Applications of finite geometries to designs and codes

This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures. A central question in the study of finite geometry designs is Hamada’s conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamada’s conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples. We begin by constructing an infinite family of counterexamples to Hamada’s conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamada’s conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs. Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes.

The Planned and the Emergent: An Alternative Model of Learning and Literacy

Within academic institutions, writing centers are uniquely situated, socially rich sites for exploring learning and literacy. I examine the work of the Michigan Tech Writing Center's UN 1002 World Cultures study teams primarily because student participants and Writing Center coaches are actively engaged in structuring their own learning and meaning-making processes. My research reveals that learning is closely linked to identity formation and leading the teams is an important component of the coaches' educational experiences. I argue that supporting this type of learning requires an expanded understanding of literacy and significant changes to how learning environments are conceptualized and developed. This ethnographic study draws on data collected from recordings and observations of one semester of team sessions, my own experiences as a team coach and UN 1002 teaching assistant, and interviews with Center coaches prior to their graduation. I argue that traditional forms of assessment and analysis emerging from individualized instruction models of learning cannot fully account for the dense configurations of social interactions identified in the Center's program. Instead, I view the Center as an open system and employ social theories of learning and literacy to uncover how the negotiation of meaning in one context influences and is influenced by structures and interactions within as well as beyond its boundaries. I focus on the program design, its enaction in practice, and how engagement in this type of writing center work influences coaches' learning trajectories. I conclude that, viewed as participation in a community of practice, the learning theory informing the program design supports identity formation —a key aspect of learning as argued by Etienne Wenger (1998). The findings of this study challenge misconceptions of peer learning both in writing centers and higher education that relegate peer tutoring to the role of support for individualized models of learning. Instead, this dissertation calls for consideration of new designs that incorporate peer learning as an integral component. Designing learning contexts that cultivate and support the formation of new identities is complex, involves a flexible and opportunistic design structure, and requires the availability of multiple forms of participation and connections across contexts.