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.

Quantum modeling of bioconjugated nanomaterials

In recent years, the bio-conjugated nanostructured materials have emerged as a new class of materials for the bio-sensing and medical diagnostics applications. In spite of their multi-directional applications, interfacing nanomaterials with bio-molecules has been a challenge due to somewhat limited knowledge about the underlying physics and chemistry behind these interactions and also for the complexity of biomolecules. The main objective of this dissertation is to provide such a detailed knowledge on bioconjugated nanomaterials toward their applications in designing the next generation of sensing devices. Specifically, we investigate the changes in the electronic properties of a boron nitride nanotube (BNNT) due to the adsorption of different bio-molecules, ranging from neutral (DNA/RNA nucleobases) to polar (amino acid molecules). BNNT is a typical member of III-V compounds semiconductors with morphology similar to that of carbon nanotubes (CNTs) but with its own distinct properties. More specifically, the natural affinity of BNNTs toward living cells with no apparent toxicity instigates the applications of BNNTs in drug delivery and cell therapy. Our results predict that the adsorption of DNA/RNA nucleobases on BNNTs amounts to different degrees of modulation in the band gap of BNNTs, which can be exploited for distinguishing these nucleobases from each other. Interestingly, for the polar amino acid molecules, the nature of interaction appeared to vary ranging from Coulombic, van der Waals and covalent depending on the polarity of the individual molecules, each with a different binding strength and amount of charge transfer involved in the interaction. The strong binding of amino acid molecules on the BNNTs explains the observed protein wrapping onto BNNTs without any linkers, unlike carbon nanotubes (CNTs). Additionally, the widely varying binding energies corresponding to different amino acid molecules toward BNNTs indicate to the suitability of BNNTs for the biosensing applications, as compared to the metallic CNTs. The calculated I-V characteristics in these bioconjugated nanotubes predict notable changes in the conductivity of BNNTs due to the physisorption of DNA/RNA nucleobases. This is not the case with metallic CNTs whose transport properties remained unaltered in their conjugated systems with the nucleobases. Collectively, the bioconjugated BNNTs are found to be an excellent system for the next generation sensing devices.

TUNING, AVO, AND FLAT-SPOT EFFECTS IN A SEISMIC ANALYSIS OF NORTH SEA BLOCK F3

Reflection seismic data from the F3 block in the Dutch North Sea exhibits many large-amplitude reflections at shallow horizons, typically categorized as “brightspots ” (Schroot and Schuttenhelm, 2003), mainly because of their bright appearance. In most cases, these bright reflections show a significant “flatness” contrasting with local structural trends. While flatspots are often easily identified in thick reservoirs, we have often occasionally observed apparent flatspot tuning effects at fluid contacts near reservoir edges and in thin reservoir beds, while only poorly understanding them. We conclude that many of the shallow large-amplitude reflections in block F3 are dominated by flatspots, and we investigate the thin-bed tuning effects that such flatspots cause as they interact with the reflection from the reservoir’s upper boundary. There are two possible effects to be considered: (1) the “wedge-model” tuning effects of the flatspot and overlying brightspots, dimspots, or polarity-reversals; and (2) the stacking effects that result from possible inclusion of post-critical flatspot reflections in these shallow sands. We modeled the effects of these two phenomena for the particular stratigraphic sequence in block F3. Our results suggest that stacking of post-critical flatspot reflections can cause similar large-amplitude but flat reflections, in some cases even causing an interface expected to produce a ‘dimspot’ to appear as a ‘brightspot’. Analysis of NMO stretch and muting shows the likely exclusion of critical offset data in stacked output. If post-critical reflections are included in stacking, unusual results will be observed. In the North Sea case, we conclude the tuning effect was the primary reason causing for the brightness and flatness of these reflections. However, it is still important to note that care should be taken while applying muting on reflections with wide range of incidence angles and the inclusion of critical offset data may cause some spurious features in the stacked section.