Algebraic and geometric perspectives on Sidon spaces and cyclic subspace codes

This thesis is devoted to the study of Sidon spaces and cyclic subspace codes over finite fields, with a focus on both structural properties and explicit constructions. It aims to explore a part of the existing bridge between additive combinatorics and its linear analogues, and the consequent implications in coding theory. This thesis consists of two main parts which will turn out to be strictly related with each other. The first part presents properties and new results regarding Sidon spaces in finite field extensions over finite fields. These objects have been introduced by Bachoc, Serra and Zémor in 2017 as the linear analogue of Sidon sets, classical objects of additive combinatorics. For this reason, we give an overview of some classical and central results of additive combinatorics related to the sumset of subsets of an abelian group and the related theorems which give lower bounds on their cardinality, such as the Cauchy-Davenport inequality and the Kneser's theorem. We then move to their corresponding linear analogues in field extensions and we use some of them to present new bounds on the dimension of the i-span of a Sidon space, as well as, a new algebraic and geometric proof of the lower bound on the dimension of the square-span of a Sidon space proved in 2017 by Bachoc, Serra and Zémor. Also, we characterize four-dimensional Sidon spaces with minimal square-span. In addition, we introduce the notion of semilinear equivalence for subspaces of finite field extensions of finite fields, under which the Sidon property is preserved. Building on this equivalence, we present new constructions of both Sidon and r-Sidon spaces, a generalization of Sidon spaces which can be seen as the linear analogue of Br-sets. The second part of the thesis explores subspace codes. We describe the theoretical foundations of subspace codes and we focus on a particular class of them, the constant dimension cyclic subspace codes. Indeed, the link between the two parts of this thesis relies on the equivalence between Sidon spaces and optimal one-orbit cyclic subspace codes, pointed out by Roth, Raviv and Tamo in 2018, i.e. cyclic orbit codes with optimal minimum distance. Also, the property of being a Sidon space, as well as, some parameters associated to an Fq-subspace of Fqn, such as the dimension of its square-span, turn out to be invariants under semilinear equivalence. Since the notion of semilinearly equivalent Fq-subspaces of Fqn relies on the concept of Fp-Frobenius isometric cyclic orbit codes provided by Gluesing Luerssen and Lehmann, we can use some of the properties of the Fq-subspaces representing the cyclic orbit codes for distinguishing non-isometric classes of codes. Basing on this invariants, we classify one-orbit cyclic subspace codes of small dimensions and we provide classification results of optimal codes in the Grassmannians G_q(6,3) and G_q(n,4), requiring in the latter case that the representative of the orbit has minimal square-span. We also analyze the case of quasi-optimal codes, establishing existence results, bounds on the parameters defining their distance distributions, and we provide explicit constructions, analyzing the associated isometry classes. We then move to considering a class of cyclic one-orbit codes with another extremal property, that is maximizing the length of the distance distribution of the code. We characterize full-weight spectrum codes, i.e. cyclic orbit codes with non-zero weights in their weight distributions. Finally, we introduce a new method for constructing cyclic subspace codes with efficient parameters. In particular, we provide constructions of cyclic subspace codes of optimal minimum distance and whose sizes asymptotically attain the Johnson-type bound, thereby achieving a near-optimal balance between code size and error-correcting capability.