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| NOTE ON REGULAR D-OPTIMAL MATRICES |
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Citation： |
LI Qiaoliang.NOTE ON REGULAR D-OPTIMAL MATRICES[J].Chinese Annals of Mathematics B,2003,24(2):215~220 |
| Page view： 1140
Net amount： 917 |
Authors： |
LI Qiaoliang; |
Foundation： |
Project supported by the Science Foundation of China for Postdoctor. |
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| Abstract： |
Let $ A $ be a $ j\times d $ $ (0, 1) $
matrix. It is
known that if $ j=2k-1 $ is odd, then det$(AA^T)\leq
(j+1)((j+1)d/4j)^j$; if $ j $ is even, then det$(AA^T)\leq
(j+1)((j+2)d/4(j+1))^j.$ $ A $ is called a regular $ D$-optimal matrix if
it satisfies the equality of the above bounds. In this note, it is
proved that if $ j=2k-1 $ is odd, then $A $ is a regular $
D$-optimal matrix if and only if $ A $ is the adjacent matrix of a
$ (2k-1, k, (j+1)d/4j)$-BIBD; if $ j=2k $ is even, then $ A $ is a
regular $ D$-optimal matrix if and only if $ A $ can be obtained
from the adjacent matrix $ B $ of a $ (2k+1, k+1,
(j+2)d/4(j+1))$-BIBD by deleting any one row from $ B.$
Three $ 21\times 42 $ regular $ D$-optimal matrices, which
were unknown in [11], are also provided. |
Keywords： |
Regular D-optimal matrices, Simplex, Weighing design |
Classification： |
62K05, 05B05, 05B20, 15A15 |
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