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Study of proper circulant weighing matrices with weight 9. (English) Zbl 1153.05012

A weighing matrix of order \(n\) and weight \(w\) is an \(n \times n\) matrix \(W\) with entries from \(\{-1,0,1\}\) satisfying \(WW^T = wI\), where \(I\) is the identity matrix. Such a matrix is circulant if its rows are obtained by shifting the first row in a cyclical manner.
Y. Strassler [PhD Thesis, Bar-Ilan University (1998)] obtained a computer-aided proof that a weighing matrix of order \(n\) and weight 9 exists if and only if \(13 \mid n\) or \(24 \mid n\). Here the same result is proved with algebraic tools.

MSC:

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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References:

[1] M.H. Ang, Group weighing matrices, Ph.D. Thesis, National University of Singapore, Singapore, 2003.; M.H. Ang, Group weighing matrices, Ph.D. Thesis, National University of Singapore, Singapore, 2003.
[2] Arasu, K. T.; Dillon, J. F., (Perfect Ternary Arrays: Different Sets, Sequences and their Correlation Properties. Perfect Ternary Arrays: Different Sets, Sequences and their Correlation Properties, NATO Advanced Science Institute Series C Mathematical and Physical Sciences, vol. 542 (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 1-15 · Zbl 0940.05019
[3] Arasu, K. T.; Ma, S. L., Some new results on circulant weighing matrices, J. Algebraic Combin., 14, 91-101 (2001) · Zbl 1007.05029
[4] Arasu, K. T.; Seberry, J., Circulant weighing designs, J. Combin. Designs, 4, 439-447 (1996) · Zbl 0912.05024
[5] Arasu, K. T.; Seberry, J., On circulant weighing matrices, Australasian J. Combin., 17, 21-37 (1998) · Zbl 0918.05030
[6] Mullin, R. C., A note on balanced weighing matrices, in Combinatorial Mathematics III, (Proceedings of the Third Australian Conference, Lecture Notes in Mathematics, vol. 452 (1975), Springer: Springer Berlin, Heidelberg, New York), 28-41
[7] Y. Strassler, The classification of circulant weighing matrices of weight 9, Ph.D. Thesis, Bar-Ilan University, Israel, 1998.; Y. Strassler, The classification of circulant weighing matrices of weight 9, Ph.D. Thesis, Bar-Ilan University, Israel, 1998. · Zbl 0936.05021
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