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On Goppa Codes and Weierstrass Gaps at Several Points

C�cero Carvalho, Fernando Torres
2005 Designs, Codes and Cryptography  
We generalize results of Homma and Kim [2001, J. Pure Appl. Algebra 162, 273-290] concerning an improvement on the Goppa bound on the minimum distance of certain Goppa codes. MSC: 94B27; 14H55; 14G50.  ...  Basic facts on Weierstrass gaps at several points The aim of this section is to recall the basic definitions and prove some facts on Weierstrass semigroups at several points.  ...  This characterization holds true for pure Weierstrass gaps at several points as we shall see in Lemma 2.5.  ... 
doi:10.1007/s10623-005-6403-4 fatcat:i4jz2uq6kjaxlm7aggmvj4z33i

A generalized floor bound for the minimum distance of geometric Goppa codes

Benjamin Lundell, Jason McCullough
2006 Journal of Pure and Applied Algebra  
We include examples of the bound applied to one-and two-point codes over certain Suzuki and Hermitian curves.  ...  We prove a new bound for the minimum distance of geometric Goppa codes that generalizes two previous improved bounds.  ...  Acknowledgments The authors would like to thank the referee for helping to improve the presentation, and Professor Iwan Duursma for his advising and tutelage through the course of this project.  ... 
doi:10.1016/j.jpaa.2005.09.016 fatcat:cc7hr4mwdjas7cvfhgnk2vstvq

Page 2407 of Mathematical Reviews Vol. , Issue 94d [page]

1994 Mathematical Reviews  
F. (1-LAS; Baton Rouge, LA) Consecutive Weierstrass gaps and minimum distance of Goppa codes.  ...  points, which do not lie in the support of G, one constructs two geometric Goppa codes as follows [cf.  ... 

A Generalized Floor Bound for the Minimum Distance of Geometric Goppa Codes and its Application to Two-Point Codes [article]

Benjamin Lundell, Jason McCullough
2004 arXiv   pre-print
We include examples of the bound to one and two point codes over both the Suzuki and Hermitian curves.  ...  We prove a new bound for the minimum distance of geometric Goppa codes that generalizes two previous improved bounds.  ...  Nor does it improve on the F-R bound for one-point codes.  ... 
arXiv:math/0408341v2 fatcat:oahyid2rsvcphpvaytwyw6h33e

Page 6617 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
When F = mQ, where Q is a Weierstrass point and m is ex- pressed in terms of two Weierstrass gaps at Q, or when F = m,Q | +m2Q>2, where m),m 2 are expressed in terms of Weierstrass pure gaps at (Q),Q2)  ...  For a rational point Q on 2, a natural number y is a Weierstrass gap at P if there is no rational function f € F,(X) such that yP is the pole divisor of /.  ... 

Weierstrass pure gaps on curves with three distinguished points [article]

Herivelto Borges, Gregory Duran
2021 arXiv   pre-print
For such curves, we determine the dimension of certain special divisors supported on {P_1,P_2,P_3}, as well as an explicit description of all pure gaps at any subset of {P_1,P_2,P_3}.  ...  In this paper, we consider the class of smooth plane curves of degree n+1>3 over 𝕂, containing three points, P_1,P_2, and P_3, such that nP_1+P_2, nP_2+P_3, and nP_3+P_1 are divisors cut out by three  ...  Acknowledgment: The first author thanks the financial support of CNPq-Brazil (grants 421440/2016-3 and 311572/2019-7).  ... 
arXiv:2107.08290v1 fatcat:rpfukcdjirfrpirv6voime6y7y

Explicit bases of the Riemann-Roch spaces on divisors on hyperelliptic curves [article]

Giovanni Falcone, Ágota Figula, Carolin Hannusch
2020 arXiv   pre-print
As an application, we determine a generator matrix of a Goppa code for j=g=3 and n=4.  ...  point, taken as the point at infinity.  ...  We want to construct the (m, 2, d)-Goppa code (where m−4 ≤ d ≤ m−1) 1 . Therefore we need the equation of H, so that we can take m points on H.  ... 
arXiv:2012.08870v1 fatcat:fg72rw3forfapaghpxzs7g3yme

Weierstrass Semigroup and Pure Gaps at several points on the GK curve [article]

Alonso S. Castellanos, Guilherme Tizziotti
2017 arXiv   pre-print
We determine the Weierstrass semigroup H(P_∞, P_1, ... , P_m) at several points on the GK curve. In addition, we present conditions to find pure gaps on the set of gaps G(P_∞, P_1, ... , P_m).  ...  Finally, we apply the results to obtain AG codes with good relative parameters.  ...  WEIERSTRASS SEMIGROUP AND PURE GAPS AT SEVERAL POINTS ON THE GK CURVE 11 Example 4.5. Consider the GK curve over F 3 6 with affine equations Z 7 = Y (2 + X 2 + 2X 4 + X 6 ) , X 3 + X = Y 4 .  ... 
arXiv:1705.05814v1 fatcat:jcuxrs6itvewrpd7gbczmgdpe4

Two-Point Codes for the Generalized GK curve [article]

Elise Barelli, Peter Beelen, Mrinmoy Datta, Vincent Neiger, Johan Rosenkilde
2017 arXiv   pre-print
Castellanos and Tizziotti recently described such bounds for two-point codes coming from the Giulietti-Korchmaros curve (GK).  ...  Our method builds on the order bound for AG codes: to enable this, we study certain Weierstrass semigroups. This allows an efficient algorithm for computing our improved bounds.  ...  The set Γ(Q; G) := Z ≥vQ(G)−deg(G) \ H(Q; G) is called the set of G-gaps at Q. Note that if G = 0 we obtain H(Q; 0) = H(Q), the Weierstrass semigroup of Q, and Γ(Q; 0) = Γ(Q), the set of gaps at Q.  ... 
arXiv:1706.00800v2 fatcat:u3shjv63y5b4nipeaka5bnx5vi

Two-Point Codes for the Generalized GK Curve

Elise Barelli, Peter Beelen, Mrinmoy Datta, Vincent Neiger, Johan Rosenkilde
2018 IEEE Transactions on Information Theory  
Castellanos and Tizziotti recently described such bounds for two-point codes coming from the Giulietti-Korchmaros curve (GK).  ...  Our method builds on the order bound for AG codes: to enable this, we study certain Weierstrass semigroups. This allows an efficient algorithm for computing our improved bounds.  ...  The set Γ(Q; G) := Z ≥v Q (G)−deg(G) \ H(Q; G) is called the set of G-gaps at Q. Note that if G = 0 we obtain H(Q; 0) = H(Q), the Weierstrass semigroup of Q, and Γ(Q; 0) = Γ(Q), the set of gaps at Q.  ... 
doi:10.1109/tit.2017.2763165 fatcat:afi64frwhvcurorbie3cz42emq

Constructing Small-Bias Sets from Algebraic-Geometric Codes

Avraham Ben-Aroya, Amnon Ta-Shma
2009 2009 50th Annual IEEE Symposium on Foundations of Computer Science  
Studying the limits of our technique, we arrive at a hypothesis that if true implies the existence of -biased sets with parameters nearly matching the lower bound, and in particular giving binary error  ...  The construction builds on an algebraic-geometric code. However, unlike previous constructions we use low-degree divisors whose degree is significantly smaller than the genus.  ...  In Subsections 3.1 and 3.1.1 we give the necessary background on algebraic function fields and geometric Goppa codes.  ... 
doi:10.1109/focs.2009.44 dblp:conf/focs/Ben-AroyaT09 fatcat:wnt72w2q6zguje66thafigfdte

An Introduction to Algebraic Geometry codes [article]

Carlos Munuera, Wilson Olaya-León
2015 arXiv   pre-print
Starting from evaluation codes and codes from order and weight functions, special attention is given to one-point codes and, in particular, to the family of Castle codes.  ...  We present an introduction to the theory of algebraic geometry codes.  ...  Rao in [13] for the duals of one-point algebraic geometry codes. At the same time, R. Matsumoto and S. Miura independently developed many of the same ideas for duals of one-point codes.  ... 
arXiv:1505.03020v1 fatcat:hywg6b3eevcqrnj7po46wfgt5u

Algebraic Geometric codes from Kummer Extensions [article]

Daniele Bartoli, Luciane Quoos, Giovanni Zini
2016 arXiv   pre-print
For many totally ramified places we give a criterion to find pure gaps at these points and present families of pure gaps.  ...  For Kummer extensions defined by y^m = f (x), where f (x) is a separable polynomial over the finite field F_q, we compute the number of Weierstrass gaps at two totally ramified places.  ...  Algebraic and Geometric Structures and their Applications (GNSAGA -INdAM). The second author L. Quoos was partially supported by CNPq, PDE grant number 200434/2015-2.  ... 
arXiv:1606.04143v2 fatcat:ng5ykbghjfbezoc57rfqax4qgi

Multi-point Codes from the GGS Curves [article]

Chuangqiang Hu, Shudi Yang
2019 arXiv   pre-print
Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps.  ...  Finally, we apply these results to find multi-point codes with excellent parameters. As one of the examples, a presented code with parameters [216,190,≥ 18] over F_64 yields a new record.  ...  The authors are very grateful to the editor and the anonymous reviewers for their valuable comments and suggestions that improved the quality of this paper.  ... 
arXiv:1706.00313v4 fatcat:akmpaia5dnbtxo2j42erpuzjfe

Multi-point codes from the GGS curves

Chuangqiang Hu, ,Yau Mathematical Sciences Center, Tsinghua University, Peking, 100084, China, Shudi Yang, ,School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, China
2019 Advances in Mathematics of Communications  
Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves.  ...  Multi-point codes with excellent parameters are found, among which, a presented code with parameters [216, 190, 18] over F 64 yields a new record.  ...  Acknowledgments The authors are very grateful to the editor and the anonymous reviewers for their valuable comments and suggestions that improved the quality of this paper.  ... 
doi:10.3934/amc.2020020 fatcat:fet76o7n5vbgppp2rclwzo5tli
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