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.

SDC: Scalable description coding for adaptive streaming media

Video compression techniques enable adaptive media streaming over heterogeneous links to end-devices. Scalable Video Coding (SVC) and Multiple Description Coding (MDC) represent well-known techniques for video compression with distinct characteristics in terms of bandwidth efficiency and resiliency to packet loss. In this paper, we present Scalable Description Coding (SDC), a technique to compromise the tradeoff between bandwidth efficiency and error resiliency without sacrificing user-perceived quality. Additionally, we propose a scheme that combines network coding and SDC to further improve the error resiliency. SDC yields upwards of 25% bandwidth savings over MDC. Additionally, our scheme features higher quality for longer durations even at high packet loss rates.

Study abroad and complexity, accuracy and fluency (CAF) development: a longitudinal investigation of French and Chinese Learners of L2 English

This longitudinal study tracked third-level French (n=10) and Chinese (n=7) learners of English as a second language (L2) during an eight-month study abroad (SA) period at an Irish university. The investigation sought to determine whether there was a significant relationship between length of stay (LoS) abroad and gains in the learners' oral complexity, accuracy and fluency (CAF), what the relationship was between these three language constructs and whether the two learner groups would experience similar paths to development. Additionally, the study also investigated whether specific reported out-of-class contact with the L2 was implicated in oral CAF gains. Oral data were collected at three equidistant time points; at the beginning of SA (T1), midway through the SA sojourn (T2) and at the end (T3), allowing for a comparison of CAF gains arising during one semester abroad to those arising during a subsequent semester. Data were collected using Sociolinguistic Interviews (Labov, 1984) and adapted versions of the Language Contact Profile (Freed et al., 2004). Overall, the results point to LoS abroad as a highly influential variable in gains to be expected in oral CAF during SA. While one semester in the TL country was not enough to foster statistically significant improvement in any of the CAF measures employed, significant improvement was found during the second semester of SA. Significant differences were also revealed between the two learner groups. Finally, significant correlations, some positive, some negative, were found between gains in CAF and specific usage of the L2. All in all, the disaggregation of the group data clearly illustrates, in line with other recent enquiries (e.g. Wright and Cong, 2014) that each individual learner's path to CAF development was unique and highly individualised, thus providing strong evidence for the recent claim that SLA is "an individualized nonlinear endeavor" (Polat and Kim, 2014: 186).