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Theses

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

Isabella Panaccione 1 
1 GRACE - Geometry, arithmetic, algorithms, codes and encryption
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : This thesis deals with algebraic geometric (AG) codes and their decoding. Those codes are composed of vectors constructed by evaluating specific functions at points of an algebraic curve. The underlying algebraic structure of these codes made it possible to design several decoding algorithms. A first one, for codes from plane curves is proposed in 1989 by Justesen, Larsen, Jensen, Havemose and Hoholdt. It is then extended to any curve by Skorobatov and Vladut and called ”basic algorithm” in the literature. A few years later, Pellikaan and independently Koetter, give a formulation without algebraic geometry using simply the language of codes. This new interpretation, takes the name ”Error Correcting Pairs” (ECP) algorithm and represents a breakthrough in coding theory since it applies to every code having a certain structure which is described only in terms of component-wise products of codes. The decoding radius of this algorithm depends on the code to which it is applied. For Reed-Solomon codes, it reaches half the minimum distance, which is the threshold for the solution to be unique. For AG, the algorithm almost always manages to decode a quantity of errors equal to half the designed distance. However, the success of the algorithm is only guaranteed for a quantity of errors less than half the designed distance minus some multiple curve’s genus. Several attempts were then made to erase this genus-proportional penalty. A first decisive result was that of Pellikaan, who proved the existence of an algorithm with a decoding radius equal to half the designed distance. Then in 1993 Ehrhard obtained an effective procedure for constructing such an algorithm. In addition to the algorithms for unique decoding, AG codes have algorithms correcting amount of errors greater than half the designed distance. Beyond this quantity, the uniqueness of the solution may not be guaranteed. We then use a so-called ”list decoding” algorithm which returns the list of any possible solutions. This is the case of Sudan’s algorithm for Reed-Solomon codes. Another approach consists in designing algorithms, which returns a single solution but may fail. This is the case of the ”power decoding”. Sudan’s and power decoding algorithms have first been designed for Reed-Solomon codes, then extended to AG codes. We observe 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 new algorithm that we call ”power error locating pairs” which, like the ECP algorithm, can be applied to any code with a certain structure described in terms of component-wise products. Compared to the ECP algorithm, this algorithm can correct errors beyond half the designed distance of the code. Applied to Reed-Solomon or to AG codes, it is equivalent to the power decoding algorithm. But it can also be applied to specific cyclic codes for which it can be used to decode beyond half the Roos bound. Moreover, this algorithm applied to AG codes disregards the underlying geometric structure which opens up interesting applications in cryptanalysis. The second result aims to erase the penalty proportional to the genus in the decoding radius of Sudan’s algorithm for AG codes. First, by following Pellikaan’s method, we prove that such an algorithm exists. Then, by combining and generalizing the works of Ehrhard and Sudan, we give an effective procedure to build this algorithm.
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Submitted on : Wednesday, January 5, 2022 - 12:13:13 PM
Last modification on : Sunday, June 26, 2022 - 3:30:49 AM
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Isabella Panaccione. On decoding algorithms for algebraic geometry codes beyond half the minimum distance. Algebraic Geometry [math.AG]. École polytechnique, 2021. English. ⟨tel-03512261v1⟩

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