Abstract
A weighing matrix W(n, k) of order n with weight k is an n × n matrix with entries from {0, 1, - 1} which satisfies WWT = kIn. Such a matrix is group-developed if its rows and columns can be indexed by elements of a finite group G so that ω g,h = ωgf, hf for all g, h, and f in G. Group-developed weighing matrices are a natural generalization of perfect ternary arrays and Hadamard matrices. They are closely related to difference sets. We describe a search for weighing matrices with order 60 and weight 25, developed over solvable groups. There is one known example of a W(60, 25) developed over a nonsolvable group; no solvable examples are known. We use techniques from representation theory, including a new viewpoint on complementary quotient images, to restrict solvable examples. We describe a computer search strategy which has eliminated two of twelve possible cases. We summarize plans to complete the search.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 65-84 |
| Number of pages | 20 |
| Journal | Journal of Combinatorial Mathematics and Combinatorial Computing |
| Volume | 62 |
| State | Published - Aug 1 2007 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
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A search for solvable weighing matrices. / Becker, Paul; Houghten, Sheridan; Haas, Wolfgang.
In: Journal of Combinatorial Mathematics and Combinatorial Computing, Vol. 62, 01.08.2007, p. 65-84.Research output: Contribution to journal › Article
TY - JOUR
T1 - A search for solvable weighing matrices
AU - Becker, Paul
AU - Houghten, Sheridan
AU - Haas, Wolfgang
PY - 2007/8/1
Y1 - 2007/8/1
N2 - A weighing matrix W(n, k) of order n with weight k is an n × n matrix with entries from {0, 1, - 1} which satisfies WWT = kIn. Such a matrix is group-developed if its rows and columns can be indexed by elements of a finite group G so that ω g,h = ωgf, hf for all g, h, and f in G. Group-developed weighing matrices are a natural generalization of perfect ternary arrays and Hadamard matrices. They are closely related to difference sets. We describe a search for weighing matrices with order 60 and weight 25, developed over solvable groups. There is one known example of a W(60, 25) developed over a nonsolvable group; no solvable examples are known. We use techniques from representation theory, including a new viewpoint on complementary quotient images, to restrict solvable examples. We describe a computer search strategy which has eliminated two of twelve possible cases. We summarize plans to complete the search.
AB - A weighing matrix W(n, k) of order n with weight k is an n × n matrix with entries from {0, 1, - 1} which satisfies WWT = kIn. Such a matrix is group-developed if its rows and columns can be indexed by elements of a finite group G so that ω g,h = ωgf, hf for all g, h, and f in G. Group-developed weighing matrices are a natural generalization of perfect ternary arrays and Hadamard matrices. They are closely related to difference sets. We describe a search for weighing matrices with order 60 and weight 25, developed over solvable groups. There is one known example of a W(60, 25) developed over a nonsolvable group; no solvable examples are known. We use techniques from representation theory, including a new viewpoint on complementary quotient images, to restrict solvable examples. We describe a computer search strategy which has eliminated two of twelve possible cases. We summarize plans to complete the search.
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M3 - Article
AN - SCOPUS:78651587005
VL - 62
SP - 65
EP - 84
JO - Journal of Combinatorial Mathematics and Combinatorial Computing
JF - Journal of Combinatorial Mathematics and Combinatorial Computing
SN - 0835-3026
ER -