@MISC{Conder_markingsof, author = {Marston Conder and John Mckay}, title = {Markings of the Golay Code}, year = {} }

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Abstract

The simple Mathieu group M 24 is the automorphism group of a Steiner system S(5; 8; 24), acting naturally on its 24 points. A marking of M 24 (or the associated Golay code) is defined to be a linear ordering x 1 ! x 2 ! : : : ! x 24 of these points, and with any such marking one may associate a 5-tuple (c 0 ; c 1 ; c 2 ; c 3 ; c 4 ), where c k denotes the number of blocks of the given Steiner system containing exactly k pairs of the form fx 2i\Gamma1 ; x 2i g. In this paper we give a solution to the marking problem for M 24 , namely the determination of the spectrum of these 5-tuples, showing that precisely 90 different 5-tuples occur. x1. Introduction The Steiner system S(5; 8; 24) is a block design made up of 24 points and 759 blocks, each of size 8, with the property that every 5 points lie in exactly one block. This design is naturally associated with the Golay code, and its automorphism group is the simple Mathieu group M 24 ; see [3; Ch.11]. The following problem was communicat...