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New families of self-dual codes

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Abstract

Recently, the author has constructed families of MDS Euclidean self-dual codes from genus zero algebraic geometry (AG) codes. In the present correspondence, more families of optimal Euclidean self-dual codes from AG codes are explored. New families of MDS Euclidean self-dual codes of odd characteristic and those of almost MDS Euclidean self-dual codes are constructed explicitly from genus zero and genus one curves, respectively. More families of Euclidean self-dual codes are constructed from algebraic curves of higher genus.

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References

  1. Ball S.: On sets of vectors of a finite space in which every subset of basis size is a basis. J. Eur. Soc. 14, 733–748 (2012).

    Article  MathSciNet  Google Scholar 

  2. Betsumiya K., Georgiou S., Gulliver T.A., Harada M., Koukouvinos C.: On self-dual codes over some prime fields. Discrete Math. 262, 37–58 (2003).

    Article  MathSciNet  Google Scholar 

  3. Bosma W., Cannon J.: Handbook of Magma Functions, Sydney (1995).

  4. Cramer R., Daza V., Gracia I., Urroz J.J., Leander G., Marti-Farre J., Padro C.: On codes, matroids, and secure multiparty computation from linear secret-sharing schemes. IEEE Trans. Inform. Theory 54(6), 2647–2657 (2008).

    Article  MathSciNet  Google Scholar 

  5. Database of best known linear codes, http://www.codetables.de.

  6. de Boer M.A.: Almost MDS codes. Des. Codes Cryptogr. 9, 143–155 (1996).

    Article  MathSciNet  Google Scholar 

  7. Dodunekov S.M., Landjev I.N.: Near-MDS codes over some small fields. Discrete Math. 213, 55–65 (2000).

    Article  MathSciNet  Google Scholar 

  8. Dougherty S.T., Mesnager S., Solé P.: Secret-sharing schemes based on self-dual codes. In: Proc. Inf. Theory Workshop, May, pp. 338–342 (2008).

  9. Driencourt Y., Stichtenoth H.: A criterion for self-duality of geometric codes. Commun. Algebra 17(4), 885–898 (1989).

    Article  MathSciNet  Google Scholar 

  10. Fang W., Fu F.: New constructions of MDS Euclidean self-dual codes from GRS codes and extended GRS codes. IEEE Trans. Inform. Theory 65(9), 5574–5579 (2019).

    Article  MathSciNet  Google Scholar 

  11. Georgiou S., Koukouvinos C.: MDS self-dual codes over large prime fields. Finite Fields Appl. 8, 455–470 (2002).

    Article  MathSciNet  Google Scholar 

  12. Goppa V.D.: Algebraico-geometric codes. Math. USSR-lvz. 21(1), 75–91 (1983).

    Article  Google Scholar 

  13. Grassl M., Gulliver T.A.: On self-dual MDS codes, ISIT 2008, Toronto, Canada, July 6 –11 (2008).

  14. Guenda K.: New MDS self-dual codes over finite fields. Des. Codes Cryptogr. 62, 31–42 (2012).

    Article  MathSciNet  Google Scholar 

  15. Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).

    Book  Google Scholar 

  16. Jin L.F., Xing C.P.: New MDS self-dual codes from generalized Reed-Solomon codes. IEEE Trans. Inform. Theory 63(3), 1434–1438 (2017).

    Article  MathSciNet  Google Scholar 

  17. Kim J.-L., Lee Y.: Construction of MDS self-dual codes over Galois rings. Des. Codes Cryptogr. 45, 247–258 (2007).

    Article  MathSciNet  Google Scholar 

  18. Kim J.-L., Lee Y.: Euclidean and Hermitian self-dual MDS codes over large finite fields. J. Comb. Theory Ser. A 105, 79–95 (2004).

    Article  MathSciNet  Google Scholar 

  19. MacWilliams F.J., Sloane N.J.A., Thompson J.G.: Good self-dual codes exist. Discrete Math. 3, 153–162 (1972).

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  21. Massey J.: Some applications of coding theory in cryptography. In: Proc. 4th IMA Conf. Cryptogr. Coding, pp. 33–47 (1995).

  22. Sok L.: Explicit constructions of MDS self-dual codes. IEEE Trans. Inform. Theory 66(6), 3603–3615 (2020).

    Article  MathSciNet  Google Scholar 

  23. Sok, L.: On Euclidean self-dual codes and isometry codes. Applicable Algebra in Engineering, Communication and Computing. https://doi.org/10.1007/s00200-020-00434-y.

  24. Stichtenoth H.: Self-dual Goopa codes. J. Pure Appl. Algebra 55, 199–211 (1988).

    Article  MathSciNet  Google Scholar 

  25. Stichtenoth H.: Algebraic Function Fields and Codes. Springer, New York (2008).

    MATH  Google Scholar 

  26. Tsfasman M.A., Vlǎduţ S.G.: Algebraic-geometric codes. Kluwer Academic Publication, Mathematics and Its Applications, vol. 58 (1991).

  27. Tong H., Wang X.: New MDS Euclidean and Hermitian self-dual codes over finite fields. Adv. Pure Math. 7, 325–333 (2017).

    Article  Google Scholar 

  28. Yan H.: A note on the constructions of MDS self-dual codes. Cryptogr. Commun. 11, 259–268 (2019).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research work is supported by Anhui Provincial Natural Science Foundation with Grant Number 1908085MA04. The author would like to thank anonymous referees for their constructive comments which improves the quality of the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, 230601, Anhui, China

    Lin Sok

  2. Department of Mathematics, Royal University of Phnom Penh, 12156, Phnom Penh, Cambodia

    Lin Sok

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  1. Lin Sok
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Correspondence to Lin Sok.

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Communicated by I. Landjev.

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Sok, L. New families of self-dual codes. Des. Codes Cryptogr. 89, 823–841 (2021). https://doi.org/10.1007/s10623-021-00847-x

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