Some New Results on Circulant Weighing Matrices
K.T. Arasu
and Siu Lun Ma2
2dagger
DOI: 10.1023/A:1011903510338
Abstract
We obtain a few structural theorems for circulant weighing matrices whose weight is the square of a prime number. Our results provide new schemes to search for these objects. We also establish the existence status of several previously open cases of circulant weighing matrices. More specifically we show their nonexistence for the parameter pairs ( n, k) (here n is the order of the matrix and k its weight) = (147, 49), (125, 25), (200, 25), (55, 25), (95, 25), (133, 49), (195, 25), (11 w, 121) for w <>
Pages: 91–101
Keywords: weighing matrices; circulant; group rings; characters
Full Text: PDF
References
1. K.T. Arasu, “A reduction theorem for circulant weighing matrices,” Australas. J. Combin. 18 (1998), 111-114.
2. K.T. Arasu and J.F. Dillon, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Vol. 542, Kluwer Academic Publisher, Dordrecht, 1999, pp. 1-15.
3. K.T. Arasu and S.L. Ma, “Abelian difference sets without self-conjugacy,” Designs, Codes & Cryptography 15 (1998), 223-230.
4. K.T. Arasu and S.L. Ma, “A nonexistence result on difference sets, partial difference sets and divisible difference sets,” J. Stat. Planning and Inference, to appear.
5. K.T. Arasu, S.L. Ma, and Y. Strassler, “Possible orders of a circulant weighing matrix of weight 9,” in preparation.
6. K.T. Arasu and J. Seberry, “Circulant weighing designs,” J. Comb. Designs 4(6) (1996), 439-447.
7. P. Eades and R.M. Hain, “On circulant weighing matrices,” Ars Combin. 2 (1976), 265-284.
8. A.V. Geramita and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel, 1979.
9. R.C. Mullin, “A note on balanced weighing matrices, Combinatorial Mathematics III,” in Proceedings of the Third Australian Conference, Lecture Notes in Mathematics, Vol. 452, Springer, Berlin-Heidelberg-New York, 1975, pp. 28-41.
10. A. Pott, Finite Geometry and Character Theory, Lecture Notes in Mathematics, Vol. 1601, Springer, Berlin- Heidelberg-New York, 1991.
11. Y. Strassler, “The classification of circulant weighing matrices of weight 9,” Ph.D. Thesis, Bar-Ilan University, 1997.
2. K.T. Arasu and J.F. Dillon, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Vol. 542, Kluwer Academic Publisher, Dordrecht, 1999, pp. 1-15.
3. K.T. Arasu and S.L. Ma, “Abelian difference sets without self-conjugacy,” Designs, Codes & Cryptography 15 (1998), 223-230.
4. K.T. Arasu and S.L. Ma, “A nonexistence result on difference sets, partial difference sets and divisible difference sets,” J. Stat. Planning and Inference, to appear.
5. K.T. Arasu, S.L. Ma, and Y. Strassler, “Possible orders of a circulant weighing matrix of weight 9,” in preparation.
6. K.T. Arasu and J. Seberry, “Circulant weighing designs,” J. Comb. Designs 4(6) (1996), 439-447.
7. P. Eades and R.M. Hain, “On circulant weighing matrices,” Ars Combin. 2 (1976), 265-284.
8. A.V. Geramita and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel, 1979.
9. R.C. Mullin, “A note on balanced weighing matrices, Combinatorial Mathematics III,” in Proceedings of the Third Australian Conference, Lecture Notes in Mathematics, Vol. 452, Springer, Berlin-Heidelberg-New York, 1975, pp. 28-41.
10. A. Pott, Finite Geometry and Character Theory, Lecture Notes in Mathematics, Vol. 1601, Springer, Berlin- Heidelberg-New York, 1991.
11. Y. Strassler, “The classification of circulant weighing matrices of weight 9,” Ph.D. Thesis, Bar-Ilan University, 1997.