Advanced Search
Article Contents
Article Contents

Locally recoverable codes from algebraic curves with separated variables

  • * Corresponding author

    * Corresponding author 

The first author was supported by Spanish Ministerio de Economía y Competitividad under grant MTM2015-65764-C3-1-P MINECO/FEDER. The second author was supported by CNPq-Brazil, under grants 201584/2015-8 and 159852/2014-5. The third author was supported by CNPq-Brazil under grant 310623/2017-0

Abstract Full Text(HTML) Figure(0) / Table(1) Related Papers Cited by
  • A Locally Recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. We study Locally Recoverable Algebraic Geometry codes arising from certain curves defined by equations with separated variables. The recovery of erasures is obtained by means of Lagrangian interpolation in general, and simply by one addition in some particular cases.

    Mathematics Subject Classification: 94B27, 11G20, 11T71, 14G50, 94B05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Table 1.  Hamming weights of Example 5

    $ t $ 1 2 3 4 5 6
    lower bound (9) 10 12 13 14 15 16
    upper bound (8) 11 12 14 15 17 18
     | Show Table
    DownLoad: CSV
  • [1] E. Ballico and C. Marcolla, Higher Hamming weights for locally recoverable codes on algebraic curves, Finite Fields and Their Applications, 40 (2016), 61-72.  doi: 10.1016/j.ffa.2016.03.004.
    [2] A. BargI. Tamo and S. Vlǎduţ, Locally recoverable codes on algebraic curves, IEEE Trans. Inform. Theory, 63 (2015), 4928-4939.  doi: 10.1109/TIT.2017.2700859.
    [3] D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, Springer, New York, 1992. doi: 10.1007/978-1-4757-2181-2.
    [4] P. GopalanC. HuangH. Simitci and S. Yekhanin, On the locality of codeword symbols, IEEE Transactions on Information Theory, 58 (2012), 6925-6934.  doi: 10.1109/TIT.2012.2208937.
    [5] K. HaymakerB. Malmskog and G. L. Matthews, Locally recoverable codes with availability $t\geq 2$ from fiber products of curves, Advances in Mathematics of Communications, 12 (2018), 317-336.  doi: 10.3934/amc.2018020.
    [6] J. W. P. HirschfeldG. Korchmáros and  F. TorresAlgebraic Curves Over a Finite Field, Princeton University Press, Princeton, NJ, 2008. 
    [7] S. KondoT. Katagiri and T. Ogihara, Automorphism groups of one-point codes from the curves $y^{q}+y = x^{q^r+1}$, IEEE Transactions on Information Theory, 47 (2001), 2573-2579.  doi: 10.1109/18.945272.
    [8] O. Kolosov, A. Barg, I. Tamo and G. Yadgar, Optimal LRC codes for all lengths $n\leq q$, preprint, (2018), arXiv: 1802.00157.
    [9] R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983.
    [10] S. Miura, Algebraic geometric codes on certain plane curves, IEICE Transactions on Fundamentals, 75 (1992), 1735-1745.  doi: 10.1002/ecjc.4430761201.
    [11] C. Munuera and W. Olaya-León, An introduction to algebraic geometry codes, in Algebra for Secure and Reliable Communication Modelling, AMS-Contemporary Mathematics, Providence, RI, 642 (2015), 87–117. doi: 10.1090/conm/642/12882.
    [12] C. MunueraA. Sepúlveda and F. Torres, Castle curves and codes, Advances in Mathematics of Communications, 3 (2009), 399-408.  doi: 10.3934/amc.2009.3.399.
    [13] C. Munuera and W. Tenório, Locally Recoverable codes from rational maps, Finite Fields and Their Applications, 54 (2018), 80-100.  doi: 10.1016/j.ffa.2018.07.005.
    [14] R. PellikaanX. W. WuS. Bulygin and  R. JurriusCodes, Cryptology and Curves with Computer Algebra, Cambridge University Press, Cambridge, 2017. 
    [15] A. Sepulveda and G. Tizziotti, Weierstrass semigroups and codes over the curve $y^{q}+y = x^{q^r+1}$, Advances in Mathematics of Communications, 8 (2014), 67-72.  doi: 10.3934/amc.2014.8.67.
    [16] H. Stichtenoth, Algebraic Function Fields and Codes, Graduate Texts in Mathematics, 254. Springer-Verlag, Berlin, 2009.
    [17] I. Tamo and A. Barg, A family of optimal locally recoverable codes, IEEE Transactions on Information Theory, 60 (2014), 4661-4676.  doi: 10.1109/TIT.2014.2321280.
    [18] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Transactions on Information Theory, 37 (1991), 1412-1418.  doi: 10.1109/18.133259.
  • 加载中



Article Metrics

HTML views(775) PDF downloads(412) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint