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On decoding algorithms for algebraic geometry codes beyond half the minimum distance

Isabella Panaccione 1
1 GRACE - Geometry, arithmetic, algorithms, codes and encryption
Inria Saclay - Ile de France, LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
Abstract : This thesis deals with algebraic geometric (AG) codes and theirdecoding. Those codes are composed of vectors constructed by evaluatingspecific functions at points of an algebraic curve. The underlyingalgebraic structure of these codes made it possible to design severaldecoding algorithms. A first one, for codes from plane curves isproposed in 1989 by Justesen, Larsen, Jensen, Havemose and Hoholdt. Itis then extended to any curve by Skorobatov and Vladut and called"basic algorithm" in the literature. A few years later, Pellikaan andindependently Koetter, give a formulation without algebraic geometryusing simply the language of codes. This new interpretation, takes thename "Error Correcting Pairs" (ECP) algorithm and represents abreakthrough in coding theory since it applies to every code having acertain structure which is described only in terms of component-wiseproducts of codes. The decoding radius of this algorithm depends onthe code to which it is applied. For Reed-Solomon codes, it reacheshalf the minimum distance, which is the threshold for the solution tobe unique. For AG, the algorithm almost always manages todecode a quantity of errors equal to half the designeddistance. However, the success of the algorithm is only guaranteed fora quantity of errors less than half the designed distance minussome multiple curve's genus. Several attempts were thenmade to erase this genus-proportional penalty. A first decisiveresult was that of Pellikaan, who proved the existence of an algorithmwith a decoding radius equal to half the designed distance. Thenin 1993 Ehrhard obtained an effective procedure for constructing such analgorithm.In addition to the algorithms for unique decoding, AG codes havealgorithms correcting amount of errors greater than half thedesigned distance. Beyond this quantity, the uniqueness of thesolution may not be guaranteed. We then use a so-called "listdecoding" algorithm which returns the list of any possiblesolutions. This is the case of Sudan's algorithm for Reed-Solomoncodes. Another approach consists in designing algorithms, whichreturns a single solution but may fail. This is the case ofthe "power decoding". Sudan's and power decoding algorithms have firstbeen designed for Reed-Solomon codes, then extended to AG codes. Weobserve that these extensions do not have the same decoding radii:that of Sudan algorithm is lower than that of the power decoding,the difference being proportional to the genus of the curve.In this thesis we present two main results. First, we propose a newalgorithm that we call "power error locating pairs" which, like theECP algorithm, can be applied to any code with a certain structuredescribed in terms of component-wise products. Compared to the ECPalgorithm, this algorithm can correct errors beyond half thedesigned distance of the code. Applied to Reed-Solomon or to AG codes,it is equivalent to the power decoding algorithm. But it can also beapplied to specific cyclic codes for which it can be used to decodebeyond half the Roos bound. Moreover, this algorithm applied to AGcodes disregards the underlying geometric structure whichopens up interesting applications in cryptanalysis.The second result aims to erase the penalty proportional to thegenus in the decoding radius of Sudan's algorithm forAG codes. First, by following Pellikaan's method, weprove that such an algorithm exists. Then, by combining andgeneralizing the works of Ehrhard and Sudan, we give aneffective procedure to build this algorithm.
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Submitted on : Wednesday, January 19, 2022 - 3:48:15 PM
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Isabella Panaccione. On decoding algorithms for algebraic geometry codes beyond half the minimum distance. Information Theory [cs.IT]. Institut Polytechnique de Paris, 2021. English. ⟨NNT : 2021IPPAX101⟩. ⟨tel-03512261v2⟩

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