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.

Add a Dash of Rhetoric, a Pinch of Technical Communication, a Handful of Pedagogy, and Mix Well: Cookbooks and Recipes as Instructional Texts, and What Domestic Texts from 1870 to 1935 Can Teach Us About Technical Communication

Through the use of rhetoric centered on authority and risk avoidance, scientific method has co-opted knowledge, especially women's everyday and experiential knowledge in the domestic sphere. This, in turn, has produced a profound affect on technical communication in the present day. I am drawing on rhetorical theory to study cookbooks and recipes for their contributions to changes in instructional texts. Using the rhetorical lenses of metis (cunning intelligence), kairos (timing and fitness) and mneme (memory), I examine the way in which recipes and cookbooks are constructed, used and perceived. This helps me uncover lost voices in history, the voices of women who used recipes, produced cookbooks and changed the way instructions read. Beginning with the earliest cookbooks and recipes, but focusing on the pivotal temporal interval of 1870-1935, I investigate the writing and rhetorical forces shaping instruction sets and domestic discourse. By the time of scientific cooking and domestic science, everyday and experiential knowledge were being excluded to make room for scientific method and the industrial values of the public sphere. In this study, I also assess how the public sphere, via Cooperative Extension Services and other government agencies, impacted the domestic sphere, further devaluing everyday knowledge in favor of the public scientific model. I will show how the changes in the production of food, cookbooks and recipes were related to changes in technical communication. These changes had wide rippling effects on the field of technical communication. By returning to some of the tenets and traditions of everyday and experiential knowledge, technical communication scholars, practitioners and instructors today can find new ways to encounter technical communication, specifically regarding the creation of instructional texts. Bringing cookbooks, recipes and everyday knowledge into the classroom and the field engenders a new realm of epistemological possibilities.