#### Year of Publication

2015

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Arts and Sciences

#### Department

Mathematics

#### First Advisor

Dr. Heide Gluesing-Luerssen

#### Abstract

Random network coding is the most effcient way to send data across a network, but it is very susceptible to errors and erasures. In 2008, Kotter and Kschischang introduced subspace codes as an algebraic approach to error correcting in random network coding. Since this paper, there has been much work in constructing large subspace codes, as well as exploring the properties of such codes. This dissertation explores properties of one particular construction and introduces a new construction for subspace codes. We begin by exploring properties of irreducible cyclic orbit codes, which were introduced in 2011 by Rosenthal et al. As the name implies, irreducible cyclic orbit codes are the orbits of a group action of the general linear group on subspaces. By studying the stabilizers of this action, we formalize the notion of the stabilizer subfield of a subspace and utilize it to gain information about cardinality and distance of the code. Additionally, I define the linkage construction, which is recursive, and compare it to other subspace code constructions. In particular, we use the linkage construction to generalize some constructions of partial spreads. Finally, we address situations for which the linkage construction is eciently decodable.

#### Recommended Citation

Troha, Carolyn E., "Analysis and Constructions of Subspace Codes" (2015). *Theses and Dissertations--Mathematics*. 26.

https://uknowledge.uky.edu/math_etds/26