American Institute of Mathematical Sciences

doi: 10.3934/amc.2021018
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The conorm code of an AG-code

 1 Researcher of CONICET at Facultad de Ingeniería Química, (UNL), Santiago del Estero 2829, (3000) Santa Fe, Argentina 2 FaMAF – CIEM (CONICET), Universidad Nacional de Córdoba, Av. Medina Allende 2144, (5000) Córdoba, Argentina 3 Facultad de Ingeniería Química, (UNL), Santiago del Estero 2829, (3000) Santa Fe, Argentina

* Corresponding author: María Chara

Received  November 2020 Revised  March 2021 Early access June 2021

Fund Project: Partially supported by CONICET, UNL CAI+D 2016, SECyT-UNC, CSIC

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.

Citation: María Chara, Ricardo A. Podestá, Ricardo Toledano. The conorm code of an AG-code. Advances in Mathematics of Communications, doi: 10.3934/amc.2021018
References:
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References:
 [1] D. Bartoli, L. 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.
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