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.

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.

Nanofabrication of Surface-Enhanced Raman Scattering Device by an Integrated Block-Copolymer and Nanoimprint Lithography Method

The integration of block-copolymers and nanoimprint lithography presents a novel and cost-effective approach to achieving nanoscale patterning capabilities. The authors demonstrate the fabrication of a surface-enhanced Raman scattering device using templates created by the block-copolymers nanoimprint lithography integrated method.

Minimal local anesthetic volume for peripheral nerve block: a new ultrasound-guided, nerve dimension-based method

BACKGROUND AND OBJECTIVES: Nerve blocks using local anesthetics are widely used. High volumes are usually injected, which may predispose patients to associated adverse events. Introduction of ultrasound guidance facilitates the reduction of volume, but the minimal effective volume is unknown. In this study, we estimated the 50% effective dose (ED50) and 95% effective dose (ED95) volume of 1% mepivacaine relative to the cross-sectional area of the nerve for an adequate sensory block. METHODS: To reduce the number of healthy volunteers, we used a volume reduction protocol using the up-and-down procedure according to the Dixon average method. The ulnar nerve was scanned at the proximal forearm, and the cross-sectional area was measured by ultrasound. In the first volunteer, a volume of 0.4 mL/mm of nerve cross-sectional area was injected under ultrasound guidance in close proximity to and around the nerve using a multiple injection technique. The volume in the next volunteer was reduced by 0.04 mL/mm in case of complete blockade and augmented by the same amount in case of incomplete sensory blockade within 20 mins. After 3 up-and-down cycles, ED50 and ED95 were estimated. Volunteers and physicians performing the block were blinded to the volume used. RESULTS: A total 17 of volunteers were investigated. The ED50 volume was 0.08 mL/mm (SD, 0.01 mL/mm), and the ED95 volume was 0.11 mL/mm (SD, 0.03 mL/mm). The mean cross-sectional area of the nerves was 6.2 mm (1.0 mm). CONCLUSIONS: Based on the ultrasound measured cross-sectional area and using ultrasound guidance, a mean volume of 0.7 mL represents the ED95 dose of 1% mepivacaine to block the ulnar nerve at the proximal forearm.

Split-mouth designs in orthodontics: an overview with applications to orthodontic clinical trials

SUMMARY Split-mouth designs first appeared in dental clinical trials in the late sixties. The main advantage of this study design is its efficiency in terms of sample size as the patients act as their own controls. Cited disadvantages relate to carry-across effects, contamination or spilling of the effects of one intervention to another, period effects if the interventions are delivered at different time periods, difficulty in finding similar comparison sites within patients and the requirement for more complex data analysis. Although some additional thought is required when utilizing a split-mouth design, the efficiency of this design is attractive, particularly in orthodontic clinical studies where carry-across, period effects and dissimilarity between intervention sites does not pose a problem. Selection of the appropriate research design, intervention protocol and statistical method accounting for both the reduced variability and potential clustering effects within patients should be considered for the trial results to be valid.