3,367 research outputs found

    Feng-Rao decoding of primary codes

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    We show that the Feng-Rao bound for dual codes and a similar bound by Andersen and Geil [H.E. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields Appl., 14 (2008), pp. 92-123] for primary codes are consequences of each other. This implies that the Feng-Rao decoding algorithm can be applied to decode primary codes up to half their designed minimum distance. The technique applies to any linear code for which information on well-behaving pairs is available. Consequently we are able to decode efficiently a large class of codes for which no non-trivial decoding algorithm was previously known. Among those are important families of multivariate polynomial codes. Matsumoto and Miura in [R. Matsumoto and S. Miura, On the Feng-Rao bound for the L-construction of algebraic geometry codes, IEICE Trans. Fundamentals, E83-A (2000), pp. 926-930] (See also [P. Beelen and T. H{\o}holdt, The decoding of algebraic geometry codes, in Advances in algebraic geometry codes, pp. 49-98]) derived from the Feng-Rao bound a bound for primary one-point algebraic geometric codes and showed how to decode up to what is guaranteed by their bound. The exposition by Matsumoto and Miura requires the use of differentials which was not needed in [Andersen and Geil 2008]. Nevertheless we demonstrate a very strong connection between Matsumoto and Miura's bound and Andersen and Geil's bound when applied to primary one-point algebraic geometric codes.Comment: elsarticle.cls, 23 pages, no figure. Version 3 added citations to the works by I.M. Duursma and R. Pellikaa

    Evaluation codes defined by finite families of plane valuations at infinity

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    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 Fq{\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. 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    List Decoding Algorithm based on Voting in Groebner Bases for General One-Point AG Codes

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    We generalize the unique decoding algorithm for one-point AG codes over the Miura-Kamiya Cab curves proposed by Lee, Bras-Amor\'os and O'Sullivan (2012) to general one-point AG codes, without any assumption. We also extend their unique decoding algorithm to list decoding, modify it so that it can be used with the Feng-Rao improved code construction, prove equality between its error correcting capability and half the minimum distance lower bound by Andersen and Geil (2008) that has not been done in the original proposal except for one-point Hermitian codes, remove the unnecessary computational steps so that it can run faster, and analyze its computational complexity in terms of multiplications and divisions in the finite field. As a unique decoding algorithm, the proposed one is empirically and theoretically as fast as the BMS algorithm for one-point Hermitian codes. As a list decoding algorithm, extensive experiments suggest that it can be much faster for many moderate size/usual inputs than the algorithm by Beelen and Brander (2010). It should be noted that as a list decoding algorithm the proposed method seems to have exponential worst-case computational complexity while the previous proposals (Beelen and Brander, 2010; Guruswami and Sudan, 1999) have polynomial ones, and that the proposed method is expected to be slower than the previous proposals for very large/special inputs.Comment: Accepted for publication in J. Symbolic Computation. LaTeX2e article.cls, 42 pages, 4 tables, no figures. Ver. 6 added an illustrative example of the algorithm executio

    Subquadratic time encodable codes beating the Gilbert-Varshamov bound

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    We construct explicit algebraic geometry codes built from the Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for alphabet sizes at least 192. Messages are identied with functions in certain Riemann-Roch spaces associated with divisors supported on multiple places. Encoding amounts to evaluating these functions at degree one places. By exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and 1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list) decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent. If \omega = 2, as widely believed, the encoding and decoding runtimes are respectively nearly linear and nearly quadratic. Prior to this work, encoding (resp. decoding) time of code families beating the Gilbert-Varshamov bound were quadratic (resp. cubic) or worse

    Cryptanalysis of public-key cryptosystems that use subcodes of algebraic geometry codes

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    We give a polynomial time attack on the McEliece public key cryptosystem based on subcodes of algebraic geometry (AG) codes. The proposed attack reposes on the distinguishability of such codes from random codes using the Schur product. Wieschebrink treated the genus zero case a few years ago but his approach cannot be extent straightforwardly to other genera. We address this problem by introducing and using a new notion, which we call the t-closure of a code

    Distance bounds for algebraic geometric codes

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    Various methods have been used to obtain improvements of the Goppa lower bound for the minimum distance of an algebraic geometric code. The main methods divide into two categories and all but a few of the known bounds are special cases of either the Lundell-McCullough floor bound or the Beelen order bound. The exceptions are recent improvements of the floor bound by Guneri-Stichtenoth-Taskin, and Duursma-Park, and of the order bound by Duursma-Park and Duursma-Kirov. In this paper we provide short proofs for all floor bounds and most order bounds in the setting of the van Lint and Wilson AB method. Moreover, we formulate unifying theorems for order bounds and formulate the DP and DK order bounds as natural but different generalizations of the Feng-Rao bound for one-point codes.Comment: 29 page
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