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On Goppa Codes and Weierstrass Gaps at Several Points
2005
Designs, Codes and Cryptography
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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
2006
Journal of Pure and Applied Algebra
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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
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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]
2004
arXiv
pre-print
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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
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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]
2021
arXiv
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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]
2020
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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
2018
IEEE Transactions on Information Theory
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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
2009
2009 50th Annual IEEE Symposium on Foundations of Computer Science
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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]
2015
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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]
2019
arXiv
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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
2019
Advances in Mathematics of Communications
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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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