Browsing by Author "Maina, Janet Lilian"

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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.
Classification of Some Internal Structures of Degree 120 Related To a Group of Extension𝑂8+ 2 : 2
(IRE Journals, 2023-08-07) Maina, Janet Lilian; Matuya, John Wanyonyi; Njuguna, Edward; Marani, Vincent Nyongesa
This paper uses the modular representation method to classify the internal structures of degree 120 related to a group of extension,𝑶𝟖+ 𝟐 : 2.Specifically, we determine the number of binary linear codes and construct their lattice structure, as well as investigate the properties of some linear codes and designs of minimum weights. Our findings reveal that there are 12 binary linear codes, consisting of 4 doubly even codes, 4 projective codes, 2 irreducible codes, and 2 decomposable codes. We also identify 2 primitive 1-designs of minimum weight. The results demonstrate the potential benefits of using linear codes and designs from finite groups of extension with modular representation methods, such as improved error correction, increased data storage capacity, improved security, efficient designs, and improved computational efficiency. However, it is important to note that this topic can be complex and technical, and we recommend that stakeholders collaborate with experts in the field to ensure the accuracy and reliability of the information being used. Overall, this study contributes to the understanding of the modular representation method and its applications in coding theory and related fields.