Generic Properties of Convolutional Codes

In this dissertation, convolutional codes possessing the maximum distance profile (MDP) and strongly maximum distance separable (sMDS) properties are studied. More specifically, ideas from linear systems theory and algebraic geometry are used to 1. give an affirmative answer to the conjecture in~cite{gl03r} that convolutional codes possessing both the MDP and sMDS properties exist for arbitrary code parameters over finite fields of every prime characteristic and 2. show that the set of such codes may be seen as a generic set in a certain Quot scheme. In order to think of all points of the aforementioned scheme as somehow representing convolutional codes of the same degree, we associate so-called homogeneous convolutional codes to them. We introduce this notion, develop a body of results similar to those that exist in the traditional (nonhomogeneous) setting, generalize the notions of MDP and sMDS to these codes, and prove existence and genericity results analogous to those mentioned above for nonhomogeneous codes. Finally, the topic of superregular matrices is addressed. Superregular matrices arise when one considers the problem of constructing codes having the MDP property. After introducing superregular matrices, we consider group actions preserving the property of superregularity. We then derive an upper bound on the smallest size a finite field can have in order that a superregular matrix of a given size can exist over that field.