δ-Sequences and Evaluation Codes de ned by Plane Valuations at Infinity

We introduce the concept of δ-sequence. A δ-sequence ∆ generates a well-ordered semigroup S in Z2 or R. We show how to construct (and compute parameters) for the dual code of any evaluation code associated with a weight function defined by ∆ from the polynomial ring in two indeterminates to a semigroup S as above. We prove that this is a simple procedure which can be understood by considering a particular class of valuations of function fields of surfaces, called plane valuations at infinity. We also give algorithms to construct an unlimited number of δ-sequences of the diferent existing types, and so this paper provides the tools to know and use a new large set of codes

On the characterization of algebraically integrable plane foliations

We give a characterization theorem for non-degenerate plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree of a non-degenerate foliation as above provides the minimum number, , of points in the projective plane through which pass infinitely many algebraic leaves of the foliatio

Discrete Equivalence of Non-positive at Infinity Plane Valuations

Non-positive at infinity valuations are a class of real plane valuations which have a nice geometrical behavior. They are divided in three types. We study the dual graphs of non-positive at infinity valuations and give an algorithm for obtaining them. Moreover we compare these graphs attending the type of their corresponding valuation

Evaluation codes defined by finite families of plane valuations at infinity

We construct evaluation codes given by weight functions defined over polynomial rings in m a parts per thousand yen 2 indeterminates. These weight functions are determined by sets of m-1 weight functions over polynomial rings in two indeterminates defined by plane valuations at infinity. Well-suited families in totally ordered commutative groups are an important tool in our procedureSupported by Spain Ministry of Education MTM2007-64704 and Bancaixa P1-1B2009-03. The authors thank to the referees for their valuable suggestions.Galindo Pastor, C.; Monserrat Delpalillo, FJ. (2014). Evaluation codes defined by finite families of plane valuations at infinity. Designs, Codes and Cryptography. 70(1-2):189-213. https://doi.org/10.1007/s10623-012-9738-7S189213701-2Abhyankar S.S.: Local uniformization on algebraic surfaces over ground field of characteristic p ≠ 0. Ann. Math. 63, 491–526 (1956)Abhyankar S.S.: On the valuations centered in a local domain. Am. J. Math. 78, 321–348 (1956)Abhyankar S.S.: Lectures on expansion techniques in algebraic geometry. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57. Tata Institute of Fundamental Research, Bombay (1977).Abhyankar S.S.: On the semigroup of a meromorphic curve (part I). In: Proceedings of the International Symposium on Algebraic Geometry (Kyoto) Kinokunio Tokio, pp. 249–414 (1977).Abhyankar S.S., Moh T.T.: Newton-Puiseux expansion and generalized Tschirnhausen transformation (I). J. Reine Angew. Math. 260, 47–83 (1973)Abhyankar S.S., Moh T.T.: Newton-Puiseux expansion and generalized Tschirnhausen transformation (II). J. Reine Angew. Math. 261, 29–54 (1973)Berlekamp E.R.: Algebraic Coding Theory. McGraw-Hill, New York (1968)Campillo A., Farrán J.I.: Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models. Finite Fields Appl. 6, 71–92 (2000)Carvalho C., Munuera C., Silva E., Torres F.: Near orders and codes. IEEE Trans. Inf. Theory 53, 1919–1924 (2007)Decker W., Greuel G.M., Pfister G., Schöenemann H.: Singular 3.1.3, a computer algebra system for polynomial computations (2011) http://www.singular.uni-kl.de .Feng G.L., Rao T.R.N.: Decoding of algebraic geometric codes up to the designed minimum distance. IEEE Trans. Inf. Theory 39, 37–45 (1993)Feng G.L., Rao T.R.N.: A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inf. Theory 40, 1003–1012 (1994)Feng G.L., Rao T.R.N.: Improved geometric Goppa codes, part I: basic theory. IEEE Trans. Inf. Theory 41, 1678–1693 (1995)Fujimoto M., Suzuki M.: Construction of affine plane curves with one place at infinity. Osaka J. Math. 39(4), 1005–1027 (2002)Galindo C.: Plane valuations and their completions. Commun. Algebra 23(6), 2107–2123 (1995)Galindo C., Monserrat F.: δ-sequences and evaluation codes defined by plane valuations at infinity. Proc. Lond. Math. Soc. 98, 714–740 (2009)Galindo C., Monserrat F.: The Abhyankar-Moh theorem for plane valuations at infinity. Preprint 2010. ArXiv:0910.2613v2.Galindo C., Sanchis M.: Evaluation codes and plane valuations. Des. Codes Cryptogr. 41(2), 199–219 (2006)Geil O.: Codes based on an $${\mathbb{F}_q}$$ -algebra. PhD Thesis, Aalborg University, June (2000).Geil O., Matsumoto R.: Generalized Sudan’s list decoding for order domain codes. Lecture Notes in Computer Science, vol. 4851, pp. 50–59 (2007)Geil O., Pellikaan R.: On the structure of order domains. Finite Fields Appl. 8, 369–396 (2002)Goppa V.D.: Codes associated with divisors. Probl. Inf. Transm. 13, 22–26 (1997)Goppa V.D.: Geometry and Codes. Mathematics and Its Applications, vol. 24. Kluwer, Dordrecht (1991).Greco S., Kiyek K.: General elements in complete ideals and valuations centered at a two-dimensional regular local ring. In: Algebra, Arithmetic, and Geometry, with Applications, pp. 381–455. Springer, Berlin (2003).Høholdt T., van Lint J.H., Pellikaan R.: Algebraic geometry codes. In: Handbook of Coding Theory, vol. 1, pp. 871–961. Elsevier, Amsterdam (1998).Jensen C.D.: Fast decoding of codes from algebraic geometry. IEEE Trans. Inf. Theory 40, 223–230 (1994)Justesen J., Larsen K.J., Jensen H.E., Havemose A., Høholdt T.: Construction and decoding of a class of algebraic geometric codes. IEEE Trans. Inf. Theory 35, 811–821 (1989)Justesen J., Larsen K.J., Jensen H.E., Høholdt T.: Fast decoding of codes from algebraic plane curves. IEEE Trans. Inf. Theory 38, 111–119 (1992)Massey J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15, 122–127 (1969)Matsumoto R.: Miura’s generalization of one point AG codes is equivalent to Høholdt, van Lint and Pellikaan’s generalization. IEICE Trans. Fundam. E82-A(10), 2007–2010 (1999)Moghaddam M.: Realization of a certain class of semigroups as value semigroups of valuations. Bull. Iran. Math. Soc. 35, 61–95 (2009)O’Sullivan M.E.: Decoding of codes defined by a single point on a curve. IEEE Trans. Inf. Theory 41, 1709–1719 (1995)O’Sullivan M.E.: New codes for the Belekamp-Massey-Sakata algorithm. Finite Fields Appl. 7, 293–317 (2001)Pinkham H.: Séminaire sur les singularités des surfaces (Demazure-Pinkham-Teissier), Course donné au Centre de Math. de l’Ecole Polytechnique (1977–1978).Sakata S.: Extension of the Berlekamp-Massey algorithm to N dimensions. Inf. Comput. 84, 207–239 (1990)Sakata S., Jensen H.E., Høholdt T.: Generalized Berlekamp-Massey decoding of algebraic geometric codes up to half the Feng-Rao bound. IEEE Trans. Inf. Theory 41, 1762–1768 (1995)Sakata S., Justesen J., Madelung Y., Jensen H.E., Høholdt T.: Fast decoding of algebraic geometric codes up to the designed minimum distance. IEEE Trans. Inf. Theory 41, 1672–1677 (1995)Sathaye A.: On planar curves. Am. J. Math. 99(5), 1105–1135 (1977)Shannon C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).Skorobogatov A.N., Vlădut S.G.: On the decoding of algebraic geometric codes. IEEE Trans. Inf. Theory 36, 1051–1060 (1990)Spivakovsky M.: Valuations in function fields of surfaces. Am. J. Math. 112, 107–156 (1990)Suzuki M.: Affine plane curves with one place at infinity. Ann. Inst. Fourier 49(2), 375–404 (1999)Tsfasman S.G., Vlăduţ T.: Zink, modular curves, Shimura curves and Goppa codes, better than Varshamov–Gilbert bound. Math. Nachr. 109, 21–28 (1982)Vlăduţ S.G., Manin Y.I. Linear codes and modular curves. In: Current problems in mathematics, vol. 25, pp. 209–257. Akad. Nauk SSSR Vseoyuz, Moscow (1984).Zariski O.: The reduction of the singularities of an algebraic surface. Ann. Math. 40, 639–689 (1939)Zariski O.: Local uniformization on algebraic varieties. Ann. Math. 41, 852–896 (1940)Zariski O., Samuel P.(1960) Commutative Algebra, vol. II. Springer, Berlin