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Decoding algorithms for lattices

Abstract : This thesis discusses two problems related to lattices, an old problem and a new one.Both of them are lattice decoding problems: Namely, given a point in the space, find the closest lattice point.The first problem is related to channel coding in moderate dimensions. While efficient lattice schemes exist in low dimensions n < 30 and high dimensions n > 1000, this is not the case of intermediate dimensions. We investigate the decoding of interesting lattices in these intermediate dimensions. We introduce new families of lattices obtained by recursively applying parity checks. These families include famous lattices, such as Barnes-Wall lattices, the Leech and Nebe lattices, as well as new parity lattices.We show that all these lattices can be efficiently decoded with an original recursive list decoder.The second problem involves neural networks. Since 2016 countless papers tried to use deep learning to solve the decoding/detection problem encountered in digital communications. We propose to investigate the complexity of the problem that neural networks should solve. We introduce a new approach to the lattice decoding problem to fit the operations performed by a neural network. This enables to better understand what a neural network can and cannot do in the scope of this problem, and get hints regarding the best architecture of the neural network. Some computer simulations validating our analysis are provided.
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Submitted on : Monday, April 19, 2021 - 3:33:25 PM
Last modification on : Tuesday, September 21, 2021 - 2:16:03 PM
Long-term archiving on: : Tuesday, July 20, 2021 - 7:05:04 PM


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  • HAL Id : tel-03202134, version 1


Vincent Corlay. Decoding algorithms for lattices. Applications [stat.AP]. Institut Polytechnique de Paris, 2020. English. ⟨NNT : 2020IPPAT050⟩. ⟨tel-03202134⟩



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