\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents
Early Access

Early Access articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Early Access publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Early Access articles via the “Early Access” tab for the selected journal.

The conorm code of an AG-code

  • * Corresponding author: María Chara

    * Corresponding author: María Chara 
Partially supported by CONICET, UNL CAI+D 2016, SECyT-UNC, CSIC
Abstract Full Text(HTML) Related Papers Cited by
  • Given a suitable extension $ F'/F $ of algebraic function fields over a finite field $ \mathbb{F}_q $, we introduce the conorm code $ \operatorname{Con}_{F'/F}( \mathcal{C}) $ defined over $ F' $ which is constructed from an algebraic geometry code $ \mathcal{C} $ defined over $ F $. We study the parameters of $ \operatorname{Con}_{F'/F}( \mathcal{C}) $ in terms of the parameters of $ \mathcal{C} $, the ramification behavior of the places used to define $ \mathcal{C} $ and the genus of $ F $. In the case of unramified extensions of function fields we prove that $ \operatorname{Con}_{F'/F}( \mathcal{C})^\perp = \operatorname{Con}_{F'/F}( \mathcal{C}^\perp) $ when the degree of the extension is coprime to the characteristic of $ \mathbb{F}_q $. We also study the conorm of cyclic algebraic-geometry codes and we show that some repetition codes, Hermitian codes and all Reed-Solomon codes can be represented as conorm codes.

    Mathematics Subject Classification: Primary: 94B27; Secondary: 94B15, 14H05.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] D. BartoliL. Quoos and G. Zini, Algebraic geometric codes on many points from Kummer extensions, Finite Fields and Their Applications, 52 (2018), 319-335.  doi: 10.1016/j.ffa.2018.04.008.
    [2] A. Couvreur, I. Márquez-Corbella and R. Pellikaan, A polynomial time attack against algebraic geometry code based public key cryptosystem, IEEE International Symposium on Information Theory, (2014), 1446–1450. doi: 10.1109/ISIT.2014.6875072.
    [3] C. Faure and H. Minder, Cryptanalysis of the McEliece cryptosystem over hyperelliptic codes, 11th Int. Workshop Algebraic and Combinat. Coding Theory, Pamporovo Bulgaria, 8 (2008), 99-107. 
    [4] A. Garcia and H. Stichtenoth, On the asymptotic behaviour of some towers of function fields over finite fields, Journal of Number Theory, 61:2 (1996), 248-273.  doi: 10.1006/jnth.1996.0147.
    [5] H. Janwa and O. Moreno, McEliece public crypto system using algebraic-geometric codes, Designs, Codes and Cryptography, 8 (1996), 293-307.  doi: 10.1023/A:1027351723034.
    [6] I. Márquez-Corbella, E. Martínez-Moro, R. Pellikaan and D. Ruano, Computational aspects of retrieving a representation of an algebraic geometry code, Journal of Symbolic Computation, 64, (2014) 67–87. doi: 10.1016/j.jsc.2013.12.007.
    [7] C. Munuera and R. Pellikaan, Equality of geometric Goppa codes and equivalence of divisors, Journal of Pure and Applied Algebra, 90 (1993) 229–252. doi: 10.1016/0022-4049(93)90043-S.
    [8] H. Stichtenoth, Algebraic Function Fields and Codes, 2$^{nd}$ edition, Graduate Texts in Mathematics, 254, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-76878-4.
    [9] C. Voss and T. Hoholdt, An explicit construction of a sequence of codes attaining the Tsfasman-Vladut-Zink bound. The first steps, IEEE Transactions on Information Theory, 43:1 (1997), 128-135.  doi: 10.1109/18.567659.
    [10] J. Wülftange, On the construction of some towers over finite fields, in Finite Fields and Applications. Fq 2003, Lecture Notes in Computer Science, 2948, Springer, Berlin, Heidelberg, 2004. doi: 10.1007/978-3-540-24633-6_13.
  • 加载中
SHARE

Article Metrics

HTML views(710) PDF downloads(526) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return