The classical binary error correcting codes, and subspace codes for error correction in random network coding are two different forms of error control coding. We identify common features between these two forms and study the relations between them using the aid of lattices. Lattices are partial ordered sets where every pair of elements has a least upper bound and a greatest lower bound in the lattice. We shall demonstrate that many questions that connect these forms have a natural motivation from the viewpoint of lattices. We shall show that a lattice framework captures the notion of Singleton bound where the bound is on the size of the code as a function of its parameters. For the most part, we consider a special type of a lattice which has the geometric modular property. We will use a lattice framework to combine the two different forms. And then, in order to demonstrate the utility of this binding view, we shall derive a general version of Singleton bound. We will note that the Singleton bounds behave differently in certain respects because the binary coding framework is associated with a lattice that is distributive. We shall demonstrate that lack of distributive gives rise to a weaker bound. We show that Singleton bound for classical binary codes, subspace codes, rank metric codes and Ferrers diagram rank metric codes can be derived using a common technique. In the literature, Singleton bounds are derived for Ferrers diagram rank metric codes where the rank metric codes are linear. We introduce a generalized version of Ferrers diagram rank metric codes and obtain a Singleton bound for this version. Next, we shall prove a conjecture concerning the constraints of embedding a binary coding framework into a subspace framework. We shall prove a conjecture by Braun, Etzion and Vardy, which states that any such embedding which contains the full space in its range is constrained to have a particular size. Our proof will use a theorem due to Lovasz, a subspace counting theorem for geometric modular lattices, to prove the conjecture. We shall further demonstrate that any code that achieves the conjectured size must be of a particular type. This particular type turns out to be a natural distributive sub-lattice of a given geometric modular lattice.

We establish the existence of a symmetry within the Gromov-Witten theory of CPn and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of the permutohedron. This symmetry expresses some difficult to compute invariants in terms of others less difficult to compute. We focus on enumerative implications; in particular this technique yields a one line proof of the uniqueness of the rational normal curve. Our method involves a study of the toric geometry of the permutohedron, and degeneration of Gromov-Witten invariants. / En este trabajo establecemos la existencia de una simetra en el marco de la teora de Gromov-Witten para CPn y su explosion a lo largo de puntos. La naturaleza de esta simetra queda codicada en la transformacion de Cremona y su resolucion en una variedad torica del permutoedro. Esta simetra expresa algunos invariantes difciles de calcular junto con otros que no lo son tanto. Nos centramos en implicaciones enumerativas; en particular esta tecnica ofrece una prueba enuna lnea de la unicidad de la curva racional normal. Nuestro metodo involucra un estudio de la geometra torica del permutoedro, as como el de la degeneracion de los invariantes de Gromov-Witten.

Cette thèse étudie les équidistributions de zéros de sections holomorphesaléatoires de fibrés en droites pour les mesures modérées. Elle consiste en deuxparties.Dans la première partie, nous construisons une famille étendue de mesuressingulières modérées sur des espaces projectifs. Ces mesures sont générées pardes fonctions quasi-plurisousharmoniques avec les potentiels höldériens.Le deuxième partie traite une propriété d' équidistribution dans un contextegénéral. Nous établissons un théorème d'équidistribution dans le cas dequelques fibrés en droites gros munis de métriques singulières. Une vitesse deconvergence précise pour l'équidistribution est obtenue. / This thesis investigates the equidistributions of zeros of random holomorphic sections of line bundles for moderate measures. It consists of two parts. In the first part, we construct a large family of singular moderate measures on projective spaces. These measures are generated by quasi-plurisubharmonic functions with Holder potentials.The second part deals with an equidistribution property in general settings. We establish an equidistribution theorem in the case of several big line bundles endowed with singular metrics. A precise convergence speed for the equidistribution is obtained.