论文信息 - Tight t-Designs and Squarefree Integers

Tight t-Designs and Squarefree Integers

The authors prove, using a variety of number-theoretical methods, that tight t-designs in the projective spaces FPn of ‘lines’ through the origin in Fn+1 (F = ℂ, or the quarternions H) satisfy t ⩽ 5. Such a design is a generalisation of a combinatorial t-design. It is known that t ⩽ 5 in the cases F = ℝ , O (the octonions) and that t ⩽ 11 for tight spherical t-designs; hence the author's result essentially completes the classification of tight t-designs in compact connected symmetric spaces of rank 1.

Eiichi Bannai | Stuart G. Hoggar

[1] W. Ljunggren. On the Diophantine Equation $Ax^4-By^2=C$, ($C=1,4$). , 1967 .

[2] Paul Erdös. On a Diophantine Equation , 1951 .

[3] G. Dumas,et al. Sur quelques cas d'irréductibilité des polynomes à coefficients rationnels , 1906 .

[4] S. G. Hoggar,et al. t-Designs in Projective Spaces , 1982, Eur. J. Comb..

[5] Paul Erdös,et al. The product of consecutive integers is never a power , 1975 .

[6] Eiichi Bannai,et al. On tight $t$-designs in compact symmetric spaces of rank one , 1985 .

[7] Stuart G. Hoggar. Parameters of t-Designs in 𝔽Pd-1 , 1984, Eur. J. Comb..

[8] J. J. Seidel,et al. Cubature Formulae, Polytopes, and Spherical Designs , 1981 .

[9] Ove Hemer. Notes on the diophantine equationy2 −k =x3 , 1954 .

[10] Paul Erdös,et al. A Theorem of Sylvester and Schur , 1934 .

[11] Hans Rohrbach,et al. Zum finiten Fall des Bertrandschen Postulats. , 1964 .

[12] Paul Erdös. Note on Products of Consecutive Integers , 1939 .

[13] Eiichi Bannai,et al. Tight spherical designs, I , 1979 .

[14] J. Seidel,et al. Spherical codes and designs , 1977 .

[15] R. M. Damerell,et al. Tight Spherical Disigns, II , 1980 .