We define new metrics for linear codes over the ring Fq[u]/(ut) via an Fq-module monomorphism on linear codes over Fq. The construction generalizes the Gray map, Gray weight, and Lee weight; and the technique allows us to find some new optimal linear codes and their weight enumerator polynomial.

It is believed that quantum computers would be able to solve complex problems more quickly than any other deterministic or probabilistic computer. Quantum computers basically exploit the rules of quantum mechanics for speeding up computations. However, building a quantum computer remains a daunting task. A quantum computer, as in any quantum mechanical system, is susceptible to decohorence of quantum bits resulting from interaction of the stored information with the environment. Error correction is then required to restore a quantum bit, which has changed due to interaction with external state, to a previous non-erroneous state in the coding subspace. Until now the methods for quantum error correction were mostly based on stabilizer codes over finite fields. The aim of this thesis is to construct quantum error correcting codes over finite Frobenius rings. We introduce stabilizer codes over quadratic algebra, which allows one to use the hamming distance rather than some less known notion of distance. We also develop propagation rules to build new codes from existing codes. Non binary codes have been realized as a gray image of linear Z4 code, hence the most natural class of ring that is suitable for coding theory is given by finite Frobenius rings as it allow to formulate the dual code similar to finite fields. At the end we show some examples of code construction along with various results of quantum codes over finite Frobenius rings, especially codes over Zm.

Cette thèse porte sur les liens entre les codes correcteurs d'erreurs et les matrices de diffusion linéaires utilisées en cryptographie symétrique. L'objectif est d'étudier les constructions possibles de codes MDS additifs définis sur le groupe (Fm2, +) des m-uplets binaires et de minimiser le coût de l'implémentation matérielle ou logicielles de ces matrices de diffusion. Cette thèse commence par l'étude des codes définis sur un anneau de polynômes du type F[x]/f(x), qui généralisent les codes quasi-cycliques. Elle se poursuit par l'étude des codes additifs systématiques définis sur (Fm2, +) et leur lien avec la diffusion linéaire en cryptographie symétrique. Un point important de la thèse est l'introduction de codes à coefficient dans l'anneau des endomorphismes de Fm2. Le lien entre les codes qui sont des sous-modules à gauche et les codes additifs est mis en évidence. La dernière partie porte sur l'étude et la construction de matrices de diffusion MDS ayant de bonnes propriétés pour la cryptographie, à savoir les matrices circulantes, les matrices dyadiques, ainsi que les matrices ayant des représentations creuses minimisant leur implémentation. / This PhD focuses on the links between error correcting codes and diffusion matrices used in cryptography symmetric. The goal is to study the possible construction of additives MDS codes defined over the group (Fm2, +) of binary m-tuples and minimize cost of hardware or software implementation of these diffusion matrices. This thesis begins with the study of codes defined over the polynomial ring F[x]/f(x), these codes are a generalization of quasi-cyclic codes, and continues with the study of additive systematic codes over (Fm2, +) and there relation with linear diffusion on symmetric cryptography. An important point of this thesis is the introduction of codes with coefficients in the ring of endomorphisms of Fm2. The link between codes which are a left-submodules and additive codes have been identified. The last part focuses on the study and construction of efficient diffusion MDS matrices for the cryptographic applications, namely the circulantes matrices, dyadic matrices, and matrices with hollow representation, in ordre to minimize their implementations.