Constructions of maximum rank distance codes, cyclic constant dimension codes, and subspace packings

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2018
Otal, Kamil
In this thesis, we aim to introduce main contributions to solve three main problems in coding theory. The first problem investigates the construction of inequivalent maximum rank distance (MRD) codes. Namely, we look for the constructions of the largest possible sets of $mtimes n$ matrices over a finite field $F_q$, such that the rank of the subtraction of any two different matrices in the set cannot be smaller than a certain number. Constructions of such codes under a suitable equivalence notion have taken a worthwhile attention in the last decade due to their applications in many areas, and most of the constructions including also our works have been discovered in last few years. We introduce these outcomes classifying them considering the main and most general equivalence idea. We basically use the language of linearized polynomials in this direction as usual in many works in the literature. The second problem, which is originated from an application related to the efficiency in random network coding, concerns the construction of large cyclic subspace codes of constant dimension. In this set up, we aim to construct large sets of $k$-dimensional subspaces of $F_q^n$ in a way that any two distinct subspaces cannot be close to each other more than a certain number in terms of the subspace distance, and the cyclic shifts of each subspace must be included in the set. We give the only systematic construction of such sets in the literature utilizing linearized polynomials again but in a slightly different way. We note that the basic structure of this construction was proposed in our another work. Additionally, we summarize the history of the solution and some further remarks. In the last problem, we focus on the constructions of subspace packings, which are the $q$-analogue of packing designs. This notion is a natural generalization of constant dimension codes, and has applications in the analysis of different network codes. We give a recursive construction of such codes using a generalization of the linkage construction rather than linearized polynomials. In particular, we make use of the matrix version of MRD codes together with some facts from linear algebra. This result is one of the main outcomes of our recent work. We express that these problems are different from but substantially related to each other. Connections among them are also expressed in related places. Furthermore, we remark that various areas of mathematics are used to solve these problems in general, e.g. finite geometry, algebraic geometry, algebra, and linear algebra. Therefore, it is not easy to introduce all advances properly with their complete preliminary information here. Moreover, we try to keep our language as simple as possible, and follow the historical journey of the advances. In that way, we target that this thesis can addresses to a more general reader group.