Research Interests My current research interests are in algebra, with applications in classical and quantum coding theory. More specifically, I am interested in: General Algebra Finite modules. Finite commutative and non-commutative rings Frobenius rings and modules Representation Theory of finite groups Algebraic/arithmetic geometry Algebraic Coding Theory Linear codes over rings and modules Isometries of codes with respect to various weights Self-orthogonal codes Quantum Information Theory Quantum error-correction Quantum stabilizer codes Symplectic isometries of quantum codes Post quantum cryptography Publications and Preprints Laplacian Simplices II: A Coding Theoretic Approach (with Marie Meyer) Submitted [arXiv] [Show Abstract] Abstract: This paper further investigates $\textit{Laplacian simplices}$. A construction by Braun and the first author associates to a simple connected graph $G$ a simplex $\mathcal{P}_G$ whose vertices are the rows of the Laplacian matrix of $G$. In this paper we associate to a reflexive $\mathcal{P}_G$ a duality-preserving linear code $\mathcal{C}(\mathcal{P}_G)$. This new perspective allows us to build upon previous results relating graphical properties of $G$ to properties of the polytope $\mathcal{P}_G$. In particular, we make progress towards a graphical characterization of reflexive $\mathcal{P}_G$ using techniques from Ehrhart theory. We provide a systematic investigation of $\mathcal{C}(\mathcal{P}_G)$ for cycles, complete graphs, and graphs with a prime number of vertices. We construct an asymptotically good family of MDS codes. In addition, we show that any rational rate is achievable by such construction. Hide Abstract On quantum stabilizer codes derived from local Frobenius rings (with Heide Gluesing-Luerssen) Submitted [arXiv] [Show Abstract] Abstract: In this paper we consider stabilizer codes over local Frobenius rings. First, we study the minimum relative distances of a stabilizer code and its reduction onto the residue field. We show that for various scenarios, a free stabilizer code over the ring does not underperform the according stabilizer code over the field. This leads us to conjecture that the same is true for all free stabilizer codes. Secondly, we focus on isometries of stabilizer codes. We present some preliminary results and introduce some interesting open problems. Hide Abstract Symplectic Isometries of Stabilizer Codes To appear in Journal of Algebra and its Applications [arXiv] [Show Abstract] Abstract: In this paper we study the equivalence of quantum stabilizer codes via symplectic isometries of stabilizer codes. We define monomially and symplectically equivalent stabilizer codes and determine how different the two notions can be. Further, we show that to monomial maps correspond local Clifford operators. We relate the latter with the LU-LC Conjecture. Hide Abstract Extension theorems for various weight functions over Frobenius bimodules (with Heide Gluesing-Luerssen) Journal of Algebra and its Applications 17(3):1850052, 2018 (28 pages) [Journal] [arXiv] [Show Abstract] Abstract: In this paper we study codes where the alphabet is a finite Frobenius bimodule over a finite ring. We discuss the extension property for various weight functions. Employing an entirely character-theoretic approach and a duality theory for partitions on Frobenius bimodules we derive alternative proofs for the facts that the Hamming weight and the homogeneous weight satisfy the extension property. We also use the same techniques to derive the extension property for other weights, such as the Rosenbloom-Tsfasman weight. Hide Abstract Lecture Notes/Slides/Posters January 9, 2019:, Thesis Defense, University of Kentucky - Slides November 13, 2018: Nonlinear Algebra in Applications, ICERM, Brown University - Poster October 20, 2018: AMS Sectional Meetings, University of Michigan - Slides August 31, 2018: CIMPA Research School, Middle East Technical University, Ankara, Turkey - Slides July 22, 2018: Fourier Analysis on Finte Abealian Groups and Applications - Lecture Notes May 19, 2018: 30th Cumberland Conference, Marshall University - Slides March 18, 2018: AMS Sectional Meetings, Ohio State University - Slides January 12, 2018: Joint Mathematics Meetings - Slides October 4, 2015: AMS Sectional Meetings, Loyola University Chicago - Slides