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Self-dual Codes

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Algebraic Coding Theory Over Finite Commutative Rings

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Abstract

In this chapter, we describe self-dual codes over Frobenius rings. We give constructions of self-dual codes over any Frobenius ring. We describe connections to unimodular lattices, binary self-dual codes and to designs. We also describe linear complementary dual codes and make a new definition of a broad generalization encompassing both self-dual and linear complementary codes.

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References

  1. Assmus Jr., E.F., Key, J.D.: Designs and their Codes. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  2. Assmus, E.F., Mattson, H.F.: New 5-designs. J. Comb. Theory 6, 122–151 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bannai, A., Dougherty, S.T., Harada, M., Oura, M.: Type II codes, even unimodular lattices, and invariant rings. IEEE-IT 45(4), 1194–1205 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carlet, C., Guilley, S.: Complementary dual codes for counter-measures to sidechannel attacks. In: Proceedings of the 4th ICMCTA Meeting, Palmela, Portugal (2014)

    Google Scholar 

  5. Conway, J.H., Pless, V., Sloane, N.J.A.: The binary self-dual codes of length up to \(32\): a revised enumeration. J. Combin. Theory Ser. A 60, 183–195 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Conway, J.H., Sloane, N.J.A.: A new upper bound on the minimal distance of self-dual codes. IEEE Trans. Inform. Theory IT-36, 1319 - 1333 (1990)

    Google Scholar 

  7. Conway, J.H., Sloane, N.J.A.: Self-dual codes over the integers modulo 4. J. Combin. Theory Ser. A 62(1), 30–45 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Conway, J. H., Sloane, N. J. A., Sphere packings, lattices and groups. Third edition. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 290. Springer-Verlag, New York, (1999)

    Google Scholar 

  9. Choie, Y.J., Dougherty, S.T.: Codes over \(\Sigma _{2m}\) and Jacobi forms over the quaternions. Appl. Algebra. Eng. Commun. Comput. 15(2), 129–147 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Choie, Y.J., Dougherty, S.T.: Codes over rings, complex lattices, and Hermitian modular forms. Eur. J. Comb. 26(2), 145–165 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dougherty, S.T.: A new construction of self-dual codes from projective planes. Australas. J. Comb. 31, 337–348 (2005)

    MathSciNet  MATH  Google Scholar 

  12. Dougherty, S.T., Fernandez-Cordoba, C.: Codes over \({\mathbb{Z}}_{2^k}\), Gray maps and self-dual codes. Adv. Math. Commun. 54 (2011)

    Google Scholar 

  13. Dougherty, S.T., Harada, M., Gulliver, T.A.: Extremal binary self-dual codes, IEEE Trans. Inf. Theory, 2036–2047 (1997)

    Google Scholar 

  14. Dougherty, S.T., Gulliver, A., Ramadurai, R.: Symmetric designs and self-dual codes over rings. Ars Comb. 85, 149–161 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Dougherty, S.T., Harada, M., Solé, P.: Self-dual codes over rings and the Chinese remainder theorem. Hokkaido Math. J. 28, 253–283 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dougherty, S.T., Kim, J.L., Kulosman, H., Liu, H.: Self-dual codes over Frobenius rings. Finite Fields Appl. 16, 14–26 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dougherty, S.T., Kim, J., Ozkaya, B., Sok, L., Solé, P.: The combinatorics of LCD codes: linear programming bound and orthogonal matrices, to appear in IJICOT

    Google Scholar 

  18. Dougherty, S.T., Kim, J.L., Solé, P.: Open problems in coding theory, Contemp. Math. 634, 79–99 (2015)

    Google Scholar 

  19. Dougherty, S., Yildiz, B., Karadeniz, S.: Self-dual codes over \(R_k\) and binary self-dual codes. Eur. J. Pure Appl. Math. 6(1), 89–106 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Harada, M., Miezaki, T.: An optimal odd unimodular lattice in dimension 72. Arch. Math. (Basel) 97(6), 529–533 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Klemm, M.: Selbstduale code über dem ring der ganzen zahlen modulo \(4\). Arch. Math. (Basel) 53, 201–207 (1989)

    Google Scholar 

  22. Karadeniz, S., Dougherty, S.T., Yildiz, B.: Constructing formally self-dual codes over \(R_k\). Discrete Appl. Math. 167, 188–196 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Karadeniz, S., Yildiz, B.: New extremal binary self-dual codes of length 64 from \(R_3\)-lifts of the extended binary Hamming code. Des. Codes Cryptogr. 74(3), 673–680 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gleason, A.M.: Weight polynomials of self-dual codes and the MacWilliams identities. Actes Congres Internl. Math. 3, 211–215 (1970)

    Google Scholar 

  25. Glynn, D.: The construction of self-dual binary codes from projective planes of odd order. Australas. J. Comb. 4, 277–284 (1991)

    MathSciNet  MATH  Google Scholar 

  26. Lam, C.W.H.: The search for a finite projective plane of order 10. Amer. Math. Monthly 98(4), 305–318 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Nebe, G.: An even unimodular 72-dimensional lattice of minimum 8. J. Reine Angew. Math. 673, 237–247 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Nebe, G., Rains, E. M., Sloane, N. J. A.: Self-dual codes and invariant theory. Algorithms Comput. Math. 17. Springer, Heidelberg (2006)

    Google Scholar 

  31. Rains, E. M., Sloane, N. J. A.: Self-Dual Codes. Handbook of Coding Theory, vol. I, II, pp. 177 –294. North-Holland, Amsterdam (1998)

    Google Scholar 

  32. Pless, V., Sloane, N.J.A.: On the classification and enumeration of self-dual codes. J. Combin. Theory Ser. A 18, 313–335 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sloane, N. J. A.: Is there a \((72,36), d=16\) self-dual code? IEEE Trans. Inf. Theory IT 192, 251 (1973)

    Google Scholar 

  34. Tanabe, K.: An Assmus-Mattson theorem for \({\mathbb{Z}}_4\)-codes. IEEE Trans. Inf. Theory 46(1), 48–53 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Scranton, Scranton, PA, USA

    Steven T. Dougherty

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  1. Steven T. Dougherty
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Correspondence to Steven T. Dougherty .

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Dougherty, S.T. (2017). Self-dual Codes. In: Algebraic Coding Theory Over Finite Commutative Rings. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-59806-2_5

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