Landscapes of Codes: rank distance codes and intersection problems in finite projective spaces
Giovanni Longobardi · 2019
The aim of this thesis is to highlight once again how Geometry, and in particular Combinatorics, is visual knowledge.\\ In this work the main results in my PhD research period are collected. More precisely, it is divided into three blocks. In the first, we investigate the theory of maximum rank distance (MRD) codes whose codewords are symmetric, alternating or Hermitian matrices. In the linearized polynomials setting, we explore how the already known classes of such codes can be seen as the intersection of an appropriate code with the restricted ambient in which ‘they live'. We solve the equivalence issues and we compute their automorphisms group. Moreover, we characterize these latters and present a new class of maximum symmetric codes.\\ In the second part, we recall some notions about finite projective spaces and in this context we introduce the $q$-analogue of the Erd\H{o}s-Ko-Rado problem originally stated in set theory. After having retraced the known results in this topic, we study maximal families of $k$-dimensional subspaces in $\PG(n,q)$, $n \geq k+2$ and $k \geq 3$, pairwise intersecting in at least a $(k-2)$-space. We also give some upper bounds on the size of relevant families, exploring the largest examples.\\ In the last part, we introduce the subspace codes theory as the geometrical counterpart of the intersection problems with assigned size arisen from the set theory. Finally, we generalize the concept of equidistant constant-dimension codes with the notion of SPID (\textit{Subspace Pre-assigned Intersection Dimensions}). The \textit{junta code}, i.e. a highly regular structure that extends the notion of \textit{sunflower}, is defined. In a vector setting, we analyze the space spanned by the elements of a SPID with two intersection dimensions and determinine a \textit{geometrical junta bound}. In particular for two consecutive assigned values of the intersection, we show that this threshold is sharp.