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.

MAXIMAL ARCS, ABOVE AND BEYOND

This report explores combinatorial structures in Finite Geometries by giving known constructions of maximal arcs; using maximal arcs to construct two-weight codes, partial geometries, strongly regular graphs and LDPC codes; a review on how to generalize maximal arcs to higher dimensions through Perp-Systems; and an effort in finding constructions of new Perp-Systems.

Strategic Optimization Techniques For FRTU Deployment and Chip Physical Design

Combinatorial optimization is a complex engineering subject. Although formulation often depends on the nature of problems that differs from their setup, design, constraints, and implications, establishing a unifying framework is essential. This dissertation investigates the unique features of three important optimization problems that can span from small-scale design automation to large-scale power system planning: (1) Feeder remote terminal unit (FRTU) planning strategy by considering the cybersecurity of secondary distribution network in electrical distribution grid, (2) physical-level synthesis for microfluidic lab-on-a-chip, and (3) discrete gate sizing in very-large-scale integration (VLSI) circuit. First, an optimization technique by cross entropy is proposed to handle FRTU deployment in primary network considering cybersecurity of secondary distribution network. While it is constrained by monetary budget on the number of deployed FRTUs, the proposed algorithm identi?es pivotal locations of a distribution feeder to install the FRTUs in different time horizons. Then, multi-scale optimization techniques are proposed for digital micro?uidic lab-on-a-chip physical level synthesis. The proposed techniques handle the variation-aware lab-on-a-chip placement and routing co-design while satisfying all constraints, and considering contamination and defect. Last, the first fully polynomial time approximation scheme (FPTAS) is proposed for the delay driven discrete gate sizing problem, which explores the theoretical view since the existing works are heuristics with no performance guarantee. The intellectual contribution of the proposed methods establishes a novel paradigm bridging the gaps between professional communities.