204 research outputs found

    Bounds on the size of codes

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    In this dissertation we determine new bounds and properties of codes in three different finite metric spaces, namely the ordered Hamming space, the binary Hamming space, and the Johnson space. The ordered Hamming space is a generalization of the Hamming space that arises in several different problems of coding theory and numerical integration. Structural properties of this space are well described in the framework of Delsarte's theory of association schemes. Relying on this theory, we perform a detailed study of polynomials related to the ordered Hamming space and derive new asymptotic bounds on the size of codes in this space which improve upon the estimates known earlier. A related project concerns linear codes in the ordered Hamming space. We define and analyze a class of near-optimal codes, called near-Maximum Distance Separable codes. We determine the weight distribution and provide constructions of such codes. Codes in the ordered Hamming space are dual to a certain type of point distributions in the unit cube used in numerical integration. We show that near-Maximum Distance Separable codes are equivalently represented as certain near-optimal point distributions. In the third part of our study we derive a new upper bound on the size of a family of subsets of a finite set with restricted pairwise intersections, which improves upon the well-known Frankl-Wilson upper bound. The new bound is obtained by analyzing a refinement of the association scheme of the Hamming space (the Terwilliger algebra) and intertwining functions of the symmetric group. Finally, in the fourth set of problems we determine new estimates on the size of codes in the Johnson space. We also suggest a new approach to the derivation of the well-known Johnson bound for codes in this space. Our estimates are often valid in the region where the Johnson bound is vacuous. We show that these methods are also applicable to the case of multiple packings in the Hamming space (list-decodable codes). In this context we recover the best known estimate on the size of list-decodable codes in a new way

    The extended codes of a family of reversible MDS cyclic codes

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    A linear code with parameters [n,k,nβˆ’k+1][n, k, n-k+1] is called a maximum distance separable (MDS for short) code. A linear code with parameters [n,k,nβˆ’k][n, k, n-k] is said to be almost maximum distance separable (AMDS for short). A linear code is said to be near maximum distance separable (NMDS for short) if both the code and its dual are AMDS. MDS codes are very important in both theory and practice. There is a classical construction of a [q+1,2uβˆ’1,qβˆ’2u+3][q+1, 2u-1, q-2u+3] MDS code for each uu with 1≀uβ‰€βŒŠq+12βŒ‹1 \leq u \leq \lfloor\frac{q+1}2\rfloor, which is a reversible and cyclic code. The objective of this paper is to study the extended codes of this family of MDS codes. Two families of MDS codes and several families of NMDS codes are obtained. The NMDS codes have applications in finite geometry, cryptography and distributed and cloud data storage systems. The weight distributions of some of the extended codes are determined

    Generalized weights and bounds for error probability over erasure channels

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    New upper and lower bounds for the error probability over an erasure channel are provided, making use of Wei's generalized weights, hierarchy and spectra. In many situations the upper and lower bounds coincide and this allows improvement of existing bounds. Results concerning MDS and AMDS codes are deduced from those bounds

    Funnel Plots for Assessing Institutional Performance

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    This thesis presents a comparison of maternal outcomes for births in New Zealand District Health Boards (DHBs).This is carried out through analysis of the National Minimum Dataset collected by the Ministry of Health for 2007. The outcome compared is postpartum haemorrhage (PPH) the results are displayed using funnel plots, a useful tool for displaying unbiased information on performance outcomes when comparing institutions. Exploration of the data found that there are differences in the demographics, maternal and birth characteristics among DHBs. The rates of PPH are different and the population mixes are made up of a range of different proportions of ethnic groups, ages and deprivation indexes. The exploratory analysis found that a large number of factors are associated with PPH. And that birth weight, parity and gestation had a large number of missing observations. These factors are not missing at random and require imputing prior to constructing the funnel plots. Results show that there is divergence amongst DHBs in the postpartum haemorrhage rate. First a raw PPH rate was plotted and the results indicated there were differences among DHBs. As there are many potential predictors for PPHa logistic regression model was applied to find the most important factors related to PPH. This allows us to apply an adjusted rate for the funnel plot. The risk adjusted funnel plot also indicated differences among DHBs. Two approaches are taken to account for the overdispersion. A winsorised estimate and a winsorised estimate with a random effects term are applied to the data. The approaches produced different results. The winsorised estimate widened the control limits and the random effects term narrowed the control limits. All four plots identified an extreme outlier and this was later removed from the analysis and the winsorisation funnel plots were rerun. The influential outlier made a difference and from this we can concluded that 2 out 20 DHBs lie outside the 95% control limits. These two DHBs could be stated as having a very low rate of PPH

    Microform-scale variations in peatland permeability and their ecohydrological implications

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    1. The acrotelm-catotelm model of peatland hydrological and biogeochemical processes posits that the permeability of raised bogs is largely homogenous laterally but varies strongly with depth through the soil profile; uppermost peat layers are highly permeable while deeper layers are, effectively, impermeable. 2. We measured down-core changes in peat permeability, plant macrofossil assemblages, dry bulk density and degree of humification beneath two types of characteristic peatland microform – ridges and hollows – at a raised bog in Wales. Six 1424 C dates were also collected for one hollow and an adjacent ridge. 3. Contrary to the acrotelm-catotelm model, we found that deeper peat can be as highly permeable as near-surface peat and that its permeability can vary by more than an order of magnitude between microforms over horizontal distances of 1-5 metres. 4. Our palaeo-ecological data paint a complicated picture of microform persistence. Some microforms can remain in the same position on a bog for millennia, growing vertically upwards as the bog grows. However, adjacent areas on the bog (< 10 m distant) show switches between microform type over time, indicating a lack of persistence. 5. Synthesis. We suggest that the acrotelm-catotelm model should be used cautiously; spatial variations in peatland permeability do not fit the simple patterns suggested by the model. To understand how peatlands as a whole function both hydrologically and ecologically it is necessary to understand how patterns of peat physical properties and peatland vegetation develop and persist

    South Atlantic Stream Fish Assemblages: Multi-Scale Structuring Factors, Trait Associations and Channelization, and Responses to Dam Removal

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    South Atlantic coastal plain wadeable streams are unique and understudied freshwater environments that provide crucial habitats for a wide range of aquatic taxa. In Chapter 1, we investigated patterns in fish assemblages across South Carolina\u27s coastal plain, and developed statistical models to identify the dominant multi-scale abiotic environmental factors that influence assemblage structure. Our analyses indicated the presence of four predominant fish assemblages that commonly occur in the coastal plain, which we termed the: 1) fluvial, 2) eastern mudminnow, 3) centrarchid, and 4) non-fluvial assemblages. Natural geographic gradients and instream habitat parameters associated with velocity, channel form, stream size, and depth played a greater role in distinguishing fish assemblages than catchment land cover, and these instream habitat parameters showed weak relationships with anthropogenic land cover conditions. This study provides a better understanding how coastal plain fish assemblages respond to multi-scale abiotic factors, and will help improve conservation and management strategies, as well as assist in the development of appropriate indicators for standardized evaluations of ecological integrity. In Chapter 2, we investigated the impacts of channelization on South Atlantic coastal plain fish assemblages using both taxonomic and trait-based analyses. We categorized sampled streams a priori into 4 channel types based on observations of their gross channel morphology: 1) single channel non-channelized streams, 2) maintained channelized streams, 3) unmaintained (\u3e5years) channelized streams, and 4) braided swamp-like non-channelized streams. We found no difference in fish taxonomic diversity metrics among channel types, and taxonomic assemblage-based analyses revealed limited information regarding structural associations. In contrast, our trait-based analysis elucidated species differences among all channelized and non-channelized channel types; principal differences were found in habitat preference, and body size/reproductive ecology. Our study suggests that trait-based analyses may be particularly well suited to elucidating information on ecological response to environmental disturbances in the South Atlantic coastal plain, and their use in conjunction with taxonomic analyses should provide a fruitful avenue for developing and testing ecological theory of fish assemblage organization in this region. In Chapter 3, we examined the effects of two dam removals on instream habitat, fish metrics, and fish assemblage structure of Twelvemile Creek, located in Pickens County, South Carolina. Our results indicated that the bulk of instream habitat changes occurred within 1-year of each dam removal. Previously lentic-dominated fish assemblages at former impounded sites generally shifted to a lotic-dominated structure within 6-months (upper-removed dam), and 1-1.5 years (lower-removed dam) after dam removal. Despite these prominent assemblage shifts, we found impacts on benthic invertivore density at sites flanking the upper-removed dam at 2.5-years post dam removal, and impacts on total density, richness, benthic invertivore density, and native centrarchid density at sites flanking the lower-removed dam at 2-years post dam removal. These findings suggested that multiple dam removals had a cumulative downstream increase in negative impacts on fish assemblages. Although dam removal can have ecological trade-offs and short-term disturbance impacts, we demonstrated that dam removal can also reverse many of the negative impacts dams have on fish assemblages, primarily through the restoration of high-quality lotic habitats required by native riverine species
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