### 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 WW^{T} = kI_{n}. 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} = ω_{gf,} _{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) |
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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)

### Cite this

*Journal of Combinatorial Mathematics and Combinatorial Computing*,

*62*, 65-84.

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*Journal of Combinatorial Mathematics and Combinatorial Computing*, vol. 62, pp. 65-84.

**A search for solvable weighing matrices.** / Becker, Paul; Houghten, Sheridan; Haas, Wolfgang.

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 -