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Báo cáo toán học: "Classification of Generalized Hadamard Matrices H(6, 3) and Quaternary Hermitian Self-Dual Codes of Length 18"

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: Classification of Generalized Hadamard Matrices H(6, 3) and Quaternary Hermitian Self-Dual Codes of Length 18. | Classification of Generalized Hadamard Matrices H 6 3 and Quaternary Hermitian Self-Dual Codes of Length 18 Masaaki Harada Department of Mathematical Sciences Yamagata University Yamagata 990-8560 Japan mharada@sci.kj.yamagata-u.ac.jp Akihiro Munemasa Graduate School of Information Sciences Tohoku University Sendai 980-8579 Japan munemasa@math.is.tohoku.ac.jp Clement Lam Department of Computer Science Concordia University Montreal QC Canada H3G 1M8 lam@cse.concordia.ca Vladimir D. Tonchev Mathematical Sciences Michigan Technological University Houghton MI 49931 USA tonchev@mtu.edu Submitted Jan 30 2010 Accepted Nov 24 2010 Published Dec 10 2010 Mathematics Subject Classifications 05B20 94B05 Abstract All generalized Hadamard matrices of order 18 over a group of order 3 H 6 3 are enumerated in two different ways once as class regular symmetric 6 3 -nets or symmetric transversal designs on 54 points and 54 blocks with a group of order 3 acting semi-regularly on points and blocks and secondly as collections of full weight vectors in quaternary Hermitian self-dual codes of length 18. The second enumeration is based on the classification of Hermitian self-dual 18 9 codes over GF 4 completed in this paper. It is shown that up to monomial equivalence there are 85 generalized Hadamard matrices H 6 3 and 245 inequivalent Hermitian selfdual codes of length 18 over GF 4 . 1 Introduction A generalized Hadamard matrix H ụ g hjj of order n gụ over a multiplicative group G of order g is a gụ X gụ matrix with entries from G with the property that for PRESTO Japan Science and Technology Agency Kawaguchi Saitama 332-0012 Japan THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R171 1 every i j 1 i j gụ each of the multi-sets hish 1 1 s gụ contains every element of G exactly ụ times. It is known 12 Theorem 2.2 that if G is abelian then H ụ g T is also a generalized Hadamard matrix where H ụ g T denotes the transpose of H ụ g see also 5 Theorem 4.11 . This result does not generalize to .