2011, 5(3): 489-504. doi: 10.3934/amc.2011.5.489

On the order bounds for one-point AG codes

1. 

Department of Mathematical Sciences, Aalborg University, Fr. Bajersvej 7G, 9220 Aalborg Øst, Denmark

2. 

Department of Applied Mathematics, University of Valladolid, Avda Salamanca SN, 47014 Valladolid, Castilla, Spain

3. 

Department of Mathematical Sciences, Aalborg University, Fr. Bajersvej 7G, 9220-Aalborg Øst, Denmark

4. 

Institute of Mathematics, Statistics and Computer Science, P.O. Box 6065, University of Campinas, 13083-970, Campinas, SP, Brazil

Received  November 2010 Revised  November 2010 Published  August 2011

The order bound for the minimum distance of algebraic geometry codes was originally defined for the duals of one-point codes and later generalized for arbitrary algebraic geometry codes. Another bound of order type for the minimum distance of general linear codes, and for codes from order domains in particular, was given in [1]. Here we investigate in detail the application of that bound to one-point algebraic geometry codes, obtaining a bound d* for the minimum distance of these codes. We establish a connection between d* and the order bound and its generalizations. We also study the improved code constructions based on d*. Finally we extend d* to all generalized Hamming weights.
Citation: 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
References:
[1]

H. Andersen and O. Geil, Evaluation codes from order domain theory,, Finite Fields Appl., 14 (2008), 92. doi: 10.1016/j.ffa.2006.12.004.

[2]

P. Beelen, The order bound for general algebraic geometric codes,, Finite Fields Appl., 13 (2007), 665. doi: 10.1016/j.ffa.2006.09.006.

[3]

I. Duursma, Algebraic geometry codes: general theory,, in, (2008), 1.

[4]

I. Duursma and R. Kirov, An extension of the order bound for AG codes,, in, (2009), 11.

[5]

I. Duursma, R. Kirov and S. Park, Distance bounds for algebraic geometric codes,, preprint, ().

[6]

I. Duursma and S. Park, Coset bounds for algebraic geometric codes,, Finite Fields Appl., 16 (2010), 36. doi: 10.1016/j.ffa.2009.11.006.

[7]

G. L. Feng and T. N. T. Rao, Improved geometric Goppa codes. Part I: basic theory,, IEEE Trans. Inform. Theory, 41 (1995), 1678. doi: 10.1109/18.476241.

[8]

J. Hansen, Codes on the Klein quartic, ideals, and decoding,, IEEE Trans. Inform. Theory, 33 (1987), 923. doi: 10.1109/TIT.1987.1057365.

[9]

P. Heijnen and R. Pellikaan, Generalized Hamming weights of $q$-ary Reed-Muller codes,, IEEE Trans. Inform. Theory, 44 (1998), 181. doi: 10.1109/18.651015.

[10]

T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes,, in, (1998), 871.

[11]

C. Munuera, Generalized Hamming weights and trellis complexity,, in, (2008), 363.

[12]

C. Munuera and R. Pellikaan, Equality of geometric Goppa codes and equivalence of divisors,, J. Pure Appl. Algebra, 90 (1993), 229. doi: 10.1016/0022-4049(93)90043-S.

[13]

C. Munuera, A. Sepúlveda and F. Torres, Algebraic geometry codes from Castle curves,, in, (2008), 117. doi: 10.1007/978-3-540-87448-5_13.

[14]

H. Stichtenoth, "Algebraic Function Fields and Codes,'', Springer, (1993).

show all references

References:
[1]

H. Andersen and O. Geil, Evaluation codes from order domain theory,, Finite Fields Appl., 14 (2008), 92. doi: 10.1016/j.ffa.2006.12.004.

[2]

P. Beelen, The order bound for general algebraic geometric codes,, Finite Fields Appl., 13 (2007), 665. doi: 10.1016/j.ffa.2006.09.006.

[3]

I. Duursma, Algebraic geometry codes: general theory,, in, (2008), 1.

[4]

I. Duursma and R. Kirov, An extension of the order bound for AG codes,, in, (2009), 11.

[5]

I. Duursma, R. Kirov and S. Park, Distance bounds for algebraic geometric codes,, preprint, ().

[6]

I. Duursma and S. Park, Coset bounds for algebraic geometric codes,, Finite Fields Appl., 16 (2010), 36. doi: 10.1016/j.ffa.2009.11.006.

[7]

G. L. Feng and T. N. T. Rao, Improved geometric Goppa codes. Part I: basic theory,, IEEE Trans. Inform. Theory, 41 (1995), 1678. doi: 10.1109/18.476241.

[8]

J. Hansen, Codes on the Klein quartic, ideals, and decoding,, IEEE Trans. Inform. Theory, 33 (1987), 923. doi: 10.1109/TIT.1987.1057365.

[9]

P. Heijnen and R. Pellikaan, Generalized Hamming weights of $q$-ary Reed-Muller codes,, IEEE Trans. Inform. Theory, 44 (1998), 181. doi: 10.1109/18.651015.

[10]

T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes,, in, (1998), 871.

[11]

C. Munuera, Generalized Hamming weights and trellis complexity,, in, (2008), 363.

[12]

C. Munuera and R. Pellikaan, Equality of geometric Goppa codes and equivalence of divisors,, J. Pure Appl. Algebra, 90 (1993), 229. doi: 10.1016/0022-4049(93)90043-S.

[13]

C. Munuera, A. Sepúlveda and F. Torres, Algebraic geometry codes from Castle curves,, in, (2008), 117. doi: 10.1007/978-3-540-87448-5_13.

[14]

H. Stichtenoth, "Algebraic Function Fields and Codes,'', Springer, (1993).

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