Doctor of Philosophy in Pure Mathematics
|Cedric Wanjala Ndarinyo||PHD/PM/001/15||Binary Linear Codes Obtained From The Alternating Group (An , n= 5,7,9).|
Given a primitive representation of a group G on a finite set , defining a multiplication on the group ring V by elements of the group and extending it linearly makes V an FG-module. The invariant submodules are all the binary codes for which G is a subgroup of the automorphism group of the code. We attempt to determine and examine all the binary invariant codes under the prescribed group. This is an enumeration and classification problem which reveals interplay between coding theory, representation theory and combinatorial structures. Considering the simple alternating group An 𝑛 = 5, 7, 9, we shall construct binary linear codes. We investigate the properties of these codes, in particular the weight distribution and the automorphism groups of these codes and their interplay between other mathematical objects like designs, graphs and finite geometries. We shall also uncover the lattice structure of the modules. We shall make use of Magma database to search for the irreducible modules under this group. We shall use the recursive searches together with Meataxe to help determine the irreducibility of the modules. We also make an in depth use of the database of irreducible faithful representations available in Magma and Wilson’s webpage with the Brauer character tables and the atlas of finite groups. These codes will be used in communication channels for error detection and error correction.