Construction of designs and codes invariant under the group PSp4 (q), where q is a prime power

Abstract

In this thesis, we aim to construct some designs and their codes from the projective symplectic group PSp4(q), where “q” is a prime power. We study the primitive permutation representations and conjugacy classes of projective symplectic groups and construct binary and ternary codes invariant under the group PSp4(q). To achieve this, we examine the structures of each maximal subgroup and study the conjugacy class table and character table within the group PSp4(q). We identify fixed points of the primitive actions of PSp4(q) and compute the designs parameters. We then construct codes that are invariant under the action of PSp4(q). Our investigation focuses on specific classes of maximal subgroups of PSp4(q), where we determine the design parameters using two methods known as Key-Moori Method 1 and 2. Finally, we construct codes based on these design parameters. The goal is to find the suborbits corresponding to specific maximal subgroups. To do this, we first identify some maximal subgroups of the group, then examine how the group action partitions the set into suborbits under the action of each maximal subgroup. These suborbits can help us understand the structure of the group and its actions more effectively, as outlined in Method 1. We examine the character table of the group PSp4(q), which provides information about the irreducible representations and their degrees. Using this information, we then find the corresponding permutation characters, which describe how the group acts on sets, such as conjugacy classes or coset spaces, under Method 2. By employing the techniques of examining suborbits and character table of the group PSp4(q), we are able to derive block-primitive and point-transitive 1−(v, k, λ) designs from the conjugacy classes and maximal subgroups of the group PSp4(q). We analyze 1 − (v, k, λ) design properties and construct codes that are defined by the action of groups matrices of the designs. We also determine the corresponding linear codes associated with these designs in the special case where q = 3. We examine various properties of these codes, such as their dual structure, weight distribution, and error-detection and error-correction capabilities. The codes, which are constructed using the permutation group, can be viewed as submodules of the permutation module corresponding to the action of PSp4(q). By analyzing these codes, we gain insights into their invariant properties and symmetries in relation to the group action. To demonstrate this, we focus on the finite simple group G = PSp4(3) and explore the linear and ternary codes derived from its 2- and 3-dimensional representations. Additionally, we establish connections between the combinatorial designs and the codes that remain invariant under the action of PSp4(3).