Designs and codes from fixed points of finite groups

Abstract

We previously have developed two methods (Key–Moori Methods 1 and 2) for constructing codes and designs from finite groups (mostly simple finite groups). In this article, we introduce a new method (Method 3) for constructing codes and designs from fixed points of elements of finite transitive groups. We first discuss background material and results required from finite groups, permutation groups and representation theory. The main aim of this article is to discuss this new method and give some examples by applying it to the sporadic simple groups HS and J₂. In subsequent papers, we aim to apply it to several other simple groups.

Keywords:

• Codes
• conjugacy classes
• designs
• fixed points
• maximal subgroups
• simple groups
• sporadic groups

AMS Subject Classification (2010):

• 20D05
• 20D08
• 05E15
• 94B25

Acknowledgment

The author acknowledges supports from NWU (Mafikeng) and would like to thank the School of Mathematics at the University of Birmingham (UK).