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Special LCD codes from products of graphs

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

We examine the binary codes from the adjacency matrices of various products of graphs, and show that if the binary codes of a set of graphs have the property that their dual codes are the codes of the associated reflexive graphs, and are thus LCD, i.e. have zero hull, then, with some restrictions, the binary code of the product will have the same property. The codes are candidates for decoding using this property, or also, in the case of the direct product, by permutation decoding.

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Notes

  1. Note typographical error on p.338, l.-11, in [24]

  2. Note typographical error on p.341, l.-7, in [24]

References

  1. Assmus, E.F., Jr, Key, J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992) (Cambridge Tracts in Mathematics, vol. 103, Second printing with corrections, 1993)

  2. Barik, S., Bapat, R.B., Pati, S.: On the Laplacian spectra of product graphs. Appl. Anal. Disc. Math. 9, 39–58 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24(3/4), 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cannon, J.,  Steel, A.,  White, G.: Linear codes over finite fields. In: Cannon, J., Bosma, W. (eds.) Handbook of Magma Functions, V2.13, pp. 3951–4023. Computational Algebra Group, Department of Mathematics, University of Sydney (2006). http://magma.maths.usyd.edu.au/magma

  5. Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Math. J. 23, 298–305 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fish, W., Key, J.D., Mwambene, E., Rodrigues, B.G.: Hamming graphs and LCD codes. J. Appl. Math. Comput. 61, 461–479 (2019). https://doi.org/10.1007/s12190-019-01259-w

    Article  MathSciNet  MATH  Google Scholar 

  7. Daniel, D.M.: Minimal permutation sets for decoding the binary Golay codes. IEEE Trans. Inform. Theory 28, 541–543 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Haemers, W.H., Peeters, R., van Rijckevorsel, J.M.: Binary codes of strongly regular graphs. Des Codes Cryptogr. 17, 187–209 (1999)

  9. Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs, 2nd edn. CRC Press, Boca Raton (2011)

  10. Huffman, W.C..: Codes and groups. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, vol 2, part 2, chapt 17, pp. 1345–1440. Elsevier, Amsterdam (1998)

  11. Key, J.D., Limbupasiriporn, J.: Permutation decoding of codes from Paley graphs. Congr. Numer. 170, 143–155 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Key, J.D., McDonough, T.P., Mavron, V.C.: Partial permutation decoding for codes from finite planes. Eur. J. Combin. 26, 665–682 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Key, J.D., McDonough, T.P., Mavron, V.C.: Information sets and partial permutation decoding for codes from finite geometries. Finite Fields Appl. 12, 232–247 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Key, J.D., McDonough, T.P., Mavron, V.C.: Improved partial permutation decoding for Reed-Muller codes. Disc. Math. 340, 722–728 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Key, J.D., Moori, J., Rodrigues, B.G.: Permutation decoding for binary codes from triangular graphs. Eur. J. Combin. 25, 113–123 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Key, J.D., Rodrigues, B.G.: \({LCD}\) codes from adjacency matrices of graphs. Appl. Algebra Engrg. Comm. Comput. 29(3), 227–244 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Key, J.D., Rodrigues, B.G.: Special \({LCD}\) codes from Peisert and generalized Peisert graphs. Graphs Combin. 35, 633–652 (2019). https://doi.org/10.1007/s00373-019-02019-0

    Article  MathSciNet  MATH  Google Scholar 

  18. Key, J.D., Rodrigues, B.G.: Binary codes from \(m\)-ary \(n\)-cubes \({Q}^m_n\). Adv. Math. Commun. 15(3), 507–524 (2021). https://doi.org/10.3934/amc.2020079

    Article  MathSciNet  MATH  Google Scholar 

  19. Key, J.D., Rodrigues, B.G.: Minimal \({PD}\)-sets associated with the graphs \({Q}^m_2\), \(m\) even. Appl. Algebra Eng. Commun. Comput. (2021) (to appear). https://doi.org/10.1007/s00200-020-00481-5

  20. Kroll, H.-J., Taherian, S.-G., Vincenti, R.: Optimal antiblocking information systems for the binary codes related to triangular graphs. Adv. Math. Commun. (2020) (to appear). https://doi.org/10.3934/amc.2020107

  21. Kroll, H.-J., Vincenti, R.: PD-sets related to the codes of some classical varieties. Discr. Math. 301, 89–105 (2005)

  22. MacWilliams, F.J.: Permutation decoding of systematic codes. Bell Syst. Tech. J. 43, 485–505 (1964)

    Article  MATH  Google Scholar 

  23. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1983)

    MATH  Google Scholar 

  24. James, L.: Massey, Linear codes with complementary duals. Disc. Math. 106(107), 337–342 (1992)

    MATH  Google Scholar 

  25. Merris, R.: Laplacian matrices of graphs? A survey. Linear Algeb. Appl. 197(198), 143–176 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schönheim, J.: On coverings. Pacific J. Math. 14, 1405–1411 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shi, M., Huang, D., Sok, L., Solé, P.: Double circulant LCD codes over \(\mathbb{Z}_4\). Finite Fields Appl. 58, 133–144 (2019)

  28. Tonchev, V.D.: Combinatorial Configurations, Designs, Codes, Graphs. Pitman Monographs and Surveys in Pure and Applied Mathematics, No. 40. Longman, New York (1988) (Translated from the Bulgarian by Robert A. Melter)

  29. West, D.B.: Introduction to Graph Theory. Prentice Hall (1996)

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville, 7535, South Africa

    W. Fish, J. D. Key & E. Mwambene

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  1. W. Fish
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  2. J. D. Key
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  3. E. Mwambene
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Fish, W., Key, J.D. & Mwambene, E. Special LCD codes from products of graphs. AAECC 34, 553–579 (2023). https://doi.org/10.1007/s00200-021-00517-4

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