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Codes de Gabidulin en caractéristique nulle : application au codage espace-temps

Gwezheneg Robert 1, 2 
2 GRACE - Geometry, arithmetic, algorithms, codes and encryption
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : Space-time codes are error correcting codes dedicated to MIMO transmissions. Mathematically, a space-time code is a finite family of complex matrices. Its preformances rely on several parameters, including its minimal rank distance. Gabidulin codes are codes in this metric, famous for their optimality and thanks to efficient decoding algorithms. That's why they are used to design space-time codes. The main difficulty is to design complex matrices from binary matrices. The aim of the works collected here is to generalize Gabidulin codes to number fields, especially cyclique extesnions. We see that they have the same properties than Gabidulin codes over finite fields. We study several errors and erasures models and introduce a quadratic algorithm to recover transmitted information. When computing in finite fields, we are faced with the growing size problem. Indeed, the size of the coefficients grows exponentielly along the algorithm. To avoid this problem, it is possible to reduce the code, in order to compute in a finite field. Finally, we design a family of space-time codes, based on generalised Gabidulin codes. We see that our codes have performances similar to those of existing codes, and that they have additional structure.
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Submitted on : Friday, July 1, 2016 - 2:05:35 PM
Last modification on : Friday, May 20, 2022 - 9:04:48 AM
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  • HAL Id : tel-01308661, version 3


Gwezheneg Robert. Codes de Gabidulin en caractéristique nulle : application au codage espace-temps. Mathématiques générales [math.GM]. Université Rennes 1, 2015. Français. ⟨NNT : 2015REN1S083⟩. ⟨tel-01308661v3⟩



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