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On Quasi-Abelian Complementary Dual Codes

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Coding Theory and Applications (ICMCTA 2017)

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Abstract

Linear codes that meet their dual trivially are also known as linear complementary dual codes. Quasi-abelian complementary dual codes are characterized using a known decomposition of a semisimple group algebra. Consequently, enumeration of such codes are obtained. More explicit formulas are given for the number of quasi-abelian complementary dual codes of index 2 with respect to Euclidean and Hermitian inner products. A sequence of asymptotically good binary quasi-abelian complementary dual codes of index 3 is constructed from an existing sequence of asymptotically good binary self-dual quasi-abelian codes of index 2.

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References

  1. Bazzi, L.M.J., Mitter, S.K.: Some randomized code constructions from group actions. IEEE Trans. Inf. Theory 52, 3210–3219 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlet, C., Guilley, S.: Complementary dual codes for countermeasures to side-channel attacks. Coding Theor. Appl. 3, 97–105 (2015)

    Article  MATH  Google Scholar 

  4. Carlet, C., Daif, A., Danger, J.L., Guilley, S., Najm, Z., Ngo, X.T., Portebouef, T., Tavernier, C.: Optimized linear complementary codes implementation for hardware trojan prevention. In: Proceedings of European Conference on Circuit Theory and Design, 24–26 August 2015, Trondheim, Norway. IEEE, Piscataway (2015)

    Google Scholar 

  5. Dey, B.K.: On existence of good self-dual quasi-cyclic codes. IEEE Trans. Inform. Theory 50, 1794–1798 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dey, B.K., Rajan, B.S.: Codes closed under arbitrary abelian group of permutations. SIAM J. Discrete Math. 18, 1–18 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding, C., Kohel, D.R., Ling, S.: Split group codes. IEEE Trans. Inform. Theory 46, 485–495 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Esmaeili, M., Yari, S.: On complementary-dual quasi-cylic codes. Finite Fields Their Appl. 15, 375–386 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Etesami, J., Hu, F., Henkel, W.: LCD codes and iterative decoding by projections, a first step towards an intuitive description of iterative decoding. In: Proceedings of IEEE Globecom, 5–9 December 2011, Texas, USA. IEEE, Piscataway (2011)

    Google Scholar 

  10. Fan, Y., Lin, L.: Thresholds of random quasi-abelian codes. IEEE Trans. Inform. Theory 61, 82–90 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guneri, C., Ozkaya, B., Solé, P.: Quasi-cylic complementary dual codes. Finite Fields Their Appl. 42, 67–80 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ishai, Y., Sahai, A., Wagner, D.: Private circuits: securing hardware against probing attacks. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 463–481. Springer, Heidelberg (2003). doi:10.1007/978-3-540-45146-4_27

    Chapter  Google Scholar 

  13. Jitman, S., Ling, S., Liu, H., Xie, X.: Abelian codes in principal ideal group algebras. IEEE Trans. Inform. Theory 59, 3046–3058 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jitman, S., Ling, S.: Quasi-abelian codes. Des. Codes Crypt. 74, 511–531 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jitman, S., Ling, S., Solé, P.: Hermitian self-dual abelian codes. IEEE Trans. Inform. Theory 60, 1496–1507 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lally, K., Fitzpatrick, P.: Algebraic structure of quasicyclic codes. Discrete Appl. Math. 111, 157–175 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ling, S., Solé, P.: On the algebraic structure of quasi-cyclic codes I: finite fields. IEEE Trans. Inform. Theory 47, 2751–2760 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ling, S., Solé, P.: Good self-dual quasi-cyclic codes exist. IEEE Trans. Inform. Theory 49, 1052–1053 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ling, S., Solé, P.: On the algebraic structure of quasi-cyclic codes III: generator theory. IEEE Trans. Inform. Theory 51, 2692–2700 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ling, S., Xing, C.: Coding Theory, A First Course. Cambridge University Press, New York (2004)

    Book  Google Scholar 

  21. Massey, J.L.: Linear codes with complementary duals. Discrete Math. 106(107), 337–342 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ngo, X.T., Guilley, S., Bhasin, S., Danger, J.L., Najm, Z.: Encoding the state of integrated circuits: a proactive and reactive protection against hardware trojans horses. In: Proceedings of WESS 2014, 12–17 October 2014, New Delhi, India. ACM, New York (2014)

    Google Scholar 

  23. Ngo, X.T., Bhasin, S., Danger, J.L., Guilley, S., Najm, Z.: Linear complementary dual code improvement to strengthen encoded cirucit against Hardware Trojan Horses. In: Proceedings of IEEE International Symposium on Hardware Oriented Security and Trust (HOST): 2015 May 2015, Washington DC Metropolitan Area, USA. IEEE, Piscataway (2015)

    Google Scholar 

  24. Pei, J., Zhang, X.: \(1\)-generator quasi-cyclic codes. J. Syst. Sci. Complex. 20, 554–561 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rajan, B.S., Siddiqi, M.U.: Transform domain characterization of abelian codes. IEEE Trans. Inform. Theory 38, 1817–1821 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Séguin, G.: A class of \(1\)-generator quasi-cyclic codes. IEEE Trans. Inform. Theory 50, 1745–1753 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sendrier, N.: Linear codes with complementary duals meet the Gilber-Varshamov bound. Discrete Math. 285, 345–347 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yang, X., Massey, J.L.: The condition for a cyclic code to have a complementary dual. Discrete Math. 126, 391–393 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wan, Z.X.: Finite Fields and Galois Rings. World Scientific Pub. Co. Pte. Ltd., Singapore (2012)

    MATH  Google Scholar 

  30. Wasan, S.K.: Quasi abelian codes. Publ. Inst. Math. 35, 201–206 (1977)

    MathSciNet  Google Scholar 

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Acknowledgment

S. Jitman was supported by the Thailand Research Fund under Research Grant MRG6080012. H. S. Palines would like to extend his sincerest gratitude to the following institutions: University of the Philippines Los Ba\(\mathrm{\tilde{n}}\)os, University of the Phillipines System, Department of Science and Technology-Science Education Institute (DOST-SEI) of the Philippines, and Mathematics Department, Faculty of Science, Silpakorn University, Nakhon Pathom, Thailand.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, 73000, Thailand

    Somphong Jitman

  2. Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños, College, 4031, Laguna, Philippines

    Herbert S. Palines

  3. Institute of Mathematics, College of Science, University of the Philippines Diliman, 1101, Quezon City, Philippines

    Herbert S. Palines & Romar B. dela Cruz

Authors
  1. Somphong Jitman
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  2. Herbert S. Palines
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  3. Romar B. dela Cruz
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Corresponding author

Correspondence to Herbert S. Palines .

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Editors and Affiliations

  1. University of Valladolid, Valladolid, Spain

    Ángela I. Barbero

  2. University of Tartu, Tartu, Estonia

    Vitaly Skachek

  3. Simula@UiB and University of Bergen, Bergen, Norway

    Øyvind Ytrehus

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Jitman, S., Palines, H.S., dela Cruz, R.B. (2017). On Quasi-Abelian Complementary Dual Codes. In: Barbero, Á., Skachek, V., Ytrehus, Ø. (eds) Coding Theory and Applications. ICMCTA 2017. Lecture Notes in Computer Science(), vol 10495. Springer, Cham. https://doi.org/10.1007/978-3-319-66278-7_16

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  • DOI: https://doi.org/10.1007/978-3-319-66278-7_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66277-0

  • Online ISBN: 978-3-319-66278-7

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