Journal of the Mathematical Society of Japan

On Euclidean tight 4-designs

Eiichi BANNAI and Etsuko BANNAI

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Abstract

A spherical t -design is a finite subset X in the unit sphere S n - 1 R n which replaces the value of the integral on the sphere of any polynomial of degree at most t by the average of the values of the polynomial on the finite subset X . Generalizing the concept of spherical designs, Neumaier and Seidel (1988) defined the concept of Euclidean t -design in R n as a finite set X in R n for which i = 1 p ( w ( X i ) / ( | S i | ) ) S i f ( x ) d σ i ( x ) = x X w ( x ) f ( x ) holds for any polynomial f ( x ) of deg ( f ) t , where { S i , 1 i p } is the set of all the concentric spheres centered at the origin and intersect with X , X i = X S i , and w : X R > 0 is a weight function of X . (The case of X S n - 1 and with a constant weight corresponds to a spherical t -design.) Neumaier and Seidel (1988), Delsarte and Seidel (1989) proved the (Fisher type) lower bound for the cardinality of a Euclidean 2 e -design. Let Y be a subset of R n and let 𝒫 e ( Y ) be the vector space consisting of all the polynomials restricted to Y whose degrees are at most e . Then from the arguments given by Neumaier-Seidel and Delsarte-Seidel, it is easy to see that | X | dim ( 𝒫 e ( S ) ) holds, where S = i = 1 p S i . The actual lower bounds proved by Delsarte and Seidel are better than this in some special cases. However as designs on S , the bound dim ( 𝒫 e ( S ) ) is natural and universal. In this point of view, we call a Euclidean 2 e -design X with | X | = dim ( 𝒫 e ( S ) ) a tight 2 e -design on p concentric spheres. Moreover if dim ( 𝒫 e ( S ) ) = dim ( 𝒫 e ( R n ) ) ( = n + e e ) holds, then we call X a Euclidean tight 2 e -design. We study the properties of tight Euclidean 2 e -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 R n in the sense of Box and Hunter (1957) with the possible minimum size n + 2 2 . We also give examples of nontrivial Euclidean tight 4-designs in R 2 with nonconstant weight,which give a counterexample to the conjecture of Neumaier and Seidel (1988) that there are no nontrivial Euclidean tight 2 e -designs even for the nonconstant weight case for 2 e 4 .

Article information

Source
J. Math. Soc. Japan, Volume 58, Number 3 (2006), 775-804.

Dates
First available in Project Euclid: 23 August 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1156342038

Digital Object Identifier
doi:10.2969/jmsj/1156342038

Mathematical Reviews number (MathSciNet)
MR2254411

Zentralblatt MATH identifier
1104.05016

Subjects
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

Keywords
experimental design rotatable design tight design spherical design 2-distance set Euclidean space addition formula

Citation

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


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