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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).  ...  We improve previously known lower bounds for the minimum distance of certain two-point AG codes constructed using a Generalized Giulietti-Korchmaros curve (GGK).  ...  Two-point AG codes on the generalized GK curve. Since the curves χ e are maximal, they are good candidates to be used for the construction of error-correcting codes.  ... 
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).  ...  We improve previously known lower bounds for the minimum distance of certain two-point AG codes constructed using a Generalized Giulietti-Korchmaros curve (GGK).  ...  Two-point AG codes on the generalized GK curve. Since the curves χ e are maximal, they are good candidates to be used for the construction of error-correcting codes.  ... 
doi:10.1109/tit.2017.2763165 fatcat:afi64frwhvcurorbie3cz42emq

AG codes from the second generalization of the GK maximal curve [article]

Maria Montanucci, Vicenzo Pallozzi Lavorante
2019 arXiv   pre-print
A new proof of the fact that the first and the second generalized GK curves are not isomorphic for any n ≥ 5 is obtained.  ...  The second generalized GK maximal curves GK_2,n are maximal curves over finite fields with q^2n elements, where q is a prime power and n ≥ 3 an odd integer, constructed by Beelen and Montanucci.  ...  AG codes from the first and second generalized GK curves 6.2. AG quantum codes for the second generalized GK curve.  ... 
arXiv:1901.08897v1 fatcat:r5hphb4r5jgzlfn3xbbaczywcy

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.  ...  Section 2 contains general results about Weierstrass semigroup and discrepancy, in addition to basic facts about AG codes and 1 the GK curve.  ... 
arXiv:1705.05814v1 fatcat:jcuxrs6itvewrpd7gbczmgdpe4

Minimum weight codewords in dual Algebraic-Geometric codes from the Giulietti-Korchmáros curve [article]

Daniele Bartoli, Matteo Bonini
2018 arXiv   pre-print
In this paper we investigate the number of minimum weight codewords of some dual Algebraic-Geometric codes associated with the Giulietti-Korchm\'aros maximal curve, by computing the maximal number of intersections  ...  between the Giulietti-Korchm\'aros curve and lines, plane conics and plane cubics.  ...  Note that if the two lines are (ℓ 2 − ℓ + 1)-secants then their common point is (0, 1, 0, 0) / ∈ GK. The previous result can be generalized to a plane curve of degree α ≤ ℓ. Proposition 3.4.  ... 
arXiv:1802.03359v1 fatcat:l2e7hhtssbgdvglzgayvylerie

Two-point AG codes on the GK maximal curves [article]

Alonso Sepúlveda, Guilherme Tizziotti
2015 arXiv   pre-print
We determine de Weierstrass semigroup of a pair of certain rational points on the GK-curves.  ...  We use this semigroup to obtain two-point AG codes with better parameters than comparable one-point AG codes arising from these curves. These parameters are new records in the MinT's tables.  ...  Two-Point codes on GK curve In this section we present two-point codes over GK whose parameters are new records in the MinT's tables.  ... 
arXiv:1507.06620v1 fatcat:lm4h72u3gfcxppu3nsi3oefzpy

Griffith–Kelly–Sherman Correlation Inequalities: A Useful Tool in the Theory of Error Correcting Codes

Nicolas Macris
2007 IEEE Transactions on Information Theory  
The correlation inequality yields a sharp lower bound on the GEXIT curve.  ...  Then, considering communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code we prove that, under a natural assumption, part of the GEXIT curve (associated to MAP  ...  He also thanks Cyril Measson for many explanations on his work on the BEC and Henry Pfister for useful comments.  ... 
doi:10.1109/tit.2006.889002 fatcat:64qpbhy3cjfcrlre3ez6554ola

On the weights of dual codes arising from the GK curve [article]

Edoardo Ballico, Matteo Bonini
2019 arXiv   pre-print
We compute the minimum distance and the minimum weight codewords of such codes and we investigate the generalized hamming weights of such codes.  ...  In this paper we investigate some dual algebraic-geometric codes associated with the Giulietti-Korchm\'aros maximal curve.  ...  ON THE WEIGHTS OF DUAL CODES ARISING FROM THE GK CURVE  ... 
arXiv:1909.08126v1 fatcat:bhgugf274rb33byqyait4wd2ci

Two-point coordinate rings for GK-curves [article]

Iwan M. Duursma
2010 arXiv   pre-print
We give a new proof for the maximality of the generalized GK-curves and we outline methods to efficiently obtain their two-point coordinate ring.  ...  The generalized GK-curves have affine equations x^q+x = y^q+1 and y^q^2-y^q = z^r, for r=(q^n+1)/(q+1).  ...  For the generalized GK-curves, this is not the case. For the GK-curve itself, the rational points divide into two orbits [7] .  ... 
arXiv:1012.3682v1 fatcat:ui4wmoz4m5fhdbrnwuhn7fsuqy

AG codes on certain maximal curves [article]

Stefania Fanali, Massimo Giulietti
2009 arXiv   pre-print
Algebraic Geometric codes associated to a recently discovered class of maximal curves are investigated.  ...  As a result, some linear codes with better parameters with respect to the previously known ones are discovered, and 70 improvements on MinT's tables are obtained.  ...  AG Codes and Improved AG Codes associated to the GK-curve Throughout this section we keep the notation of the previous Sections. For q =q 3 , let X be the GK curve defined by (3.1).  ... 
arXiv:0906.2935v1 fatcat:lpuqnnqpznhk3hb45ooikyhj6i

Two-point AG codes from the Beelen-Montanucci maximal curve [article]

Leonardo Landi, Lara Vicino
2021 arXiv   pre-print
We study properties of certain two-point Weierstrass semigroups of the curve and use them for determining a lower bound on the minimum distance of such codes.  ...  In this paper we investigate two-point algebraic-geometry codes (AG codes) coming from the Beelen-Montanucci (BM) maximal curve.  ...  In Section 2, general results on AG codes will be presented, with a particular focus on two-point AG codes.  ... 
arXiv:2106.14564v2 fatcat:thrxniz3ijaepmf6xbbcbudnuu

Fuzzy image segmentation of generic shaped clusters

M.A. Ali, G.C. Karmakar, L.S. Dooley
2005 IEEE International Conference on Image Processing 2005  
This limitation was the primary motivation in our investigation into using shape information to improve the generality of these algorithms.  ...  The new algorithm has also been shown to be application independent so it can be applied in areas such as video object plane segmentation in MPEG-4 based coding.  ...  A set of significant points for a shape were then generated using the convex hull of the respective initial segmentation and either a Bezier Curve (BC) or B-spline [11] approximation used to generate  ... 
doi:10.1109/icip.2005.1530277 dblp:conf/icip/AliKD05 fatcat:mpwdfxmmrfav7fe55piw4yrpeu

On certain self-orthogonal AG codes with applications to Quantum error-correcting codes [article]

Daniele Bartoli, Maria Montanucci, Giovanni Zini
2019 arXiv   pre-print
Examples are given by Castle curves, GK curves, generalized GK curves and the Abdon-Bezerra-Quoos maximal curves. Applications of our method to these curves are provided.  ...  Several classes of well-known algebraic curves with many rational points turn out to be Swiss curves.  ...  The research of D. Bartoli  ... 
arXiv:1912.08021v1 fatcat:c4ahb2bftza6zmxrvz6fnobp6m

An Elliptic-Curve-Based Hierarchical Cluster Key Management in Wireless Sensor Network [chapter]

Srikanta Kumar Sahoo, Manmanth Narayan Sahoo
2013 Advances in Intelligent Systems and Computing  
The proposed work uses digital signature scheme and encryptiondecryption mechanisms using elliptic curve cryptography(ECC).  ...  In this paper we proposed an elliptic curve based hierarchical cluster key management scheme, which is very much secure, have better time complexity and consumes reasonable amount of energy.  ...  (b) CH generates two points on elliptic curve: C1 = r * P and C2=r * Q i +M. 4. CH generates ECDSA Signature of the message sig(M) by using its private key. 5.  ... 
doi:10.1007/978-81-322-1665-0_38 fatcat:p7nf5hmb7zgujl6zbgxqes3j7y

Locally Recoverable Codes with Availability t≥ 2 from Fiber Products of Curves [article]

Kathryn Haymaker, Beth Malmskog, Gretchen Matthews
2018 arXiv   pre-print
Employing maximal curves, we create several new families of locally recoverable codes with multiple recovery sets, including codes with two recovery sets from the generalized Giulietti and Korchmáros (  ...  GK) curves and the Suzuki curves, and new locally recoverable codes with many recovery sets based on the Hermitian curve, using a fiber product construction of van der Geer and van der Vlugt.  ...  In addition, we would like to acknowledge the hospitality of IPAM and the organizers of the 2016 Algebraic Geometry for Coding Theory and Cryptography Workshop: Everett Howe, Kristin Lauter, and Judy Walker  ... 
arXiv:1612.03841v4 fatcat:ansbd35nm5ah3lt24bieob3a7y
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