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Bush‐type Hadamard matrices and symmetric designs
Author(s) -
Janko Zvonimir,
Kharaghani Hadi,
Tonchev Vladimir D.
Publication year - 2001
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/1520-6610(2001)9:1<72::aid-jcd6>3.0.co;2-m
Subject(s) - mathematics , hadamard transform , combinatorics , integer (computer science) , hadamard matrix , block (permutation group theory) , class (philosophy) , type (biology) , matrix (chemical analysis) , order (exchange) , point (geometry) , discrete mathematics , computer science , mathematical analysis , geometry , ecology , materials science , finance , artificial intelligence , economics , composite material , biology , programming language
Abstract Abstact: A symmetric 2‐(100, 45, 20) design is constructed that admits a tactical decomposition into 10 point and block classes of size 10 such that every point is in either 0 or 5 blocks from a given block class, and every block contains either 0 or 5 points from a given point class. This design yields a Bush‐type Hadamard matrix of order 100 that leads to two new infinite classes of symmetric designs with parameters $\nu = 100(81^m+81^{m-1}+\cdots +81+1), \quad \kappa = 45(81)^m, \quad \lambda = 20(81)^m,$ and $\nu = 100(121^m+121^{m-1}+\cdots +121+1), \quad \kappa = 55(121)^m, \quad \lambda = 30(121)^m,$ where m is an arbitrary positive integer. Similarly, a Bush‐type Hadamard matrix of order 36 is constructed and used for the construction of an infinite family of designs with parameters ${v} = 36(25^m + 25^{m-1} + \cdots + 51 + 1),\quad k = 15(25)^m,\quad\lambda = 6(25)^m,$ and a second infinite family of designs with parameters ${v} = 36(49^m + 49^{m-1} + \cdots + 49 +1,\quad k = 21(49)^m,\quad\lambda = 12(49)^m,$ where m is any positive integer. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 72–78, 2001