On Euclidean tight 4-designs

スポンサーリンク

概要

• 論文の詳細を見る

• A spherical t-design is a finite subset X in the unit sphere Sn-1⊂Rn 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 Rn as a finite set X in Rn for which $¥sum$i=1p(w(Xi)/(|Si|)) ∫Sif(x)dσi(x) = $¥sum$x∈Xw(x)f(x) holds for any polynomial f(x) of deg(f)≤t, where {Si, 1≤i≤p} is the set of all the concentric spheres centered at the origin and intersect with X, Xi=X∩Si, and w:X→R>0 is a weight function of X. (The case of X⊂Sn-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 2e-design. Let Y be a subset of Rn and let $¥mathscr{P}$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($¥mathscr{P}$e(S)) holds, where S=∪i=1pSi. 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($¥mathscr{P}$e(S)) is natural and universal. In this point of view, we call a Euclidean 2e-design X with |X| = dim($¥mathscr{P}$e(S)) a tight 2e-design on p concentric spheres. Moreover if dim($¥mathscr{P}$e(S)) = dim($¥mathscr{P}$e(Rn)) (=${n+e ¥choose e}$) holds, then we call X a Euclidean tight 2e-design. We study the properties of tight Euclidean 2e-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 Rn 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 R2 with nonconstant weight, which give a counterexample to the conjecture of Neumaier and Seidel (1988) that there are no nontrivial Euclidean tight 2e-designs even for the nonconstant weight case for 2e≥4.

• 社団法人 日本数学会の論文
• 2006-07-01

著者

• 坂内 英一

  九州大学数理学研究院

• Bannai Eiichi

  Graduate School Of Mathematics Kyushu University

• BANNAI Etsuko

  Graduate School of Mathematics Kyushu University

関連論文

• Modular invariants of the modular data of finite groups (Codes, lattices, vertex operator algebras and finite groups)
• Moore Graphsについて (群論と組み合せ論)
• Generalized spin models and association schemes
• THE DECOMPOSITION OF THE PERMUTATION CHARACTER $1^{GL(2n,q)}_{GL(n,q^2)}$ (Topics in Young Diagrams and Representation Theory)
• THE TERWILLIGER ALGEBRAS OF GROUP ASSOCIATION SCHEMES
• D. G. Higmanの組合せ論・群論についての後期の仕事を巡って (有限群・頂点作用素代数と組合せ論)
• Some Remarks on Pseudo-Cyclic Association Schemes(Group Theory and Related Topics)
• On Euclidean tight 4-designs
• Extremal latticesに付随する球面デザインとextremal modular formsのFourier係数のmodulo pの性質について : 小池正夫、篠原雅史、田上真との共同研究 (代数的組合せ論)
• 球の詰め込み問題についての最近の進展 I : Cohn-Elkies-Kumarの仕事の紹介を中心にして (代数的組合せ論)
• 種々のtightデザインの存在・非存在問題について (符号と暗号の代数的数理)
• On the work of Tosiro Tsuzuku (1929-2002) on finite permutation groups (Algebraic Combinatorics)
• モジュラー形式についての考察 : 小池正夫, 宗政昭弘, 関口次郎との共同研究 (代数的組合せ論)
• Association schemes and Spin models
• 代数的組合せ論 - アソシエーションスキームの最近の話題 -
• 純代数的立場から見たfusion algebras(複素解析幾何学とその周辺の研究:数理物理と複素幾何)
• Association schemes and fusion algebras(GROUPS AND COMBINATORICS)
• Association schemeの指標表のいくつかの具体的な計算例(組合せ論とその周辺の研究)
• Tight Spherical Designs (デザインの構成と解析)
• Nonexistence Theorems of Perfect Codes and Tight Designs in Distance Transitive Graphs (デザインの構成法および不存在性)
• 多重可移置変換について (有限群の研究)
• Type II codes over $\mathbf{F}_2+u\mathbf{F}_2$ and an application to Hermitian modular forms (Algebraic Combinatorics)
• ON JACOBI FORMS OF WEIGHT 4
• 組合せ論と符号理論 (現代数学はいかに使われているか--代数編)
• 有限単純群の低いランクの極大部分群について (有限群の研究)
• 純粋数学としての組合せ論 : 代数的組合せ論のめざすもの

もっと見る 閉じる

スポンサーリンク