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### On Singleton arrays and Cauchy matrices

T. Maruta
<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We construct all Singleton arrays for any finite field GF(q). There exist 449 -1) arrays for each GF(q) where #J is the Euler function.  ...  See [3, p. 3221 for the relation between Singleton arrays and MDS codes. Our main theorem is: Theorem 1.  ...  Let A be a (5, q -4)submatrix of a Singleton array S,. Considering the (q + 1, 5)-MDS code with the generator matrix G = [I ) A], A is a Cauchy matrix by Theorem 4 and the corollary of Theorem 2.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(90)90177-j">doi:10.1016/0012-365x(90)90177-j</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ailedudbnnhetnohyvekbcblqi">fatcat:ailedudbnnhetnohyvekbcblqi</a> </span>
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### Optimal quantum error correcting codes from absolutely maximally entangled states

Zahra Raissi, Christian Gogolin, Arnau Riera, Antonio Acín
<span title="2018-01-15">2018</span> <i title="IOP Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nzcpmvl7nzgojf4q47bpg2qrdu" style="color: black;">Journal of Physics A: Mathematical and Theoretical</a> </i> &nbsp;
Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states.  ...  Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes [[n,1,n/2]]_q for n even, saturating the quantum Singleton bound.  ...  To construct suitable A matrices, we now introduce the concept of so-called Singleton arrays.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1751-8121/aaa151">doi:10.1088/1751-8121/aaa151</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/76prbfmlqrc6pdxhhvxr3rvciq">fatcat:76prbfmlqrc6pdxhhvxr3rvciq</a> </span>
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### Page 6402 of Mathematical Reviews Vol. , Issue 91M [page]

<span title="">1991</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;
(J-OKAY) On Singleton arrays and Cauchy matrices. Discrete Math. 81 (1990), no. 1, 33-36.  ...  t, ¢, 7, ¢) and the excess of Hadamard matrices.  ...
<span class="external-identifiers"> </span>
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### Matricially free random variables

Romuald Lenczewski
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/6ijda3dw7fgqbgpjopyzft3hhy" style="color: black;">Journal of Functional Analysis</a> </i> &nbsp;
At the same time, the sums of matricially free random variables, called random pseudomatrices, are closely related to Gaussian random matrices.  ...  We show that the operatorial framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication.  ...  Acknowledgments The remarks and suggestions of the anonymous referees were very helpful in the preparation of the revised version of the manuscript.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jfa.2010.03.010">doi:10.1016/j.jfa.2010.03.010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rinxko7cqbaq5ezqgxzyzash7y">fatcat:rinxko7cqbaq5ezqgxzyzash7y</a> </span>
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R. Lenczewski
<span title="2014-03-24">2014</span> <i title="Oxford University Press (OUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/mjvsxmuxeje55kwqek2yas5f7i" style="color: black;">International mathematics research notices</a> </i> &nbsp;
This approach gives a block refinement of the fundamental result of Voiculescu on asymptotic freeness of independent Gaussian random matrices.  ...  We also demonstrate that the ensemble of independent matrices of this type is asymptotically conditionally free with respect to the pair of partial traces.  ...  We assign the color 1 to all singletons of depth one and the color 2 to all remaining singletons.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1093/imrn/rnu041">doi:10.1093/imrn/rnu041</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/racaaspypvhilkaxf75iknchfm">fatcat:racaaspypvhilkaxf75iknchfm</a> </span>
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### Restricted r-Stirling Numbers and their Combinatorial Applications [article]

Beáta Bényi, Miguel Méndez, José L. Ramírez, Tanay Wakhare
<span title="2018-11-30">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Finally, we introduce related (S,r) generalizations of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on generalized combinatorial sequences.  ...  Through several examples, we demonstrate that for some S the matrices and their inverses involve the enumeration sequences of several combinatorial objects.  ...  Acknowledgments The authors would like to thank the anonymous referees for reading carefully the paper and giving helpful comments and suggestions. The research of José L.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1811.12897v1">arXiv:1811.12897v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3z42nyvaijhqzadka3e6kh3gue">fatcat:3z42nyvaijhqzadka3e6kh3gue</a> </span>
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### Lightweight MDS Involution Matrices [chapter]

Siang Meng Sim, Khoongming Khoo, Frédérique Oggier, Thomas Peyrin
<span title="">2015</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;
By proving many new properties and equivalence classes for various MDS matrices constructions such as circulant, Hadamard, Cauchy and Hadamard-Cauchy, we exhibit new search algorithms that greatly reduce  ...  In this article, we provide new methods to look for lightweight MDS matrices, and in particular involutory ones.  ...  We also wish to thank Wang HuaXiong for providing useful and valuable suggestions.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-48116-5_23">doi:10.1007/978-3-662-48116-5_23</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/sm3rhkizyrg2hd7qppicvnflcy">fatcat:sm3rhkizyrg2hd7qppicvnflcy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190308083253/http://pdfs.semanticscholar.org/efde/5723052164442782048e5e73f3549a7a9b7d.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ef/de/efde5723052164442782048e5e73f3549a7a9b7d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-48116-5_23"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

### A construction of MDS 2D convolutional codes of rate 1/n based on superregular matrices

Joan-Josep Climent, Diego Napp, Carmen Perea, Raquel Pinto
<span title="">2012</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wsx3rzhpingfvewcn5nwhfkq3e" style="color: black;">Linear Algebra and its Applications</a> </i> &nbsp;
We first establish an upper bound on the maximum possible distance for these codes.  ...  convolutional codes with finite support with the same rate and degree.  ...  Acknowledgements The authors are very grateful to the anonymous reviewers for their comments and suggestions which led to significant improvements.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.laa.2012.02.032">doi:10.1016/j.laa.2012.02.032</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jhaf7yx2brg63cxfdohhd5zjuq">fatcat:jhaf7yx2brg63cxfdohhd5zjuq</a> </span>
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### On Perron complements of totally nonnegative matrices

Shaun M. Fallat, Michael Neumann
<span title="">2001</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wsx3rzhpingfvewcn5nwhfkq3e" style="color: black;">Linear Algebra and its Applications</a> </i> &nbsp;
of totally nonnegative matrices, when the Perron complement is totally nonnegative.  ...  We also demonstrate a quotient formula for Perron complements analogous to the so-called quotient formula for Schur complements, and verify an ordering between the Perron complement and Schur complement  ...  Ando for providing some useful comments on our manuscript that, in particular, lead to a shortening of the proof of Theorem 2.4. We also thank the referees for their helpful comments.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0024-3795(00)00312-8">doi:10.1016/s0024-3795(00)00312-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xxuz5sf4y5dqrfmy7jhnhplpem">fatcat:xxuz5sf4y5dqrfmy7jhnhplpem</a> </span>
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### Privacy-Preserving Data Storage in Cloud Using Array BP-XOR Codes

Yongge Wang
<span title="2015-10-01">2015</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ic7oylwbfvfw3hf4wbnzwcqeuq" style="color: black;">IEEE Transactions on Cloud Computing</a> </i> &nbsp;
(Cft + Cft+1 )−1 (Cft + Cft+2 )−1 · · · (Cft + Cft+k )−1 dispersal and retrieval protocols and Blomer et al [11] used Cauchy matrices to define XOR-based erasure-resilient coding By the fact  ...  Specif- 10 ically, they first design Reed-Solomon codes using Cauchy parity check matrix further. matrices over F2b .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tcc.2014.2344662">doi:10.1109/tcc.2014.2344662</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ti4keutlkfedtkh7ixb3nfhfdy">fatcat:ti4keutlkfedtkh7ixb3nfhfdy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20150405232305/http://webpages.uncc.edu/yonwang/papers/ecldpc.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c9/46/c9462553fc3e11349707bc99493fbcadb677e378.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tcc.2014.2344662"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

### A construction of $\mathbb{F}_2$-linear cyclic, MDS codes

Sara D. Cardell, ,Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Campinas, Brazil, Joan-Josep Climent, Daniel Panario, Brett Stevens, ,Departament de Matemàtiques, Universitat d'Alacant, Alacant, Spain, ,School of Mathematics and Statistics, Carleton University, Ottawa, Canada
<span title="">2019</span> <i title="American Institute of Mathematical Sciences (AIMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jbbcig2iqnbuvngg7spzuoozwu" style="color: black;">Advances in Mathematics of Communications</a> </i> &nbsp;
These codes are always cyclic and the density of the parity-check and the generator matrices decreases to 0 as n grows (for a fixed r).  ...  Then we apply a Zech logarithm to the elements of these sets and use the results to construct an index array which represents the parity-check matrix of the code.  ...  Apart from Vandermonde and Cauchy matrices, there is no further information about constructions of superregular matrices in the literature.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3934/amc.2020047">doi:10.3934/amc.2020047</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wxpeusoiurbgherekheykualwy">fatcat:wxpeusoiurbgherekheykualwy</a> </span>
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### Cryptographically significant mds matrices over finite fields: A brief survey and some generalized results

Kishan Chand Gupta, Sumit Kumar Pandey, Indranil Ghosh Ray, Susanta Samanta
<span title="">2019</span> <i title="American Institute of Mathematical Sciences (AIMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jbbcig2iqnbuvngg7spzuoozwu" style="color: black;">Advances in Mathematics of Communications</a> </i> &nbsp;
In this paper we provide a brief survey on cryptographically significant MDS matrices -a first to the best of our knowledge.  ...  We exhibit some deep and nontrivial interconnections between different constructions of MDS matrices.  ...  Rana Barua and Dr. Sanjay Bhattacherjee for providing several useful and valuable suggestions.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3934/amc.2019045">doi:10.3934/amc.2019045</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3wpn73syfnbwvjlrrssiyegpei">fatcat:3wpn73syfnbwvjlrrssiyegpei</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200507210224/https://www.aimsciences.org/article/exportPdf?id=8da3c52e-04aa-435a-bcec-0ad59daf7a55" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e3/86/e386c05008eecb6ba18d7fc4380d8186f10d2ca7.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3934/amc.2019045"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

### Duality Mapping for Schatten Matrix Norms

<span title="2021-05-10">2021</span> <i title="Informa UK Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/d36ykuhgj5cshamoqjf65gt55i" style="color: black;">Numerical Functional Analysis and Optimization</a> </i> &nbsp;
We prove in our main result that, for p 2 ð1, 1Þ, the duality mapping over the space of real-valued matrices with Schatten-p norm is a continuous and single-valued function and provide an explicit form  ...  Our approach is based on the analysis of the saturation of the H€ older inequality for Schatten norms.  ...  of admissible matrices B to just one ray.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1080/01630563.2021.1922438">doi:10.1080/01630563.2021.1922438</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qqvogpjpv5apvcrxtyecjcgxwi">fatcat:qqvogpjpv5apvcrxtyecjcgxwi</a> </span>
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### New construction for k-uniform and absolutely maximally entangled states [article]

Zahra Raissi, Adam Teixido, Christian Gogolin, Antonio Acin
<span title="2020-03-17">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
To our knowledge, the only known systematic construction of these quantum states is based on classical error correction codes.  ...  Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed.  ...  Singleton arrays S q represent a special case of Cauchy matrices [2, Chapter 11] and have the property that all its square sub-matrices are non-singular.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1910.12789v2">arXiv:1910.12789v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tzwwl7mqfne4nfa7tzdjngtzfm">fatcat:tzwwl7mqfne4nfa7tzdjngtzfm</a> </span>
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### Solvers for $\mathcal{O} (N)$ Electronic Structure in the Strong Scaling Limit

Nicolas Bock, Matt Challacombe, Laxmikant V. Kalé
<span title="">2016</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wbdvoluxebgjhn3wq3qsnldey4" style="color: black;">SIAM Journal on Scientific Computing</a> </i> &nbsp;
Comput. 35 C72, 2013], and involves a recursive, task-parallel algorithm, often employed by generalized N -Body solvers, to occlusion and culling of negligible products in the case of matrices with decay  ...  This result is achieved with a nested approach to Spectral Projection and the Sparse Approximate Matrix Multiply [Bock and Challacombe, SIAM J. Sci.  ...  NB and MC thank the LDRD program for funding this research under LDRD-ER grant 20110230ER.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/140974602">doi:10.1137/140974602</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vwyzeiwsazchtcfafbibq4sdt4">fatcat:vwyzeiwsazchtcfafbibq4sdt4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190217081922/http://pdfs.semanticscholar.org/045f/bb8c0c003f53e292899ca946f865d78e716d.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/04/5f/045fbb8c0c003f53e292899ca946f865d78e716d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/140974602"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>
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