doi: 10.3934/amc.2021018

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 Published  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:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83

[2]

Kwankyu Lee. Decoding of differential AG codes. Advances in Mathematics of Communications, 2016, 10 (2) : 307-319. doi: 10.3934/amc.2016007

[3]

José Ignacio Iglesias Curto. Generalized AG convolutional codes. Advances in Mathematics of Communications, 2009, 3 (4) : 317-328. doi: 10.3934/amc.2009.3.317

[4]

Olav Geil, Carlos Munuera, Diego Ruano, Fernando Torres. On the order bounds for one-point AG codes. Advances in Mathematics of Communications, 2011, 5 (3) : 489-504. doi: 10.3934/amc.2011.5.489

[5]

Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021

[6]

Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355-365. doi: 10.3934/amc.2016010

[7]

Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028

[8]

Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004

[9]

Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177

[10]

Elisa Gorla, Felice Manganiello, Joachim Rosenthal. An algebraic approach for decoding spread codes. Advances in Mathematics of Communications, 2012, 6 (4) : 443-466. doi: 10.3934/amc.2012.6.443

[11]

Carlos Munuera, Fernando Torres. A note on the order bound on the minimum distance of AG codes and acute semigroups. Advances in Mathematics of Communications, 2008, 2 (2) : 175-181. doi: 10.3934/amc.2008.2.175

[12]

Gustavo Terra Bastos, Reginaldo Palazzo Júnior, Marinês Guerreiro. Abelian non-cyclic orbit codes and multishot subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 631-650. doi: 10.3934/amc.2020035

[13]

Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41

[14]

Yunwen Liu, Longjiang Qu, Chao Li. New constructions of systematic authentication codes from three classes of cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 1-16. doi: 10.3934/amc.2018001

[15]

Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55

[16]

Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025

[17]

Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017

[18]

Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018

[19]

San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038

[20]

Gerardo Vega, Jesús E. Cuén-Ramos. The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes. Advances in Mathematics of Communications, 2020, 14 (3) : 525-533. doi: 10.3934/amc.2020059

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (10)
  • HTML views (19)
  • Cited by (0)

[Back to Top]