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

Elise Barelli, Peter Beelen, Mrinmoy Datta, Vincent Neiger, Johan Rosenkilde
<span title="2017-10-07">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1706.00800v2">arXiv:1706.00800v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/u3shjv63y5b4nipeaka5bnx5vi">fatcat:u3shjv63y5b4nipeaka5bnx5vi</a> </span>
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Two-Point Codes for the Generalized GK Curve

Elise Barelli, Peter Beelen, Mrinmoy Datta, Vincent Neiger, Johan Rosenkilde
<span title="">2018</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tit.2017.2763165">doi:10.1109/tit.2017.2763165</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/afi64frwhvcurorbie3cz42emq">fatcat:afi64frwhvcurorbie3cz42emq</a> </span>
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AG codes from the second generalization of the GK maximal curve [article]

Maria Montanucci, Vicenzo Pallozzi Lavorante
<span title="2019-01-25">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1901.08897v1">arXiv:1901.08897v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r5hphb4r5jgzlfn3xbbaczywcy">fatcat:r5hphb4r5jgzlfn3xbbaczywcy</a> </span>
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Weierstrass Semigroup and Pure Gaps at several points on the GK curve [article]

Alonso S. Castellanos, Guilherme Tizziotti
<span title="2017-05-16">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1705.05814v1">arXiv:1705.05814v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jcuxrs6itvewrpd7gbczmgdpe4">fatcat:jcuxrs6itvewrpd7gbczmgdpe4</a> </span>
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Minimum weight codewords in dual Algebraic-Geometric codes from the Giulietti-Korchmáros curve [article]

Daniele Bartoli, Matteo Bonini
<span title="2018-02-09">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1802.03359v1">arXiv:1802.03359v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/l2e7hhtssbgdvglzgayvylerie">fatcat:l2e7hhtssbgdvglzgayvylerie</a> </span>
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Two-point AG codes on the GK maximal curves [article]

Alonso Sepúlveda, Guilherme Tizziotti
<span title="2015-07-23">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1507.06620v1">arXiv:1507.06620v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lm4h72u3gfcxppu3nsi3oefzpy">fatcat:lm4h72u3gfcxppu3nsi3oefzpy</a> </span>
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Griffith–Kelly–Sherman Correlation Inequalities: A Useful Tool in the Theory of Error Correcting Codes

Nicolas Macris
<span title="">2007</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tit.2006.889002">doi:10.1109/tit.2006.889002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/64qpbhy3cjfcrlre3ez6554ola">fatcat:64qpbhy3cjfcrlre3ez6554ola</a> </span>
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On the weights of dual codes arising from the GK curve [article]

Edoardo Ballico, Matteo Bonini
<span title="2019-09-17">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1909.08126v1">arXiv:1909.08126v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bhgugf274rb33byqyait4wd2ci">fatcat:bhgugf274rb33byqyait4wd2ci</a> </span>
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Two-point coordinate rings for GK-curves [article]

Iwan M. Duursma
<span title="2010-12-16">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1012.3682v1">arXiv:1012.3682v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ui4wmoz4m5fhdbrnwuhn7fsuqy">fatcat:ui4wmoz4m5fhdbrnwuhn7fsuqy</a> </span>
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AG codes on certain maximal curves [article]

Stefania Fanali, Massimo Giulietti
<span title="2009-06-16">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0906.2935v1">arXiv:0906.2935v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lpuqnnqpznhk3hb45ooikyhj6i">fatcat:lpuqnnqpznhk3hb45ooikyhj6i</a> </span>
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Two-point AG codes from the Beelen-Montanucci maximal curve [article]

Leonardo Landi, Lara Vicino
<span title="2021-11-02">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2106.14564v2">arXiv:2106.14564v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/thrxniz3ijaepmf6xbbcbudnuu">fatcat:thrxniz3ijaepmf6xbbcbudnuu</a> </span>
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Fuzzy image segmentation of generic shaped clusters

M.A. Ali, G.C. Karmakar, L.S. Dooley
<span title="">2005</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/anlh4tvwprcrtoxv5d4h6a7rye" style="color: black;">IEEE International Conference on Image Processing 2005</a> </i> &nbsp;
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  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/icip.2005.1530277">doi:10.1109/icip.2005.1530277</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/icip/AliKD05.html">dblp:conf/icip/AliKD05</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mpwdfxmmrfav7fe55piw4yrpeu">fatcat:mpwdfxmmrfav7fe55piw4yrpeu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180719095742/http://oro.open.ac.uk/11403/1/ICIP_2005_Ameer_Karmakar_and_Dooley.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/16/f4/16f413b15ab724d9e4aa0568e564448dd9a50b6d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/icip.2005.1530277"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

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

Daniele Bartoli, Maria Montanucci, Giovanni Zini
<span title="2019-12-17">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1912.08021v1">arXiv:1912.08021v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/c4ahb2bftza6zmxrvz6fnobp6m">fatcat:c4ahb2bftza6zmxrvz6fnobp6m</a> </span>
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An Elliptic-Curve-Based Hierarchical Cluster Key Management in Wireless Sensor Network [chapter]

Srikanta Kumar Sahoo, Manmanth Narayan Sahoo
<span title="2013-12-18">2013</span> <i title="Springer India"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/kax3wwzwmncwhi472pxbzqsjja" style="color: black;">Advances in Intelligent Systems and Computing</a> </i> &nbsp;
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-81-322-1665-0_38">doi:10.1007/978-81-322-1665-0_38</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/p7nf5hmb7zgujl6zbgxqes3j7y">fatcat:p7nf5hmb7zgujl6zbgxqes3j7y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170812090420/http://dspace.nitrkl.ac.in/dspace/bitstream/2080/1953/1/ICACNI-209.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/8e/e7/8ee7e5d22dcee55fe9e553d1f629fc02cdb4801d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-81-322-1665-0_38"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

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

Kathryn Haymaker, Beth Malmskog, Gretchen Matthews
<span title="2018-08-16">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1612.03841v4">arXiv:1612.03841v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ansbd35nm5ah3lt24bieob3a7y">fatcat:ansbd35nm5ah3lt24bieob3a7y</a> </span>
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