv   pre-print
In this article we present a construction of error correcting codes, that have representation as very sparse matrices and belong to the class of Low Density Parity Check Codes.  ...  We also show results of computer simulations of BER (bit error rate) of the obtained codes in order to compare them with other known LDPC codes.  ...  In order to minimize the number of errors during transmission, we can use error correcting codes.  ... 
arXiv:1612.03279v1 fatcat:s2hbq2ln7ndy5nkj2grr4b5zmu

Quantum error-correcting codes associated with graphs

D. Schlingemann, R. F. Werner
2001 Physical Review A. Atomic, Molecular, and Optical Physics  
We present a construction scheme for quantum error correcting codes. The basic ingredients are a graph and a finite abelian group, from which the code can explicitly be obtained.  ...  We prove necessary and sufficient conditions for the graph such that the resulting code corrects a certain number of errors.  ...  INTRODUCTION From the beginning of Quantum Information Theory it was recognized that error correcting codes play a crucial role.  ... 
doi:10.1103/physreva.65.012308 fatcat:umipescac5dt3iqrul73xqwzei

Guaranteed error correction capability of codes on graphs

S.K. Chilappagari, B. Vasic, M.W. Marcellin
2009 2009 Information Theory and Applications Workshop  
A summary of recent results relating the column weight and girth of the Tanner graph to the guaranteed error correction capability is provided.  ...  The guaranteed error correction capability of left regular LDPC codes under different hard decision decision algorithms is investigated.  ...  The following fact from [5] relates the expansion and error correction capability of an (n, d v , d c ) LDPC code with Tanner graph G when decoded using the parallel bit flipping decoding algorithm.  ... 
doi:10.1109/ita.2009.5044922 fatcat:hlcahfbdozdodka7axc4472oj4
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