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Dual Lattice of ℤ-module Lattice Cover

Dual Lattice of ℤ-module Lattice

Formalized Mathematics
Volume 25 (2017): Issue 2 (July 2017)
By:
Open Access
|Sep 2017

References

  1. [1] Grzegorz Bancerek. Cardinal numbers. <em>Formalized Mathematics</em>, 1(<bold>2</bold>):377–382, 1990.
    Search in Google ScholarBack to article
  2. [2] Grzegorz Bancerek. Cardinal arithmetics. <em>Formalized Mathematics</em>, 1(<bold>3</bold>):543–547, 1990.
    Search in Google ScholarBack to article
  3. [3] Grzegorz Bancerek. The fundamental properties of natural numbers. <em>Formalized Mathematics</em>, 1(<bold>1</bold>):41–46, 1990.
    Search in Google ScholarBack to article
  4. [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. <em>Formalized Mathematics</em>, 1(<bold>1</bold>):107–114, 1990.
    Search in Google ScholarBack to article
  5. [5] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, <em>Intelligent Computer Mathematics</em>, volume 9150 of <em>Lecture Notes in Computer Science</em>, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: <a href="https://doi.org/10.1007/978-3-319-20615-817." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1007/978-3-319-20615-817.</a><pub-id pub-id-type="doi"><a href="https://doi.org/10.1007/978-3-319-20615-817" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1007/978-3-319-20615-817</a></pub-id>
    Open DOISearch in Google ScholarBack to article
  6. [6] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. <em>Formalized Mathematics</em>, 1(<bold>3</bold>):529–536, 1990.
    Search in Google ScholarBack to article
  7. [7] Czesław Byliński. Functions and their basic properties. <em>Formalized Mathematics</em>, 1(<bold>1</bold>): 55–65, 1990.
    Search in Google ScholarBack to article
  8. [8] Czesław Byliński. Functions from a set to a set. <em>Formalized Mathematics</em>, 1(<bold>1</bold>):153–164, 1990.
    Search in Google ScholarBack to article
  9. [9] Czesław Byliński. Some basic properties of sets. <em>Formalized Mathematics</em>, 1(<bold>1</bold>):47–53, 1990.
    Search in Google ScholarBack to article
  10. [10] Wolfgang Ebeling. <em>Lattices and Codes</em>. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1007/978-3-658-00360-9" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1007/978-3-658-00360-9</a></dgdoi:pub-id>
    Search in Google ScholarBack to article
  11. [11] Yuichi Futa and Yasunari Shidama. Lattice of ℤ-module. <em>Formalized Mathematics</em>, 24 (<bold>1</bold>):49–68, 2016. doi: <a href="https://doi.org/10.1515/forma-2016-0005." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1515/forma-2016-0005.</a><pub-id pub-id-type="doi"><a href="https://doi.org/10.1515/forma-2016-0005" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1515/forma-2016-0005</a></pub-id>
    Open DOISearch in Google ScholarBack to article
  12. [12] Yuichi Futa and Yasunari Shidama. Embedded lattice and properties of Gram matrix. <em>Formalized Mathematics</em>, 25(<bold>1</bold>):73–86, 2017. doi: <a href="https://doi.org/10.1515/forma-2017-0007." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1515/forma-2017-0007.</a><pub-id pub-id-type="doi"><a href="https://doi.org/10.1515/forma-2017-0007" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1515/forma-2017-0007</a></pub-id>
    Open DOISearch in Google ScholarBack to article
  13. [13] Yuichi Futa and Yasunari Shidama. Divisible ℤ-modules. <em>Formalized Mathematics</em>, 24 (<bold>1</bold>):37–47, 2016. doi: <a href="https://doi.org/10.1515/forma-2016-0004." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1515/forma-2016-0004.</a><pub-id pub-id-type="doi"><a href="https://doi.org/10.1515/forma-2016-0004" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1515/forma-2016-0004</a></pub-id>
    Open DOISearch in Google ScholarBack to article
  14. [14] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. <em>Formalized Mathematics</em>, 20(<bold>1</bold>):47–59, 2012. doi: <a href="https://doi.org/10.2478/v10037-012-0007-z." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10037-012-0007-z.</a><pub-id pub-id-type="doi"><a href="https://doi.org/10.2478/v10037-012-0007-z" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10037-012-0007-z</a></pub-id>
    Open DOISearch in Google ScholarBack to article
  15. [15] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. <em>Formalized Mathematics</em>, 20(<bold>3</bold>):205–214, 2012. doi: <a href="https://doi.org/10.2478/v10037-012-0024-y." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10037-012-0024-y.</a><pub-id pub-id-type="doi"><a href="https://doi.org/10.2478/v10037-012-0024-y" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10037-012-0024-y</a></pub-id>
    Open DOISearch in Google ScholarBack to article
  16. [16] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Matrix of ℤ-module. <em>Formalized Mathematics</em>, 23(<bold>1</bold>):29–49, 2015. doi: <a href="https://doi.org/10.2478/forma-2015-0003." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/forma-2015-0003.</a><pub-id pub-id-type="doi"><a href="https://doi.org/10.2478/forma-2015-0003" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/forma-2015-0003</a></pub-id>
    Open DOISearch in Google ScholarBack to article
  17. [17] Andrzej Kondracki. Basic properties of rational numbers. <em>Formalized Mathematics</em>, 1(<bold>5</bold>): 841–845, 1990.
    Search in Google ScholarBack to article
  18. [18] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. <em>Formalized Mathematics</em>, 1(<bold>2</bold>):335–342, 1990.
    Search in Google ScholarBack to article
  19. [19] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients. <em>Mathematische Annalen</em>, 261(4):515–534, 1982. doi: <a href="https://doi.org/10.1007/BF01457454." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1007/BF01457454.</a><pub-id pub-id-type="doi"><a href="https://doi.org/10.1007/BF01457454" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1007/BF01457454</a></pub-id>
    Open DOISearch in Google ScholarBack to article
  20. [20] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. <em>The International Series in Engineering and Computer Science</em>, 2002.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1007/978-1-4615-0897-7_8" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1007/978-1-4615-0897-7_8</a></dgdoi:pub-id>
    Search in Google ScholarBack to article
  21. [21] Andrzej Trybulec. Function domains and Frænkel operator. <em>Formalized Mathematics</em>, 1 (<bold>3</bold>):495–500, 1990.
    Search in Google ScholarBack to article
  22. [22] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. <em>Formalized Mathematics</em>, 1(<bold>3</bold>):569–573, 1990.
    Search in Google ScholarBack to article
  23. [23] Wojciech A. Trybulec. Pigeon hole principle. <em>Formalized Mathematics</em>, 1(<bold>3</bold>):575–579, 1990.
    Search in Google ScholarBack to article
  24. [24] Wojciech A. Trybulec. Linear combinations in vector space. <em>Formalized Mathematics</em>, 1 (<bold>5</bold>):877–882, 1990.
    Search in Google ScholarBack to article
  25. [25] Wojciech A. Trybulec. Basis of vector space. <em>Formalized Mathematics</em>, 1(<bold>5</bold>):883–885, 1990.
    Search in Google ScholarBack to article
  26. [26] Zinaida Trybulec. Properties of subsets. <em>Formalized Mathematics</em>, 1(<bold>1</bold>):67–71, 1990.
    Search in Google ScholarBack to article
  27. [27] Edmund Woronowicz. Relations and their basic properties. <em>Formalized Mathematics</em>, 1 (<bold>1</bold>):73–83, 1990.
    Search in Google ScholarBack to article
DOI: https://doi.org/10.1515/forma-2017-0015 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630
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Language: English
Page range: 157 - 169
Submitted on: Jun 27, 2017
Published on: Sep 23, 2017
Published by: University of Bialystok
In partnership with: Paradigm Publishing Services
Publication frequency: 1 times per year
Keywords:
ℤ-lattice,
dual lattice of ℤ-lattice,
dual basis of ℤ-lattice
Related subjects:
Mathematics,
General mathematics,
Computer sciences,
Computer sciences, other

© 2017 Yuichi Futa, Yasunari Shidama, published by University of Bialystok
This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 License.

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