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Group codes on certain algebraic curves with many rational points

Authors:

Johan P. Hansen,

Henning StichtenothAuthors Info & Claims

Applicable Algebra in Engineering, Communication and Computing, Volume 1, Issue 1

Pages 67 - 77

https://doi.org/10.1007/BF01810849

Published: 01 March 1990 Publication History

Abstract

We construct a series of algebraic geometric codes using a class of curves which have many rational points. We obtain codes of lengthq2 over $$\mathbb{F}$$ q, whereq = 2q02 andq0 = 2n, such that dimension + minimal distance ?q2 + 1 ? q0(q ? 1). The codes are ideals in the group algebra $$\mathbb{F}$$ q[S], whereS is a Sylow-2-subgroup of orderq2 of the Suzuki-group of orderq2(q2 + 1)(q ? 1). The curves used for construction have in relation to their genera the maximal number of $$\mathbb{F}$$ GFq-rational points. This maximal number is determined by the explicit formulas of Weil and is effectively smaller than the Hasse--Weil bound.

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Navarro H(2024)Bases for Riemann–Roch spaces of linearized function fields with applications to generalized algebraic geometry codesDesigns, Codes and Cryptography10.1007/s10623-024-01426-692:10(3033-3048)Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s10623-024-01426-6
Matthews GPiñero F(2020)Codes with locality from cyclic extensions of Deligne–Lusztig curvesDesigns, Codes and Cryptography10.1007/s10623-020-00767-288:9(1909-1924)Online publication date: 3-Jul-2020
https://dl.acm.org/doi/10.1007/s10623-020-00767-2
Eid AHasson HKsir APeachey J(2016)Suzuki-invariant codes from the Suzuki curveDesigns, Codes and Cryptography10.1007/s10623-015-0164-581:3(413-425)Online publication date: 1-Dec-2016
https://dl.acm.org/doi/10.1007/s10623-015-0164-5
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Group codes on certain algebraic curves with many rational points
1. Mathematics of computing
  1. Information theory
2. Theory of computation
  1. Formal languages and automata theory
    1. Formalisms

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cover image Applicable Algebra in Engineering, Communication and Computing

Applicable Algebra in Engineering, Communication and Computing Volume 1, Issue 1

March 1990

72 pages

ISSN:0938-1279

EISSN:1432-0622

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 March 1990

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Navarro H(2024)Bases for Riemann–Roch spaces of linearized function fields with applications to generalized algebraic geometry codesDesigns, Codes and Cryptography10.1007/s10623-024-01426-692:10(3033-3048)Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s10623-024-01426-6
Matthews GPiñero F(2020)Codes with locality from cyclic extensions of Deligne–Lusztig curvesDesigns, Codes and Cryptography10.1007/s10623-020-00767-288:9(1909-1924)Online publication date: 3-Jul-2020
https://dl.acm.org/doi/10.1007/s10623-020-00767-2
Eid AHasson HKsir APeachey J(2016)Suzuki-invariant codes from the Suzuki curveDesigns, Codes and Cryptography10.1007/s10623-015-0164-581:3(413-425)Online publication date: 1-Dec-2016
https://dl.acm.org/doi/10.1007/s10623-015-0164-5
Ben-Sasson EGabizon AKaplan YKopparty SSaraf SBoneh DRoughgarden TFeigenbaum J(2013)A new family of locally correctable codes based on degree-lifted algebraic geometry codesProceedings of the forty-fifth annual ACM symposium on Theory of Computing10.1145/2488608.2488714(833-842)Online publication date: 1-Jun-2013
https://dl.acm.org/doi/10.1145/2488608.2488714
Drake NMatthews G(2010)Minimum distance decoding of general algebraic geometry codes via listsIEEE Transactions on Information Theory10.1109/TIT.2010.205467056:9(4335-4340)Online publication date: 1-Sep-2010
https://dl.acm.org/doi/10.1109/TIT.2010.2054670
Fanali SGiulietti M(2010)One-point AG codes on the GK maximal curvesIEEE Transactions on Information Theory10.1109/TIT.2009.203482656:1(202-210)Online publication date: 1-Jan-2010
https://dl.acm.org/doi/10.1109/TIT.2009.2034826
Giulietti MKorchmáros G(2008)On automorphism groups of certain Goppa codesDesigns, Codes and Cryptography10.1007/s10623-007-9110-547:1-3(177-190)Online publication date: 1-Jun-2008
https://dl.acm.org/doi/10.1007/s10623-007-9110-5
Maharaj HMatthews GPirsic G(2005)Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequencesJournal of Pure And Applied Algebra10.1016/j.jpaa.2004.06.010195:3(261-280)Online publication date: 1-Feb-2005
https://dl.acm.org/doi/10.1016/j.jpaa.2004.06.010
Munuera CTorres F(2004)Bounding the Trellis State Complexity of Algebraic Geometric CodesApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-004-0150-z15:2(81-100)Online publication date: 1-Sep-2004
https://dl.acm.org/doi/10.1007/s00200-004-0150-z
Campillo AFarrán J(2002)Symbolic Hamburger-Noether expressions of plane curves and applications to AG codesMathematics of Computation10.1090/S0025-5718-01-01390-471:240(1759-1780)Online publication date: 1-Oct-2002
https://dl.acm.org/doi/10.1090/S0025-5718-01-01390-4
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Curves with Many Points and Multiplication Complexity in Any Extension of Fq

Dual codes of systematic group codes over abelian groups

Rational parametrization of conchoids to algebraic curves

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