Theses and dissertations

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    2-MODULAR REPRESENTATIONS OF UNITARY GROUP U3(4) AS LINEAR CODES
    (Kibabii University, 2019-12-01) Maina, Janet Lilian
    A monumental achievement in group theory was done with the announcement of the completion of classification of simple finite groups in 2004. The proof of this work which was termed, a theorem, consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published between 1995 and 2004. Such voluminous work cannot be understood by any single person. Attempts to simplify the proof has already been embarked on. It is thought that a knowledge of internal structures associated with the groups and more so representation theoretic methods, could go along way to help simplify the proof. This has sparked research of combinatorial objects like codes obtained from groups and their interplay. This thesis is a study of linear binary codes obtained from primitive permutation representations of the simple finite classical group U3(4). Using the established magma databases and the Meataxe software, we consider for each primitive representation over F2, the permutation module obtained from the action of the group on the cosets of its maximal subgroups and the subsequent maximal submodules. Each submodule constitutes a binary code invariant under the group. In this thesis we study linear binary codes, designs and graphs obtained from the group U3(4). Using modular theoretic methods , we construct and enumerate all linear binary codes and designs from primitive permutation representations of degrees 208 and 416 and classify most of the codes. Furthermore, we determine their properties and establish the interplay between these codes and other combinatorial objects like designs and graphs. In the process, we have uncovered the lattice structure of the submodules. We have also determined the full automorphism groups of the codes and designs. Codes are applied in many areas particularly in error correction, storage and transmission of data. The properties of a code determines its usage. We found some codes with good parameters. We found some self-orthogonal, doubly even codes, irreducible and decomposable codes.
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    Analysis of student characteristics and school predictors on academic achievement in Kenya certificate of secondary education examination in Busia county, Kenya
    (2017-12) Echaune, Manasi
    Students’ academic achievement in secondary school examination is critical in preparing students for further education and the world of work. This is the very reason unsatisfactory academic achievement in Kenya Certificate of Secondary Education (KCSE) examination that has been witnessed in Busia County over the last three years should be a cause worry to the stakeholders in education. This study analyzed the student characteristics and school predictors on academic achievement in KCSE examination in Busia County. The specific objectives of the study were; to establish whether student characteristics influence students’ academic achievement in KCSE in Busia County, to determine the effect of school characteristics on academic achievement in KCSE examination in Busia County, to investigate the effect of teacher inputs on academic achievement in KCSE examination in Busia County and to examine the effect of non-teacher school inputs on academic achievement in KCSE examination in Busia County. The study employed the production function theory. A descriptive survey design was employed. The study targeted 152 principals, 2360 teachers and 7550 students. A sample of 100 secondary schools and 1091 respondents was used. The respondents comprised of 100 principals, 236 teachers and 755 students. The study employed simple random sampling to select teachers and students. Purposive sampling was used to select principals while stratified random sampling was used to select schools. Self administered questionnaires and document analysis guide were used to collect data. Questionnaires were piloted in fifteen secondary schools within Busia County and the schools used to pilot the instruments were not included in the actual study. Test re-test technique was used to ascertain reliability of the questionnaires. The test re-tests of the questionnaires used to collect data yielded reliability coefficients of 0.810 and 0.873 for the student and teacher questionnaire respectively. Experts in the Department of Educational Planning and Management were requested to ascertain both the face and content validity of the instruments. Data was analyzed using STATA version 14. Descriptive statistics including percentages, frequencies, means, and standard deviations were used in the preliminary analysis of data. Inferential statistics including t-tests, ANOVA, correlation and regression were used to test the study hypotheses. Hierarchical linear modeling was used to test the relationship between academic achievement in KCSE examination, school predictors and students characteristics. The findings of the study were presented in tables and figures. The study revealed that school predictors accounted for the largest variation in academic achievement in KCSE examination (56.86 percent) compared to student characteristics which accounted for 43.13 percent of the variation in academic achievement in the same examination. School characteristic and teacher inputs had no statistically significant effects on academic achievement in KCSE examination. Non teacher school inputs particularly adequacy of physical facilities and textbooks had statistically significant effects on academic achievement in KCSE examination. The study concluded that variation in students academic achievement in KCSE had more to do with school predictors than students’ characteristics. National government, County government, educational planners, principals and School Boards of Management will find the findings of the study useful in planning for education and improvement of academic achievement in KCSE examination. The study recommended that the focus should be on the school predictors such as physical facilities and text books which accounted for a larger variation in academic achievement in KCSE examination.
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    Codes, designs and graphs obtained from some projective symplectic group
    (KIBU, 2019-11) Kananu, Rukaria Lydiah
    After the classification of finite simple groups, there is still much work to be done to give a clear geometric identification of the finite simple groups. There are also many problems in enumerating and characterizing a structure which either has a particular group acting on it or which has some degree of symmetry from a group action. It has been shown that there exists interplay between finite simple groups and codes. In this thesis we construct and enumerate binary linear codes for the projective symplectic group S8(2) from the permutation representations of degree 120, 136, 255, 2295, 5355, 5440 and 11475. We find that the support of codewords of a given weight in a code hold a combinatorial design, or represent points of a projective space PG(2m−1,q), or represent the rows of the adjacency matrix of a graph or equivalently are the incidence vectors of the blocks of a design. Through coding theory, the interplay between the combinatorial objects is enhanced and the internal structures of the group characterized.
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    SOME LINEAR CODES, GRAPHS AND DESIGNS FROM MATHIEU GROUPS M24 AND M23
    (Kibabii University, 2019-05-01) Marani, Vincent
    In this thesis, we have used four steps to determine G-invariant codes from primitive permutation representations of Mathieu groups M24 and M23 . We constructed all G-invariant codes from primitive representations of degree 24, 276, 759, and 1288 from the simple group M24. We found one self dual [24, 12, 8] code, three irreducible codes; [276,11,128], [759,11,352] and [1288,11,648]. There were several decomposable, self orthogonal and projective linear binary codes. There were two strongly regular graphs from a representation of degree 276 and 759. These graphs are known. We determined designs from some binary codes using codewords of minimum weight. All the designs constructed were primitive. We constructed symmetric 1-designs from the primitive permutation representations of degree 24, 276, 759, 1771, 2024 and 3795 defined by the action of a group G on a set Ω = G/Gα. In most cases the full automorphism group of the design was M24 while in some cases the full automorphism group of the design was either S24 or S276. We also constructed all G-invariant codes from primitive representations of degree 23, 253, and 253 from the simple group M23. There was no self dual linear code. There were four irreducible codes [23,11,8], [253,11,112],[253,44] and [253,11,112] . There were several decomposable, self orthogonal and projective linear binary codes. There was no strongly regular graph from the three representations. We determined designs from some binary codes using codewords of minimum weight. All the designs constructed were primitive. We constructed symmetric 1-designs from the primitive permutation representations of degree 23, 253 and 253 defined by the action of a group G on a set Ω = G/Gα. In most cases the full automorphism group of the design was M23 while in some cases the full automorphism group of the design was either S23, S253 or S506