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European Journal of Combinatorics a Survey on Spherical Designs And View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector European Journal of Combinatorics 30 (2009) 1392–1425 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc A survey on spherical designs and algebraic combinatorics on spheres Eiichi Bannai a, Etsuko Bannai b a Faculty of Mathematics, Graduate School, Kyushu University, Hakozaki 6-10-1, Higashu-ku, Fukuoka, 812-8581, Japan b Misakigaoka 2-8-21, Maebaru-shi, Fukuoka, 819-1136, Japan article info a b s t r a c t Article history: This survey is mainly intended for non-specialists, though we try to Available online 12 February 2009 include many recent developments that may interest the experts as well. We want to study ``good'' finite subsets of the unit sphere. To consider ``what is good'' is a part of our problem. We start with n−1 n the definition of spherical t-designs on S in R : After discussing some important examples, we focus on tight spherical t-designs on Sn−1: Tight t-designs have good combinatorial properties, but they rarely exist. So, we are interested in the finite subsets on Sn−1, which have properties similar to tight t-designs from the various viewpoints of algebraic combinatorics. For example, rigid t-designs, universally optimal t-codes (configurations), as well as finite sets which admit the structure of an association scheme, are among them. We will discuss various results on the existence and the non-existence of special spherical t-designs, as well as general spherical t-designs, and their constructions. We will discuss the relations between spherical t-designs and many other branches of mathematics. For example: by considering the spherical designs which are orbits of a finite group in the real orthogonal group O.n/, we get many connections with group theory; by considering those which are shells of Euclidean lattices, we get many unexpected connections with number theory, such as modular forms and Lehmer's conjecture about the zeros of the Ramanujan τ function. Spherical t-designs and Euclidean t-designs are special cases of cubature formulas in approximation theory, and thus we get many connections with analysis and statistics, and in particular with orthogonal polynomials, and moment problems. Moreover, Delsarte's linear programming method and many recent generalizations, including E-mail addresses: [email protected] (Eiichi Bannai), [email protected] (Etsuko Bannai). 0195-6698/$ – see front matter ' 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2008.11.007 Eiichi Bannai, Etsuko Bannai / European Journal of Combinatorics 30 (2009) 1392–1425 1393 the work of Musin and the subsequent progress in using semi- definite programming, have strong connections with geometry (in particular sphere packing problems) and the theory of optimizations. These various connections explain the reason of the charm of algebraic combinatorics on spheres. At the same time, these theories of spherical t-designs and related topics have strong roots in the developments of algebraic combinatorics in general, which was started as Delsarte theory of codes and designs in the framework of association schemes. ' 2008 Elsevier Ltd. All rights reserved. 1. Introduction The purpose of this paper is to give a survey on the study of spherical designs and codes, in particular from the viewpoint of ``algebraic combinatorics''. n−1 n Our aim is to study ``good'' finite sets of points on the unit sphere S in the Euclidean space R . The natural question is what does ``good'' mean. There is yet no final answer known. (Also, it is unrealistic to expect a single good answer.) To consider this question of ``what is good'' is an important part of our problem. Let X be a finite subset of Sn−1. Spherical codes and spherical designs are nothing but finite subsets of Sn−1. Roughly speaking, the code theoretical viewpoint is to try to find X, whose points n−1 are scattered on S as far as possible, i.e. the minimum distance dmin of X is as large as possible for a given size of X. (In some other cases, we impose some other conditions, e.g. there are only s kinds of distances between two distinct points of X, i.e. X is an s-distance subset, and then try to increase the size of X as large as possible.) On the other hand, the design theoretical viewpoint is to try to find X which globally approximates the sphere Sn−1 very well. Of course, ``What does approximate the sphere Sn−1 well mean?'' is also an interesting question. There is one very reasonable answer introduced by Delsarte–Goethals–Seidel [88] in 1977. Namely, a finite subset X on Sn−1 n−1 is called a spherical t-design on S , if for any polynomial f .x/ D f .x1; x2;:::; xn/ of degree at most t; the value of the integral of f .x/ on Sn−1 (divided by the volume of Sn−1) is just the average value of f .x/ on the finite set X. As is obvious from the definition, a spherical t-design is better if t is larger, and usually a spherical t-design X is better if the cardinality jXj is smaller. It seems that the concept of spherical t-designs caught the hearts of combinatorialists, as a natural analogue of the classical concept of combinatorial t-designs (t–.v; k; λ) designs) which appear in the traditional theory of designs in combinatorics. Actually there is a nice analogy between the theories of codes and designs in the frame work of association schemes as formulated in Delsarte [87] and the theory of spherical codes and designs as formulated in Delsarte–Goethals–Seidel [88]. These are called Delsarte theory on association schemes and Delsarte theory on spheres, or slightly more broadly, algebraic combinatorics on association schemes or algebraic combinatorics on spheres. In Section 2, we consider algebraic combinatorics on spheres starting from the definition of spherical t-designs. Our main focus is on the interplay between the design theoretical viewpoint and the code theoretical viewpoint. We will discuss examples of spherical t-designs, including tight t- designs, i.e. t-designs whose size attain the natural lower bounds (called Fisher type bounds). Also, we see that tight t-designs have good extremal properties in the interplay of code theory and design theory viewpoints. (Tight 2s-designs are s-distance sets, and tight .2s − 1/-designs are antipodal s- distance sets, etc.) If X ⊂ Sn−1 is a t-design and an s-distance set, then we always have t ≤ 2s, so the designs with t close to 2s are interesting. In particular if X is a tight t-design, or more generally if t ≥ 2s−2, then X has the structure of a Q-polynomial association scheme [88]. Then we discuss topics, such as classification problems of tight t-designs, rigid spherical t-designs, the existence theorem (of Seymour–Zaslavsky [164]) of spherical t-designs on Sn−1 for any t and n; and the explicit construction problems of spherical t-designs (and interval t-designs). Most of the material treated in Section 2 is 1394 Eiichi Bannai, Etsuko Bannai / European Journal of Combinatorics 30 (2009) 1392–1425 standard and well known, and not particularly new to experts. But some of them are very recent, and may be new to them. In Section 3, we give more examples of spherical designs. It seems that the most natural way to get a good distribution of finitely many points on Sn−1 is to take a finite subgroup G of the real orthogonal group O.n/, and a point u on Sn−1, and take the orbit of u by G. Many good examples are obtained this 3 4 way. For example, taking G D W .H4/, the real reflection group of type H4 and a suitable u 2 S ⊂ R , 3 4 we can get a spherical 19-design on S ⊂ R . Many more examples of spherical t-designs are obtained by this method. However, it seems difficult to obtain t-designs for large t by this method for n ≥ 3. In Section 3.1, we mention what kinds of conditions of the group G insure that such orbits are t- designs for certain t. For example, the irreducibility of certain representations of G is an important factor. Various related materials will also be surveyed. Somewhat older work in this direction are due to Sobolev [171], Goethals–Seidel [97,98], Bannai [22–24] and many others. Newer results include the work of: Sidelnikov [168,169], Nebe–Rains–Sloane [138], de la Harpe–Pache [84], and some others. Another natural way to get examples of spherical t-designs is to take a shell (layer, i.e. the set of n points in the lattice with a fixed distance from the origin) of a lattice in R . For example, any shell of 8 7 8 the E8-lattice (Korkine–Zorotaleff lattice) in R is a spherical 7-design on S ⊂ R , and any shell of 24 23 24 the Leech lattice in R is a spherical 11-design on S ⊂ R . Many other examples are also obtained from other lattices. In Section 3.2, we first review the work of Venkov which says that any shell of an 8m extremal even unimodular lattices in R is a spherical 11-(respectively, 7-, 3-)design if m is congruent to 0 (respectively, 1, 2) modulo 3. (The E8-lattice and the Leech lattice are extremal even unimodular lattices.
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