Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 58, Number 3 (2006), 775-804.
On Euclidean tight 4-designs
A spherical -design is a finite subset in the unit sphere which replaces the value of the integral on the sphere of any polynomial of degree at most by the average of the values of the polynomial on the finite subset . Generalizing the concept of spherical designs, Neumaier and Seidel (1988) defined the concept of Euclidean -design in as a finite set in for which holds for any polynomial of , where is the set of all the concentric spheres centered at the origin and intersect with , , and is a weight function of . (The case of and with a constant weight corresponds to a spherical -design.) Neumaier and Seidel (1988), Delsarte and Seidel (1989) proved the (Fisher type) lower bound for the cardinality of a Euclidean -design. Let be a subset of and let be the vector space consisting of all the polynomials restricted to whose degrees are at most . Then from the arguments given by Neumaier-Seidel and Delsarte-Seidel, it is easy to see that holds, where . The actual lower bounds proved by Delsarte and Seidel are better than this in some special cases. However as designs on , the bound is natural and universal. In this point of view, we call a Euclidean -design with a tight -design on concentric spheres. Moreover if holds, then we call a Euclidean tight -design. We study the properties of tight Euclidean -designs by applying the addition formula on the Euclidean space. Furthermore, we give the classification of Euclidean tight 4-designs with constant weight. It is possible to regard our main result as giving the classification of rotatable designs of degree 2 in in the sense of Box and Hunter (1957) with the possible minimum size . We also give examples of nontrivial Euclidean tight 4-designs in with nonconstant weight,which give a counterexample to the conjecture of Neumaier and Seidel (1988) that there are no nontrivial Euclidean tight -designs even for the nonconstant weight case for .
J. Math. Soc. Japan, Volume 58, Number 3 (2006), 775-804.
First available in Project Euclid: 23 August 2006
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Primary: 05E99: None of the above, but in this section
Secondary: 05B99: None of the above, but in this section 51M99: None of the above, but in this section 62K99: None of the above, but in this section
BANNAI, Eiichi; BANNAI, Etsuko. On Euclidean tight 4-designs. J. Math. Soc. Japan 58 (2006), no. 3, 775--804. doi:10.2969/jmsj/1156342038. https://projecteuclid.org/euclid.jmsj/1156342038