On Euclidean tight 4-designs

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


  • Tokyo Sugaku Kaisya Zasshi  

    Tokyo Sugaku Kaisya Zasshi 58(3), 775-804, 2006-07-01 

    The Mathematical Society of Japan

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