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On Goppa Codes and Weierstrass Gaps at Several Points

2005
*
Designs, Codes and Cryptography
*

We generalize results of Homma

doi:10.1007/s10623-005-6403-4
fatcat:i4jz2uq6kjaxlm7aggmvj4z33i
*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. ...##
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A generalized floor bound for the minimum distance of geometric Goppa codes

2006
*
Journal of Pure and Applied Algebra
*

We include examples of the bound applied to

doi:10.1016/j.jpaa.2005.09.016
fatcat:cc7hr4mwdjas7cvfhgnk2vstvq
*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. ...##
###
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. ...##
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A Generalized Floor Bound for the Minimum Distance of Geometric Goppa Codes and its Application to Two-Point Codes
[article]

2004
*
arXiv
*
pre-print

We include examples of the bound to

arXiv:math/0408341v2
fatcat:oahyid2rsvcphpvaytwyw6h33e
*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*. ...##
###
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]

2021
*
arXiv
*
pre-print

For such curves, we determine the dimension of certain special divisors supported

arXiv:2107.08290v1
fatcat:rpfukcdjirfrpirv6voime6y7y
*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). ...##
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Explicit bases of the Riemann-Roch spaces on divisors on hyperelliptic curves
[article]

2020
*
arXiv
*
pre-print

As an application, we determine a generator matrix of a

arXiv:2012.08870v1
fatcat:fg72rw3forfapaghpxzs7g3yme
*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. ...##
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Weierstrass Semigroup and Pure Gaps at several points on the GK curve
[article]

2017
*
arXiv
*
pre-print

We determine the

arXiv:1705.05814v1
fatcat:jcuxrs6itvewrpd7gbczmgdpe4
*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 . ...##
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Two-Point Codes for the Generalized GK curve
[article]

2017
*
arXiv
*
pre-print

Castellanos

arXiv:1706.00800v2
fatcat:u3shjv63y5b4nipeaka5bnx5vi
*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. ...##
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Two-Point Codes for the Generalized GK Curve

2018
*
IEEE Transactions on Information Theory
*

Castellanos

doi:10.1109/tit.2017.2763165
fatcat:afi64frwhvcurorbie3cz42emq
*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. ...##
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Constructing Small-Bias Sets from Algebraic-Geometric Codes

2009
*
2009 50th Annual IEEE Symposium on Foundations of Computer Science
*

Studying the limits of our technique, we arrive

doi:10.1109/focs.2009.44
dblp:conf/focs/Ben-AroyaT09
fatcat:wnt72w2q6zguje66thafigfdte
*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*. ...##
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An Introduction to Algebraic Geometry codes
[article]

2015
*
arXiv
*
pre-print

Starting from evaluation

arXiv:1505.03020v1
fatcat:hywg6b3eevcqrnj7po46wfgt5u
*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*. ...##
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Algebraic Geometric codes from Kummer Extensions
[article]

2016
*
arXiv
*
pre-print

For many totally ramified places we give a criterion to find pure

arXiv:1606.04143v2
fatcat:ng5ykbghjfbezoc57rfqax4qgi
*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. ...##
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Multi-point Codes from the GGS Curves
[article]

2019
*
arXiv
*
pre-print

Along this line, we characterize explicitly the

arXiv:1706.00313v4
fatcat:akmpaia5dnbtxo2j42erpuzjfe
*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. ...##
###
Multi-point codes from the GGS curves

2019
*
Advances in Mathematics of Communications
*

Along this line, we characterize explicitly the

doi:10.3934/amc.2020020
fatcat:fet76o7n5vbgppp2rclwzo5tli
*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. ...
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