Gröbner basis techniques for certain problems in coding and systems theory

There is much common ground between the areas of coding theory and systems theory. Fitzpatrick has shown that a Göbner basis approach leads to efficient algorithms in the decoding of Reed-Solomon codes and in scalar interpolation and partial realization. This thesis simultaneously generalizes and simplifies that approach and presents applications to discrete-time modeling, multivariable interpolation and list decoding. Gröbner basis theory has come into its own in the context of software and algorithm development. By generalizing the concept of polynomial degree, term orders are provided for multivariable polynomial rings and free modules over polynomial rings. The orders are not, in general, unique and this adds, in no small way, to the power and flexibility of the technique. As well as being generating sets for ideals or modules, Gröbner bases always contain a element which is minimal with respect tot the corresponding term order. Central to this thesis is a general algorithm, valid for any term order, that produces a Gröbner basis for the solution module (or ideal) of elements satisfying a sequence of generalized congruences. These congruences, based on shifts and homomorphisms, are applicable to a wide variety of problems, including key equations and interpolations. At the core of the algorithm is an incremental step. Iterating this step lends a recursive/iterative character to the algorithm. As a consequence, not all of the input to the algorithm need be available from the start and different "paths" can be taken to reach the final solution. The existence of a suitable chain of modules satisfying the criteria of the incremental step is a prerequisite for applying the algorithm.

Simplified decoding techniques for linear block codes

Error correcting codes are combinatorial objects, designed to enable reliable transmission of digital data over noisy channels. They are ubiquitously used in communication, data storage etc. Error correction allows reconstruction of the original data from received word. The classical decoding algorithms are constrained to output just one codeword. However, in the late 50’s researchers proposed a relaxed error correction model for potentially large error rates known as list decoding. The research presented in this thesis focuses on reducing the computational effort and enhancing the efficiency of decoding algorithms for several codes from algorithmic as well as architectural standpoint. The codes in consideration are linear block codes closely related to Reed Solomon (RS) codes. A high speed low complexity algorithm and architecture are presented for encoding and decoding RS codes based on evaluation. The implementation results show that the hardware resources and the total execution time are significantly reduced as compared to the classical decoder. The evaluation based encoding and decoding schemes are modified and extended for shortened RS codes and software implementation shows substantial reduction in memory footprint at the expense of latency. Hermitian codes can be seen as concatenated RS codes and are much longer than RS codes over the same aphabet. A fast, novel and efficient VLSI architecture for Hermitian codes is proposed based on interpolation decoding. The proposed architecture is proven to have better than Kötter’s decoder for high rate codes. The thesis work also explores a method of constructing optimal codes by computing the subfield subcodes of Generalized Toric (GT) codes that is a natural extension of RS codes over several dimensions. The polynomial generators or evaluation polynomials for subfield-subcodes of GT codes are identified based on which dimension and bound for the minimum distance are computed. The algebraic structure for the polynomials evaluating to subfield is used to simplify the list decoding algorithm for BCH codes. Finally, an efficient and novel approach is proposed for exploiting powerful codes having complex decoding but simple encoding scheme (comparable to RS codes) for multihop wireless sensor network (WSN) applications.

The effect of positive psychological interventions on psychological and physical well-being during pregnancy

Prenatal well-being can have significant effects on the mother and developing foetus. Positive psychological interventions, including gratitude and mindfulness, consistently demonstrate benefits for well-being in diverse populations. No research has been conducted on gratitude during pregnancy; the few studies of prenatal mindfulness interventions have demonstrated well-being benefits. The current study examined the effects of gratitude and mindfulness interventions on prenatal maternal well-being, cortisol and birth outcomes. Five studies were conducted. Study 1 was a systematic review of mindfulness intervention effects on cortisol; this highlighted potential benefits of mindfulness but the need for rigorous protocols in future research. In Study 2 a gratitude and a mindfulness intervention were developed and evaluated; findings indicate usefulness of two 3 week interventions. Study 3 examined the effects of these interventions in a randomised controlled trial (RCT) of non-pregnant women, before examining a pregnant group. No significant intervention effects were found in this study, potentially due to insufficient power and poor protocol adherence. Changes in expected directions were observed for most outcomes and the potential utility of a combined gratitude and mindfulness intervention was noted. In Study 4 a gratitude during pregnancy (GDP) scale was developed and the reliability of an existing mindfulness measure (MAAS) was examined in a pregnant group. Both scales were found to be suitable and reliable measures in pregnancy. Study 5 incorporated the findings of the previous four studies to examine of the effect of a combined mindfulness and gratitude intervention with a group of pregnant women. Forty-six participants took part in a 5-week RCT that examined intervention effects on prenatal gratitude, mindfulness, happiness, satisfaction with life, social support, prenatal stress, depression and sleep. Findings indicated that the intervention improved sleep quality and that effects for prenatal distress were approaching significance. Issues of attrition and non-compliance to study protocols were problematic and are discussed. In summary, the current thesis highlights the need for robust measurement, and intervention and cortisol sampling protocols in future research, particularly with pregnant groups. Findings also demonstrate tentative benefits of a gratitude and mindfulness intervention during pregnancy.