A mƟ approach of REED-MULLER codes and an application to mƟ steganographic protocols

Résumé

In this thesis, we study the modal Ɵ-valent Reed-Muller codes and their application to mƟ steganographic protocols. We get new results on the mƟ parameters of these codes using as techniques algorithmic and arithmetic tools on the mƟ finite field F2Z. The determination of the mƟ parameters of these codes requires a rather detailed study on the classical Reed-Muller codes [75, 76], on mƟ codes, on mƟ cyclic codes and mƟ linear codes, finally on the Chrysippian modal Ɵ-valent representation of the Galois fields [30]. In addition, a precise analysis on the construction of a word of the Reed-Muller code from the Boolean function allowed us to identify the main results. We characterize, in, particular, the parameters (length, dimension, weight) of the mƟ spectrum of Reed-Muller codes. Then, we propose an algorithm of decoding of mƟ Reed-Muller codes. Later we define the mƟ generalized Reed-Muller codes from the dual codes of the rth order Generalized Reed-Muller codes [29] and the chrysippian modal Ɵ-valent representation of GF(pZ; r) [32]. Lastly, we propose an application of mƟ Reed-Muller codes to mƟ steganographic protocols.

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