The applications of generalized inverses of matrices appear in many fields like applied mathematics, statistics and engineering [2]. In this thesis, we discuss generalized inverses of matrices over Ore polynomial rings (also called Ore matrices). We first introduce some necessary and sufficient conditions for the existence of {1}-, {1,2}-, {1,3}-, {1,4}- and MP-inverses of Ore matrices, and give some explicit formulas for these inverses. Using {1}-inverses of Ore matrices, we present the solutions of linear systems over Ore polynomial rings. Next, we extend Roth's Theorem 1 and generalized Roth's Theorem 1 to the Ore matrices case. Furthermore, we consider the extensions of all the involutions ψ on R(x), and construct some necessary and sufficient conditions for ψ to be an involution on R(x)[D;σ,δ]. Finally, we obtain two different explicit formulas for {1,3}- and {1,4}-inverses of Ore matrices. The Maple implementations of our main algorithms are presented in the Appendix. / May 2017

Cyclic codes are a well-known class of linear block codes with efficient decoding algorithms. In recent years they have been generalized to skew-constacyclic codes; such a generalization has previously been shown to be useful. We begin with a study of skew-polynomial rings so that we may examine these codes algebraically as quotient modules of non-commutative skew-polynomial rings. We introduce a skew-generalized circulant matrix to aid in examining skew-constacyclic codes, and we use it to recover a well-known result on the duals of skew-constacyclic codes from Boucher/Ulmer in 2011. We also motivate and develop a notion of idempotent elements in these quotient modules. We are particularly concerned with the existence and uniqueness of idempotents that generate a given submodule; we generalize relevant results from previous work on skew-constacyclic codes by Gao/Shen/Fu in 2013 and well-known results from the classical case.

Le sujet porte sur une classe de codes correcteurs d erreurs dits codes cycliques tordus, et ses applications a l'Informatique quantique et aux codes quasi-cycliques. Les codes cycliques classiques ont une structure d'idéaux dans un anneau de polynômes. Ulmer a introduit en 2008 une généralisation aux anneaux dits de polynômes tordus, une classe d'anneaux non commutatifs introduits par Ore en 1933. Dans cette thèse on explore le cas du corps a quatre éléments et de l'anneau produit de deux copies du corps a deux éléments. / The topic of the thesis is the study of skew cyclic codes, with application to Quantum Computing and quasi-cyclic codes. Classical cyclic codes have a natural structure of ideals in a polynomial ring. This was generalized by Ulmer in 2008 to skew polynomial rings, a class of non commutative rings introduced by Ore in 1933. The latter codes are not classically cyclic if the alphabet ring admits a non trivial automorphism. In this work is explored the cases of the finite field of order four and of a product ring of two copies of the finite field of order two.