Finite Groups, Designs and Codes ∗ J Moori School of Mathematical Sciences, North-West University (Mafikeng) Mmabatho, 2375, South Africa † September 14, 2011 Abstract We introduce the reader to combinatorial structures such as Designs and Linear Codes and will discuss some of their properties, we also give few examples. We then introduce background material required from Finite Groups and Representation Theory of Finite Groups (Linear and Permuta- tion Representations). We aim to introduce two new methods for construct- ing codes and designs from finite groups (mostly simple finite groups). We outline some of recent collaborative work by the author with J D Key and B Rorigues. Keywords: Designs, codes, simple groups, maximal subgroups, conju- gacy classes. 1 Introduction Error-correcting codes that have large automorphism groups are useful in appli- cations as the group can help in determining the code’s properties, and can be useful in decoding algorithms: see Huffman [15]. In a series of 3 lectures given at the NATO Advanced Study Institute ”Infor- mation Security and Related Combinatorics” held in Croatia [28], we discussed two methods for constructing codes and designs for finite groups (mostly simple finite groups). The first method dealt with construction of symmetric 1-designs and binary codes obtained from from the action on the maximal subgroups, of a finite group G. This method has been applied to several sporadic simple groups, for example in [18], [22], [23], [29], [30], [31] and [32]. The second method intro- duces a technique from which a large number of non-symmetric 1-designs could be constructed. Let G be a finite group, M be a maximal subgroup of G and C g =[g]= nX be the conjugacy class of G containing g. We construct 1 −(v,k,λ) designs D =(P , B), where P = nX and B = {(M ∩ nX) y |y ∈ G}. The param- eters v, k, λ and further properties of D are determined. We also study codes associated with these designs. In Subsections 5.1, 5.2 and 5.3 we apply the second method to the groups A 7 , PSL 2 (q) and J 1 respectively. * AMS Subject Classification (2000): 20D05, 05B05. † Supports from NATO, NRF, University of North-Westl and AIMS are acknowledged. 1