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On a class of nonlinear arithmetic codes that are easy to decode

David Mandelbaum
<span title="">1976</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fbkexx2qg5anljkxraobuswdq4" style="color: black;">Information and Control</a> </i> &nbsp;
Improved redundancy bounds for a class of previously constructed nonlinear arithmetic multiple-error correcting codes are derived. These codes are encoded by means of the Chinese remainder theorem.  ...  It is shown that they can be simply decoded by means of the Euclidean algorithm for a slight increase in redundancy. The decoding procedure is suitable for a general purpose computer.  ...  ERROR CORRECTION PROCEDURE The Euclidean algorithm can be used for simple decoding of these arithmetic codes in an analogous fashion to the decoding of finite-field Goppa codes (see Sugiyama et al., 1974a  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0019-9958(76)90325-9">doi:10.1016/s0019-9958(76)90325-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/l3pnmw2wijfkzjnbi4fzlytevy">fatcat:l3pnmw2wijfkzjnbi4fzlytevy</a> </span>
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Linear Codes, Target Function Classes, and Network Computing Capacity

Rathinakumar Appuswamy, Massimo Franceschetti, Nikhil Karamchandani, Kenneth Zeger
<span title="">2013</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
Computing capacity bounds and achievability are given with respect to these target function classes for network codes that use routing, linear coding, or nonlinear coding.  ...  We study the use of linear codes for network computing in single-receiver networks with various classes of target functions of the source messages.  ...  We now define a class of target functions that are easily shown to not be reducible.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tit.2013.2252052">doi:10.1109/tit.2013.2252052</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hxuncoc7e5h2tawjxvzqjhyqji">fatcat:hxuncoc7e5h2tawjxvzqjhyqji</a> </span>
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Secure Multipliers Resilient to Strong Fault-Injection Attacks Using Multilinear Arithmetic Codes

Zhen Wang, Mark Karpovsky, Ajay Joshi
<span title="">2012</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uqbr2omxsbdgtaxslmblka2nnu" style="color: black;">IEEE Transactions on Very Large Scale Integration (vlsi) Systems</a> </i> &nbsp;
Public-key cryptographic devices are vulnerable to fault-injection attacks. As countermeasures, a number of secure architectures based on linear and nonlinear error detecting codes were proposed.  ...  Under mild assumptions the proposed construction achieves near nonlinear code error detection performance at a lower cost (at most 50% area overhead) due to the fact that no nonlinear operations are needed  ...  ACKNOWLEDGMENT We would like to thank Prasanna Rao from Boston University and Professor Berk Sunar from Worcester Polytechnic Institute for their contributions and valuable advices for the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tvlsi.2011.2147340">doi:10.1109/tvlsi.2011.2147340</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gl2d6jgtfbfdtgvzqg3yconcam">fatcat:gl2d6jgtfbfdtgvzqg3yconcam</a> </span>
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Design of Reliable and Secure Multipliers by Multilinear Arithmetic Codes [chapter]

Zhen Wang, Mark Karpovsky, Berk Sunar, Ajay Joshi
<span title="">2009</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Under mild assumptions the proposed construction achieves near nonlinear code error detection performance at a lower cost (about 50% area overhead) due to the fact that no nonlinear operations are needed  ...  Here we propose a novel error detection technique, based on the random selection of linear arithmetic codes.  ...  It is easy to prove that the number of errors in this class is upper bounded by 2 k−1 p .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-11145-7_6">doi:10.1007/978-3-642-11145-7_6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qycqlpgvxrcmxaz7o53yosrsge">fatcat:qycqlpgvxrcmxaz7o53yosrsge</a> </span>
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Page 1908 of Mathematical Reviews Vol. 49, Issue 5 [page]

<span title="">1975</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Classes of arithmetic codes for the detection or correction of error patterns resulting from a single fault are presented. A decoding algorithm is provided.” P.  ...  Conversely, on the basis of a given FM or AM-FM signal it is easy to find a differential operator describing the corresponding reducible system.  ... 
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Near-Lossless Distributed Coding Of Hyperspectral Images Using A Statistical Model For Probability Estimation

Marco Grangetto, E. Magli, G. Olmo
<span title="2006-09-04">2006</span> <i title="Zenodo"> Zenodo </i> &nbsp;
Publication in the conference proceedings of EUSIPCO, Florence, Italy, 2006  ...  The value of a given LLR depends on the considered bit-plane and bit class. More details on this aspect are given in Sect. 2.5.  ...  CALIC performs prediction of a pixel value based on a causal neighborhood; the neighborhood is also used to compute contexts for arithmetic coding.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5281/zenodo.39711">doi:10.5281/zenodo.39711</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/swpcmtcz5rd4fkofuzrt65z6dq">fatcat:swpcmtcz5rd4fkofuzrt65z6dq</a> </span>
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Page 714 of Mathematical Reviews Vol. 53, Issue 2 [page]

<span title="">1977</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Mandelbaum, David 5154 On a class of nonlinear arithmetic codes that are easy to decode. Information and Control 30 (1976), no. 2, 151-168.  ...  Conditions are obtained on A that permit the correction of all errors E of weight <f in the received n-tuple U=Y+E, and a decoding method is presented.  ... 
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Context-based, adaptive, lossless image coding

X. Wu, N. Memon
<span title="">1997</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/dha3bnxvkvhslj74k332qi2ewa" style="color: black;">IEEE Transactions on Communications</a> </i> &nbsp;
A unique feature of CALIC is the use of a large number of modeling contexts (states) to condition a nonlinear predictor and adapt the predictor to varying source statistics.  ...  In this learning process, CALIC estimates only the expectation of prediction errors conditioned on a large number of different contexts rather than estimating a large number of conditional error probabilities  ...  Column 2 lists the actual bit rates obtained by CALIC with the CACM++ implementation of arithmetic coding. Results with a static Huffman code are on an average 3.5% worse.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/26.585919">doi:10.1109/26.585919</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2ywiwadqijbenedbpvysv6tu24">fatcat:2ywiwadqijbenedbpvysv6tu24</a> </span>
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Linear Codes, Target Function Classes, and Network Computing Capacity [article]

Rathinakumar Appuswamy, Massimo Franceschetti, Nikhil Karamchandani,, Kenneth Zeger
<span title="2011-05-07">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Computing capacity bounds and achievability are given with respect to these target function classes for network codes that use routing, linear coding, or nonlinear coding.  ...  We study the use of linear codes for network computing in single-receiver networks with various classes of target functions of the source messages.  ...  This reveals a limitation of linear codes compared to nonlinear ones for non-injective target functions that are not reducible.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1101.0085v3">arXiv:1101.0085v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vd7xk2m55jdkjhheb6xuzmidpm">fatcat:vd7xk2m55jdkjhheb6xuzmidpm</a> </span>
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Nonlinear network coding is necessary to combat general Byzantine attacks

Oliver Kosut, Lang Tong, David Tse
<span title="">2009</span> <i title="IEEE"> 2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton) </i> &nbsp;
We show that linear codes are inadequate in general, and a slight augmentation of the class of linear codes can increase throughput.  ...  We introduce a new class of codes known as bounded-linear that make use of distributions defined over bounded sets of integers subject to linear constraints using real arithmetic.  ...  We consider a class of nonlinear network codes that we denote linearplus, in which each edge may carry, in addition to a linear function of the message, a nonlinear function occupying an arbitrarily small  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/allerton.2009.5394912">doi:10.1109/allerton.2009.5394912</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gjoknvk3jjacbbnlm5vcgtjlye">fatcat:gjoknvk3jjacbbnlm5vcgtjlye</a> </span>
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Coding for a multiple-access channel

E.J. Weldon
<span title="">1978</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fbkexx2qg5anljkxraobuswdq4" style="color: black;">Information and Control</a> </i> &nbsp;
inferior to nonlinear codes on this channel is derived.  ...  for the noiseless channel; the minimum distance of these codes is lower bounded; three decoding procedures are analyzed.  ...  Yiu discovered the codes described in Construction 1 and made several other valuable contributions to this paper. Kai Ping, thanks. RECEIVED: June 18,, 1976; REVISED: March 11, 1977  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0019-9958(78)90312-1">doi:10.1016/s0019-9958(78)90312-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3ltm3erf6nczvdhggq4wbd4e24">fatcat:3ltm3erf6nczvdhggq4wbd4e24</a> </span>
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Designing communication systems via iterative improvement: error correction coding with Bayes decoder and codebook optimized for source symbol error [article]

Chai Wah Wu
<span title="2021-10-07">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We provide numerical experiments to show that they can be superior to classical linear block codes such as Hamming codes and decoding methods such as minimum distance decoding.  ...  For this metric, the positions of the bits are not relevant to the decoding, and in many noise models, not relevant to the BER either.  ...  One of the purposes of this section is to introduce an encoding that is easy to define and implement that appears to be near optimal.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.07429v5">arXiv:1805.07429v5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ziah6lqzufaylcvhyupzli7o5e">fatcat:ziah6lqzufaylcvhyupzli7o5e</a> </span>
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LP Decodable Permutation Codes based on Linearly Constrained Permutation Matrices [article]

Tadashi Wadayama, Manabu Hagiwara
<span title="2011-07-08">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A set of linearly constrained permutation matrices are proposed for constructing a class of permutation codes.  ...  It is demonstrated that the LP decoding performance of the proposed class of permutation codes is characterized by the vertices of the code polytope of the code.  ...  Han Vinck for directing the author's attention to the field of permutation codes. The first author also wishes to thank Dr. Jun Muramatsu for inspiring discussions on permutations.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1011.6441v3">arXiv:1011.6441v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fijfh5xubfgh5dhhlh6clvt2fu">fatcat:fijfh5xubfgh5dhhlh6clvt2fu</a> </span>
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Linear coding for network computing

Rathinakumar Appuswamy, Massimo Franceschetti, Nikhil Karamchandani, Kenneth Zeger
<span title="">2011</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ux5tf45dtrgptnezraiii73msm" style="color: black;">2011 IEEE International Symposium on Information Theory Proceedings</a> </i> &nbsp;
Computing capacity bounds are given with respect to these target function classes for network codes that use routing, linear coding, or nonlinear coding.  ...  We study the use of linear codes for network computing in single-receiver networks with various classes of target functions of the source messages.  ...  This reveals a limitation of linear codes compared to nonlinear ones for non-injective target functions that are not reducible.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/isit.2011.6034052">doi:10.1109/isit.2011.6034052</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/isit/AppuswamyFKZ11.html">dblp:conf/isit/AppuswamyFKZ11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/luaqitvwkna6nmghr5zi3m7bwa">fatcat:luaqitvwkna6nmghr5zi3m7bwa</a> </span>
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Nonlinear Transform Coding

Johannes Balle, Philip A. Chou, David Minnen, Saurabh Singh, Nick Johnston, Eirikur Agustsson, Sung Jin Hwang, George Toderici
<span title="">2020</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/aznf273kcvcbfjcdeghr3xjd6i" style="color: black;">IEEE Journal on Selected Topics in Signal Processing</a> </i> &nbsp;
We review a class of methods that can be collected under the name nonlinear transform coding (NTC), which over the past few years have become competitive with the best linear transform codecs for images  ...  and provide a direct comparison of a number of methods to parameterize the rate-distortion trade-off of nonlinear transforms, introducing a simplified one.  ...  ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers, as well as Lucas Theis and Aaron Wagner for their helpful comments on the draft and insightful discussions.  ... 
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