# American Institute of Mathematical Sciences

August  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. Google Scholar [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. Google Scholar [3] I. Duursma, Algebraic geometry codes: general theory,, in, (2008), 1. Google Scholar [4] I. Duursma and R. Kirov, An extension of the order bound for AG codes,, in, (2009), 11. Google Scholar [5] I. Duursma, R. Kirov and S. Park, Distance bounds for algebraic geometric codes,, preprint, (). Google Scholar [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. Google Scholar [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. Google Scholar [8] J. Hansen, Codes on the Klein quartic, ideals, and decoding,, IEEE Trans. Inform. Theory, 33 (1987), 923. doi: 10.1109/TIT.1987.1057365. Google Scholar [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. Google Scholar [10] T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes,, in, (1998), 871. Google Scholar [11] C. Munuera, Generalized Hamming weights and trellis complexity,, in, (2008), 363. Google Scholar [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. Google Scholar [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. Google Scholar [14] H. Stichtenoth, "Algebraic Function Fields and Codes,'', Springer, (1993). Google Scholar

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##### 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. Google Scholar [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. Google Scholar [3] I. Duursma, Algebraic geometry codes: general theory,, in, (2008), 1. Google Scholar [4] I. Duursma and R. Kirov, An extension of the order bound for AG codes,, in, (2009), 11. Google Scholar [5] I. Duursma, R. Kirov and S. Park, Distance bounds for algebraic geometric codes,, preprint, (). Google Scholar [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. Google Scholar [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. Google Scholar [8] J. Hansen, Codes on the Klein quartic, ideals, and decoding,, IEEE Trans. Inform. Theory, 33 (1987), 923. doi: 10.1109/TIT.1987.1057365. Google Scholar [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. Google Scholar [10] T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes,, in, (1998), 871. Google Scholar [11] C. Munuera, Generalized Hamming weights and trellis complexity,, in, (2008), 363. Google Scholar [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. Google Scholar [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. Google Scholar [14] H. Stichtenoth, "Algebraic Function Fields and Codes,'', Springer, (1993). Google Scholar
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