# Báo cáo toán học: "Mutually Disjoint Steiner Systems S(5, 8, 24) and 5-(24, 12, 48) Designs"

## Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài:Mutually Disjoint Steiner Systems S(5, 8, 24) and 5-(24, 12, 48) Designs. | Mutually Disjoint Steiner Systems S 5 8 24 and 5- 24 12 48 Designs Makoto Araya Masaaki Harada Department of Computer Science Shizuoka University Hamamatsu 432-8011 Japan araya@ Department of Mathematical Sciences Yamagata University Yamagata 990-8560 Japan and PRESTO Japan Science and Technology Agency Kawaguchi Saitama 332-0012 Japan mharada@ Submitted Aug 4 2009 Accepted Dec 9 2009 Published Jan 5 2010 Mathematics Subject Classifications 05B05 Abstract We demonstrate that there are at least 50 mutually disjoint Steiner systems S 5 8 24 and there are at least 35 mutually disjoint 5- 24 12 48 designs. The latter result provides the existence of a simple 5- 24 12 6m design for m 24 32 40 48 56 64 72 80 112 120 128 136 144 152 160 168 200 208 216 224 232 240 248 and 256. 1 Introduction A t- v k A design D is a pair of a set X of v points and a collection B of k-subsets of X called blocks such that every t-subset of X is contained in exactly A blocks. We often denote the design D by X B . A design with no repeated block is called simple. All designs in this note are simple. A Steiner system S t k v is a t- v k A design with A 1. Two t- v k A designs with the same point set are said to be disjoint if they have no blocks in common. Two t- v k A designs are isomorphic if there is a bijection between their point sets that maps the blocks of the first design into the blocks of the second design. An automorphism of a t- v k A design D is any isomorphism of the design with itself and the set consisting of all automorphisms of D is called the automorphism group Aut D of D. The well-known Steiner system S 5 8 24 and a 5- 24 12 48 design are constructed by taking as blocks the supports of codewords of weights 8 and 12 in the extended Go-lay 24 12 8 code respectively. It is well known that there is a unique Steiner system S 5 8 24 up to isomorphism 8 and there is a unique 5- 24 12 48 design having even THE ELECTRONIC JOURNAL OF .

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