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Last modified: Feb 2016.

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Publication List @ MathSciNet
Publication List @ DBLP


    Submitted Papers/ Work in Progress

  1. Shalin Shah, Parth Dave and Manish K Gupta, Computing real numbers using DNA self-assembly, 21 typed pages, Submitted to FNANO 2016, Feb 2016 arxiv pdf file.
  2. Krishna Gopal Benerjee and Manish K Gupta, Tradeoff for Heterogeneous Distributed Storage Systems between Storage and Repair Cost, 10 typed pages, Submitted to IEEE Transcations on Information Theory, Oct 2016 arxiv pdf file.
  3. Krishna Gopal Benerjee and Manish K Gupta, On Optimal Heterogeneous Regenerating Codes, 4 typed pages, Submitted to IEEE Communications Letters, Jan 2016 arxiv pdf file.
  4. Dixita Limbachiya and Manish K Gupta, Natural Data Storage: A Review on sending Information from now to then via Nature, 18 typed pages, 2016 arxiv pdf file.
  5. Srajan Paliwal, Saurabh Tiwary, Bhaskar Chaudhury and Manish K Gupta, Generating Binary Optimal Codes Using Heterogeneous Parallel Computingi, July 2015 arxiv pdf file.
  6. Nilay Chheda and Manish K Gupta, RNA as a Permutation, 9 typed pages, 2014 arxiv pdf file.
  7. Krishna Gopal Benerjee and Manish K Gupta, On Heterogeneous Regenerating Codes and Capacity of Distributed Storage Systems, 6 typed pages, Submitted to IEEE NetCod 2014 arxiv pdf file.
  8. On weak dress codes for cloud storage, 6 typed pages, (with Anupam Agrawal and Deepak Yadav), 2013 arxiv 1302.3681 pdf file.
  9. Enumerating some fractional repetition codes, 6 typed pages, (with Srijan Anil and T. Aaron Gulliver), 2013 arxiv 1302.3681 pdf file.
  10. Biospectrogram: a tool for spectral analysis of biological sequences, preprint, 2 typed pages (with Nilay Chhedda, Naman Turakhia, Ruchin Shah and Jigar Raisinghani), 2012. arxiv 1210.1472 pdf file.
  11. Modular arithmetic expressions and primality testing via DNA self-assembly, preprint, 2012, 15 typed pages (with Abhishek Chhajer, Sandeep Vasani, and Jaley Dholakiya) arxiv 1207.1161 pdf file.
  12. Self-assembly of DNA: Overview, challenges and opportunities, Systems and Synthetic Biology Journal, Springer, to be submitted, 2014, 12 typed pages
  13. Stochastic pi-calculus modeling of HIV-1 viral dynamics with induced NF-kB activation, to be submitted, 2014 (with A. Agrawal, R. Misra and M. Kukreja), 12 typed pages

    Papers Published (or In Press) in Refereed Journals

  14. On octonary codes and their covering radii, The Australasian Journal of Combinatorics, Volume 63(2), 2015, Pages 246--261 (with Manoj Raut) pdf file.
  15. On the covering radius of some modular codes, Advances in Mathematics of Communications, vol 8, No. 2, pp.129-137, 2014 (with C. Durairajan) arxiv 1206.3038 pdf file.
  16. On Some Quaternary Self-Orthogonal Codes, Ars. Comb., vol 85, pp. 193-209, Oct 2007.
        (with D.G. Glynn and T.A. Gulliver) Abstract pdf file ps file
  17. Linear Transformations on Codes, Discrete Mathematics, Vol 306, Issue 16, August 2006, pp 1871--1880
        (with D.G. Glynn and T.A. Gulliver) Abstract pdf file ps file (Ranked 3rd most viewed article [July-Sep 2006] at journal's web site)
  18. The Quest for Error-Correction in Biology, IEEE Engineering in Medicine and Biology Magazine Special Issue on Communication Theory, Coding Theory and Molecular Biology, vol 25, Issue 1, pp 46-53, January/February 2006 (Invited Paper).
    Abstract Pub Med Entry (PMID:16485391) pdf file
    Full Issue of IEEE EMB on Coding Theory, Molecular Biology and Communication Theory is here on IEEE Explore
  19. On Linear Codes over Z_2^s , Designs Codes and Cryptography, vol 36, pp 227-244, September 2005.
        (with M.C. Bhandari and A.K. Lal) Abstract pdf file ps file (Ranked 4th most viewed article at journal's web site)
  20. On Z_4 Codes Satisfying the Chain Condition, The Australasian Journal of Combinatorics, vol 31, pp 263-272, Feb 2005. pdf file ps file
  21. Generalized Weights for Codes Z_k, The Australasian Journal of Combinatorics, vol 31, pp 231-248, Feb 2005.
        (with S.T. Dougherty and K. Shiromoto) pdf file ps file
  22. On Senary Simplex Codes, AAECC-14, Lecture Notes in Computer Science, vol 2227,   pp 112 - 121, November 2001.
        (with D.G. Glynn and T.A. Gulliver) Abstract pdf file ps file
  23. On Z_4 Simplex Codes and their Gray Images, AAECC-13, Lecture Notes in Computer Science, vol 1719,    pp 170 - 180, November 1999.
        (with M.C. Bhandari and A.K. Lal) Abstract
  24. Some Results on NQR Codes, Designs Codes and Cryptography, vol 16, Issue 1, pp 5-9, January 1999.
        (with M.C. Bhandari and A.K. Lal) Abstract

    25. Conference

  25. On Optimal Family of Codes for Archival DNA Storage, Proc. of the seventh International Workshop on Signal Design and its Applications in Communications (IWSDA 15), September 13-18, 2015, pp. 143-147, IISc Bangalore
    Abstract arxiv pdf file (with Dixita Limbachiya, Vijay Dhameliya and Madhav Khakhar)
  26. On Dress Codes with Flowers, Proc. of the seventh International Workshop on Signal Design and its Applications in Communications (IWSDA 15), September 13-18, 2015, pp. 108-112, IISc Bangalore
    Abstract arxiv pdf file (with Krishna Gopal Benerjee)
  27. Mathematical and Computational Opportunities in DNA Computing, NNMCB Symposium on Mathematical and Computational Biology, March 21-22, 2015, IIT Gandhinagar (Invited Speaker)
    Abstract
  28. How to store Elephants? Indo-US conference and workshop on synthetic and systems biology, Nov 9-12, 2014, JNU, New Delhi (Invited Speaker)
    Abstract
  29. Manish K Gupta, Self-assembly of DNA: Overview, Opportunities and Challenges, Biodesign India 1.0 symposium on synthetic biology, Oct 7-9, 2010, Trivandrum, Kerala ( Invited Speaker )
    Abstract
  30. Muskan Kukreja and Manish K Gupta, Stochastic pi-calculus simulations of few RNA-interference models and its dynamics and efficacy as anti-viral treatment against HIV-1, Proc. of 2nd International Meeting on Membrane Computing and Biologically Inspired Process Calculii, 3 - 4 September 2008, Romania, pp. 131-144, Cuza University of Iasi
    Abstract pdf Supplementary Material
  31. Error-correcting mechanisms in DNA self-assembly, preprint (with Navin Kashyap)
    Abstract
  32. Biological Coding Theory I: Introduction
    Biological Coding Theory II: Coding Theory for DNA Computing and Microarray
    Biological Coding Theory III:Coding Theory for DNA Self Assembly
    COM2MAC Workshop on Biological Coding Theory COM2MAC Center, Postech University,
    Pohang, South Korea, 27-28 June 2005 (Invited Keynote Speaker for 3 hours lectures)
  33. On Modular Gray Map, Special Session on Algebraic Coding Theory, Central Section Meeting #995, American Mathematical Society, Athens,OH, 26-27 March 2004 (Invited Speaker)
    34. Abstract (pdf)
  34. Biological Coding Theory: The Emerging Paradigm? DIMACS Workshop on Algebraic Coding Theory and Information Theory, DIMACS Center, Rutgers University, Piscataway, NJ, 15-18 December 2003 (Invited Speaker)
    35. Abstract (pdf)
  35. Coding Theory, Biology and Quantum Computers Conference and workshop on Coding Theory and Quantum Computing, University of Virginia Charlottesville, Virginia 20-24 May 2003
    36. Abstract (pdf)
  36. On Quaternary MacDonald Codes, Proc. of IEEE International Conf. on Information Technology: Coding and Computing ITCC 2003, April 28-30, 2003 Las Vegas, Nevada, pp. 212--215,
        (with Charles J. Colbourn) pdf file
  37. On Senary Simplex Codes, Proc. AAECC 14, pp. 17--18, Nov. 2001. (with D.G. Glynn and T.A. Gulliver)
  38. On Z_4 Simplex Codes and Their Gray Images, Proc. AAECC 13, Nov.1999. (with M.C. Bhandari and A.K. Lal)
  39. On Some Modular Linear Codes, Workshop on Coding and Cryptography, September 10--13 2001, Second part of Inagural programe CODING THEORY AND DATA INTEGRITY of Institue for Mathematical Sciences, National University of Singapore, Singapore. pdf file
  40. On Z_4 Codes Satisfying the Chain Condition, 25^th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing, December 4--8, 2000, University of Canterbury, Christchurch, NEW ZEALAND,
  41. Some results on NQR codes, Proc. of AMS Joint Math. Meeting #930, Baltimore, MD Jan.1998. (with M.C. Bhandari and A.K. Lal)

    42. Poster Papers

  42. Dixita Limbachiya, Krishna Gopal Benerjee, Bansari Rao and Manish K. Gupta, Computational Algebraic Perspectives of DNA Codes, International School on Computer Algebra (COCOA 2016) IIT Gandhinagar, India, Poster Abstract Feb 2016.
  43. Shikhar Kumar Gupta, Foram Joshi, Dixita Limbachiya and Manish K. Gupta, 3DNA: A tool for Lego Modeling, FNANO 2014 Conference Poster Abstract arxiv pdf file.
    3DNA Main Page
  44. Shalin Shah and Dixita Limbachiya and Manish K. Gupta, DNA Cloud: A potential tool for storing big data on DNA, FNANO 2014 Conference Poster Abstract arxiv pdf file.
    DNA Cloud Main Page
  45. Mit Sheth, Krishna Gopal Benerjee and Manish K. Gupta: Quorum sensing for Regenerating codes in distributed storage. Proc. of IEEE/ACM COMSNETS 2014: Pages 1-4.
    arxiv pdf file
  46. Arnav Goyal, Dixita Limbachiya, Shikhar Kumar Gupta, Foram Joshi, Sushant Pritmani, Akshita Sahai and Manish K. Gupta, DNA Pen: A Tool for Drawing on a Molecular Canvas, DNA 19 Conference 2013, Arizona, USA, 2013. Paper (pdf) DNA Pen Main Page (Supported by Microsoft Research India Travel Grant)
  47. Krishna Gopal Benerjee, Manish K. Gupta and Nikhil Agrawal, Reconstruction and Repair Degree of Fractional Repetition Codes, IEEE NetCod 2013 Conference, Calgary, Canada, 2013. Paper (pdf) Youtube Video
  48. Anshul Chaurasia, Sudhanshu Dwivedi, Prateek Jain and Manish K Gupta, XTile: An Error Correction Package for DNA Self-Assembly, in proceedings of 6th Annual Conference on Foundations of Nanoscience (FNANO 09): Self-Assembled Architectures and Devices, Salt Lake City, Utah, U.S.A., 20th-24th April 2009, pp. 225 - 229. Abstract Poster (pdf) Paper (pdf) XTile Main Page
    49. (Supported by Microsoft Research India Travel Grant)
  49. Stochastic pi-simulations of few HIV-1 models, poster at Computational and Systems Biology course at The Microsoft Research-University of Trento, Center of Computational and Systems Biology, Poster No 18, March 10-14, 2008 (with Muskan Kukreja)
    50. (Partially supported by Microsoft Research Italy)

    Online Wiki

  50. "DNA origami of Taj Mahal", International Bio-molecular Design Competition (Biomod), Nov 5, 2011, Harvard University, (with Mayank Kandpal, Avinash Parida and Taslimarif Saiyed), Biomod Wiki Page See http://www.biomod.net (Supported by Microsoft Research India and Kuliza Technologies)

    Books / Monographs/ Thesis

  51. D.G. Glynn, T.A. Gulliver, J.G. Maks and M.K.Gupta, "The Geometry of Additive Quantum Codes," submitted to Springer-Verlag Lacture Notes, Feb 2007 (An Unpublished Research Monograph).

  52.   On Some Linear Codes over Z_2^s, PhD Thesis, Department of Mathematics,  IIT Kanpur, India (July 2000), 98 pages
                                    (Advisors: M.C. Bhandari and A.K. Lal) pdf file of PhD Thesis | ps file of PhD Thesis


  53. A Problem Book in Mathematics for IIT JEE, In Prepration.    
    (with Shbuhra Gupta)
    About 350 original problems were invented by me for the joint entrance examination of Indian Institute of Technology, India. I am giving them a shape of a book with the help of Shubhra Gupta. A draft copy will appear here soon.

    54. Reviews

    ACM Computing Reviews

    For a recent list see
    ACM Computing Reviewes Featured Reviewer

  54. Book Review on "Bioinspired computation in combinatorial optimization : algorithms and their computational complexityi, Neumann F., Witt C. 2010, NY 216 pp", ACM Computing Reviews, May 19, 2011 (CR139071)
  55. Book Review on "Design Theory, Linder C. & Rodger C., Chapman & Hall/CRC 2008, 272 pp.", ACM Computing Reviews, April 10, 2009 (CR136674)
  56. Review on "Extracting constrained 2-interval subsets in 2-interval sets", ACM Computing Reviews, Feb 29, 2008 ( CR135317 )
  57. Review on "An accurate nonuniform fourier transform for SPRITE magnetic resonance imaging data", ACM Computing Reviews, November 16, 2007
    ( CR134953)
  58. Review on "Why simulating evolutionary processes is just as interesting as applying them", ACM Computing Reviews, March 9, 2006
    ( CR132549)
  59. Review on "L(2,1)-labeling of strong products of cycles", ACM Computing Reviews, Jan 25, 2006 ( CR132358)

    AMS Mathematical Reviews

  60. MR2494853 Albuquerque, C. D.; Palazzo, R., Jr.; Silva, E. B. Topological quantum codes on compact surfaces with genus $g\geq2$. J. Math. Phys. 50 (2009), no. 2, 023513, 20 pp. (Reviewer: Manish Kumar Gupta) 81P70 (81P68 94B60)
  61. MR2488832 (2009m:94096) de Albuquerque, Clarice Dias; Palazzo, Reginaldo, Júnior; da Silva, Eduardo Brandani On toric quantum codes. Int. J. Pure Appl. Math. 50 (2009), no. 2, 221--226. (Reviewer: Manish Kumar Gupta) 94B60 (05A18 11D09)
  62. MR2492562 (2010a:92006) Taylor, Stephanie R.; Gunawan, Rudiyanto; Petzold, Linda R.; Doyle, Francis J., III Sensitivity measures for oscillating systems: application to mammalian circadian gene network. IEEE Trans. Circuits Syst. I. Regul. Pap. 2008, Special issue on systems biology, 177--188. (Reviewer: Manish Kumar Gupta) 92B05 (34C15 92C30 92D10)
  63. MR2437821 (2009m:68089) Wiedermann, Jiří; Pardubská, Dana Wireless mobile computing and its links to descriptive complexity. Internat. J. Found. Comput. Sci. 19 (2008), no. 4, 887--913. (Reviewer: Manish Kumar Gupta) 68Q10 (68M10 68Q05 68Q15 68Q25)
  64. MR2435631 (2009j:92001) Fages, François; Soliman, Sylvain Abstract interpretation and types for systems biology. Theoret. Comput. Sci. 403 (2008), no. 1, 52--70. (Reviewer: Manish Kumar Gupta) 92B05 (92C35)
  65. MR2431821 (2009g:94139) Michel, Christian J.; Pirillo, Giuseppe; Pirillo, Mario A. A relation between trinucleotide comma-free codes and trinucleotide circular codes. Theoret. Comput. Sci. 401 (2008), no. 1-3, 17--26. (Reviewer: Manish Kumar Gupta) 94B25 (68R15 92D10)
  66. MR2415989 (2009j:94002) Calamoneri, Tiziana; Clementi, Andrea E. F.; Di Ianni, Miriam; Lauria, Massimo; Monti, Angelo; Silvestri, Riccardo Minimum-energy broadcast and disk cover in grid wireless networks. Theoret. Comput. Sci. 399 (2008), no. 1-2, 38--53. (Reviewer: Manish Kumar Gupta) 94A05 (05C90 68M10)
  67. MR2396791 (2009i:92038) Panovska, Jasmina; Byrne, Helen M.; Maini, Philip K. A theoretical study of the response of vascular tumours to different types of chemotherapy. Math. Comput. Modelling 47 (2008), no. 5-6, 560--579. (Reviewer: Manish Kumar Gupta) 92C50 (35K50 35K57 92C37)
  68. MR2385684 (2009i:92034) Hogea, Cosmina; Davatzikos, Christos; Biros, George An image-driven parameter estimation problem for a reaction-diffusion glioma growth model with mass effects. J. Math. Biol. 56 (2008), no. 6, 793--825. (Reviewer: Manish Kumar Gupta) 92C50 (35K57 74L15 92C55)
  69. MR2373031 (2009d:92015) Siebert, Tobias; Rode, Christian; Herzog, Walter; Till, Olaf; Blickhan, Reinhard Nonlinearities make a difference: comparison of two common Hill-type models with real muscle. Biol. Cybernet. 98 (2008), no. 2, 133--143. (Reviewer: Manish Kumar Gupta) 92C20 (74L15 92C10)
  70. MR2376176 (2009a:94058) Ma, Yuefei The asymptotic probability distribution of the relative distance of additive quantum codes. J. Math. Anal. Appl. 340 (2008), no. 1, 550--557. (Reviewer: Manish Kumar Gupta) 94B60 (81P68 94B65)
  71. MR2419577 (2009i:94003) Andrews, Matthew A survey of scheduling theory in wireless data networks. Wireless communications, 1--17, IMA Vol. Math. Appl., 143, Springer, New York, 2007. (Reviewer: Manish Kumar Gupta) 94A05 (68M20 94A40 94C99)
  72. MR2336381 (2008j:90109) Xiong, Yu; Ma, Jun; Shen, Hao On optimal codes with $w$-identifiable parent property. Des. Codes Cryptogr. 45 (2007), no. 1, 65--90. (Reviewer: Manish Kumar Gupta) 90C27 (68R10 90C47 94A62)
  73. MR2332313 (2008f:94041) Xu, Wen-Qing; Golomb, Solomon W. Optimal interleaving schemes for correcting two-dimensional cluster errors. Discrete Appl. Math. 155 (2007), no. 10, 1200--1212. (Reviewer: Manish Kumar Gupta) 94B60 (94B75)
  74. MR2324286 (2008c:94044) Abualrub, Taher; Siap, Irfan Reversible cyclic codes over $Z\sb 4$. Australas. J. Combin. 38 (2007), 195--205. (Reviewer: Manish Kumar Gupta) 94B15
  75. MR2277141 (2008b:92077) Daskalakis, Constantinos; Mossel, Elchanan; Roch, Sébastien Optimal phylogenetic reconstruction. STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 159--168, ACM, New York, 2006. (Reviewer: Manish Kumar Gupta) 92D15 (68Q25)
  76. MR2271506 (2008b:92085) Dinu, Liviu P.; Sgarro, Andrea A low-complexity distance for DNA strings. Fund. Inform. 73 (2006), no. 3, 361--372. (Reviewer: Manish Kumar Gupta) 92D20
  77. MR2262626 (2008a:94155) Mizuki, Takaaki; Uchiike, Fumishige; Sone, Hideaki Securely computing XOR with 10 cards. Australas. J. Combin. 36 (2006), 279--293. (Reviewer: Manish Kumar Gupta) 94A62
  78. MR2252641 (2007m:68144) Bruschi, Danilo; Pighizzini, Giovanni String distances and intrusion detection: bridging the gap between formal languages and computer security. Theor. Inform. Appl. 40 (2006), no. 2, 303--313. (Reviewer: Manish Kumar Gupta) 68Q45 (68Q15)
  79. MR2237511 (2007e:68035) Bottoni, Paolo; Labella, Anna; Manca, Vincenzo; Mitrana, Victor Superposition based on Watson-Crick-like complementarity. Theory Comput. Syst. 39 (2006), no. 4, 503--524. (Reviewer: Manish Kumar Gupta) 68Q45 (68Q05 92D10 92D20)
  80. MR2235478 (2008c:94006) Groisser, David Some differential-geometric remarks on a method for minimizing constrained functionals of matrix-valued functions. J. Math. Imaging Vision 24 (2006), no. 3, 349--358. (Reviewer: Manish Kumar Gupta) 94A08 (58E30 92C55)
  81. MR2233946 (2007f:94072) Chang, Chin-Chen; Lee, Jung-San An anonymous and flexible $t$-out-of-$n$ electronic voting scheme. J. Discrete Math. Sci. Cryptogr. 9 (2006), no. 1, 133--151. (Reviewer: Manish Kumar Gupta) 94A62 (91B12)
  82. MR2232279 (2007e:94025) Lu, Bing; Du, Hongwei; Jia, Xiaohua; Xu, Yinfeng; Zhu, Binhai On a minimum linear classification problem. J. Global Optim. 35 (2006), no. 1, 103--109. (Reviewer: Manish Kumar Gupta) 94A13
  83. MR2188057 (2006m:92020) Izhikevich, Eugene M. Polychronization: computation with spikes. Neural Comput. 18 (2006), no. 2, 245--282. (Reviewer: Manish Kumar Gupta) 92C20 (68T05)
  84. MR2175275 (2007a:94217) Zhang, Fangguo; Safavi-Naini, Reihaneh; Susilo, Willy Attack on Han et al.'s ID-based confirmer (undeniable) signature at ACM-EC'03. Appl. Math. Comput. 170 (2005), no. 2, 1166--1169. (Reviewer: Manish Kumar Gupta) 94A60 (94A62)
  85. MR2174339 (2006m:68073) Kari, Lila; Konstantinidis, Stavros; Sosà k, Petr Bond-free languages: formalizations, maximality and construction methods. Internat. J. Found. Comput. Sci. 16 (2005), no. 5, 1039--1070. (Reviewer: Manish Kumar Gupta) 68Q45
  86. MR2166915 (2006m:92096) Wang, Lusheng; Zhang, Fuji; Zhao, Hao On the ordering of benzenoid chains and cyclo-polyphenacenes with respect to their numbers of Clar aromatic sextets. J. Math. Chem. 38 (2005), no. 2, 293--309. (Reviewer: Manish Kumar Gupta) 92E10
    Those who have access to AMS mathscinet can directly access an updated list by clicking here

    Some Invited Presentations

  87. A Story of Nature: Emerging Trends in Life Sciences, Department of Biotechnology and Bioinformatics, Jaypee University of Information Technology, Waknaghat, Solan, HP, November 13, 2014.
  88. A Story of Natural Engineering, DST- Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University (BHU), May 26, 2014
  89. How to code the cloud using weak dress codes?, DST- Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University (BHU), May 27, 2014
  90. A Story of Natural Engineering, Samsung Advanced Institute of Technology (SAIT) Lab, Jan 10, 2014, Bangalore
  91. Calculus, Where are you? Pedagogy Sessions in Calculus, Gujarat Technological University, June 28, 2012
  92. Mathematics of Natural Information Processing, Scope of Research in Applied Mathematics, Nirma University, Ahmedabad, India, April 17, 2010
  93. How to count with DNA? International Institute of Information Technology, Bangalore, India, October 18, 2006
  94. How to count with DNA? Birla Institute of Technology and Science, Pilani, India, October 04, 2006
  95. How to count with DNA? Department of Mathematics, Indian Institute of Technology, Kanpur, India, September 20, 2006
  96. Structural Coding and Information Theory for Biological Systems California Institute for Quantitative Biomedical Research (QB3), University of California, San Francisco, CA, USA, May 8, 2006.
    97. Abstract
  97. Information Processing in Biology: Algorithmic Self-Assembly, Opportunities and Challenges for Coding Department of Electrical and Computer Engineering, Ohio State University, OH, USA, February 8, 2005.
    98. Abstract
  98. Codes and Biology: A Second Emerging Bridge, Department of Mathematics, San Diego State University, CA, USA, September 25, 2003.
    99. Abstract (pdf)
  99. Quantum Codes, Department of Mathematics, Indian Institute of Technology, Kanpur, India, October 4, 2001.
  100. A Story of Quantum Codes, Department of Mathematics, National University of Singapore, Singapore, August 31, 2001.


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