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@ Akihiro Munemasa Graduate School of Information Sciences Tohoku University Sendai 980-8579 Japan munemasa@ Clement Lam Department of Computer Science Concordia University Montreal QC Canada H3G 1M8 lam@ Vladimir D. Tonchev Mathematical Sciences Michigan Technological University Houghton MI 49931 USA tonchev@ 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 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 . This result does not generalize to .
