19
ITB J.
Sci.
Vol.
43 A.
No.
1, 2011, 19-42
On Tight Euclidean 6-Designs: An Experimental Result Djoko Suprijanto Combinatorial Mathematics Group Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung 40132.
INDONESIA.
Email: djoko@math.
Abstract.
A finite set X Es C with a weight function w : X C C A 0 is called Euclidean t-design in C .
upported by p concentric sphere.
if the following condition holds:
wA Xi A f A x A dA i A x A A w A x A f A x A Si S i A1 xEa X A A of degree at most t .
Here S Es C is a for any polynomial f A x A Ea Pol C sphere of radius ri C 0.
X i A X AO Si , and A i ( x ) is an O A n A -invariant measure on S i such that Si A ri n A1 n A1 , with Si is the surface area of S i and S n A1 a surface area of the unit sphere in C .
Recently.
Bajnok .
constructed tight Euclidean t-designs in the plane A n A 2 A for arbitrary t and p .
In this paper we show that for case t A 6 and p A 2 , tight Euclidean 6-designs constructed by Bajnok is the unique configuration in C , for 2 C n C 8 .
Keywords: association schemes.
distance sets.
Euclidean designs.
spherical designs.
tight designs.
Introduction and Result A combinatorial t A A v, k .
A A design X is one of important objects in It might be viewed .
), in a sense, as an approximation of the discrete sphere S k of all k-subsets by the sub-collection X of S k , where A A Sk :A x Ea Cv : x12 A x22 A AU A xv2 A k , xi Ea A0,1A .
Later.
Delsarte.
Goethals and Seidel .
introduced an analogue concept of designs for .
sphere by defining what they called spherical t-design.
Received April 6th, 2010.
Revised June 16th, 2010.
Accepted for publication June 16th, 2010.
Djoko Suprijanto This new concept might be viewed as an approximation of the unit sphere S n A1 Es C n by the subset X of S n A1 with respect to integral of polynomial functions of degree at most t.
The concept of spherical t-design was generalized by Neumaier and Seidel (.
, see also Delsarte and Seidel .
) by allowing weights and multiple spheres.
their papers.
Neumaier and Seidel and also Delsarte and Seidel conjectured the non-existence of tight Euclidean 2e-designs except the trivial ones.
Conjecture 1.
1 (Delsarte-Neumaier-Seide.
The only tight Euclidean 2edesigns in C n , for e C 2 , are regular simplices.
The first breakthrough on this area was performed by Bannai and Bannai .
Having slightly generalized the previous concept of Euclidean t-designs by dropping the condition of excluding 0 vector, they constructed a tight Euclidean 4-design in C 2 supported by two concentric spheres as a counter-example for the conjecture.
Moreover, they also completely classified tight Euclidean 4designs with constant weight in C n , for n C 2 , supported by two concentric Recently, in a joint work with Bannai and Bannai .
, the author introduced a new concept of strong non-rigidity for Euclidean t-designs.
using this new concept we also disproved Delsarte-Neumaier-SeidelAos conjecture by showing the existence of infinitely many tight Euclidean designs having certain parameters.
Going back to Delsarte.
Goethals and Seidel, regarding the spherical designs they showed that there is no tight spherical 6-designs in any Euclidean space except the one on S 1 .
unit sphere in a plane C 2 ) (.
Theorem 7.
Inspired by the situation in spherical designs, a natural question is what about tight Euclidean 6-designs? Are there any tight Euclidean 6-designs in C n ? In answering this question, we divide them into two cases: the ones with constant weight and the others with non-constant weights.
We observe that if the designs contain 0 vector, then by Lemma 2.
15 given in the next section, e A 62 should be even, which is impossible.
Moreover, if X is a tight Euclidean 6-design with constant weight, then Lemma 2.
Remark 2.
14, and Lemma 2.
15 below imply p = 2 or 3.
The purpose of this paper is to give a partial answer for the question by restricting our observation only to tight Euclidean 6-designs supported by two concentric spheres, sitting on the Euclidean spaces of small dimension n.
Namely, we prove the following main theorem.
On Tight Euclidean 6-Designs: An Experimental Result Theorem 1.
2 (Main theore.
The only tight Euclidean 6-design in C n , for 2 C n C 8 , supported by two concentric spheres is the one in C 2 :
E2jAk E E 2 j A k EE A E , rk sin E A E E :1 C j C 5,1 C k C 2 E .
E 5
E 5
X A Ebkj A E rk cos E
A A
The weight function of this design is w bkj A r15 , for k A 1, 2 .
We remark that the design in the theorem above was constructed by Bajnok .
Hence the theorem says that in the Euclidean space of small dimension.
BajnokAos construction of such designs is a unique configuration.
The paper is organized as follows.
In section 2 we lay the groundwork for our We begin with some basic facts about association schemes.
We also recall some facts about distance sets, spherical designs as well as Euclidean We proved our main theorem in Section 3.
Section 4 summarize the current status of classification of tight Euclidean designs.
We end the paper by giving a conjecture on the .
on-)existence of tight Euclidean 6-designs in C n , for n C 2 , supported by two concentric spheres.
Preliminaries This section contains some basic facts on association schemes, spherical designs, and Euclidean designs.
We begin with association schemes.
Association Schemes See .
for undefined terms in association schemes.
Let X A ( X ,{Ri }0CiCd ) be a symmetric association scheme.
Let P and Q be the A j, i A -entry is pi A j A and qi A j A , pi A 0 A A ki and qi A 0 A A mi for 0 C i C d .
Then the first and second eigenmatrix whose Denote following relation is well-known:
q j Ai A A pi A j A .
An association scheme X = A X .
ARi A0Ci C d A is imprimitive if there exists a nonempty proper subset AU A C A0AA of A0,1,AU, d A for which Ai iEaAU Ri defines an Djoko Suprijanto equivalence relation on the set X .
If X is not imprimitive, the X is called Lemma 2.
ee, e.
, .
Let X = A X .
ARi A0Ci C d A be a symmetric association scheme.
For any i such that 0 C i C d , the set X can be embedded m A1
Es Ae , where mi A rank A Ei A , by in the unit sphere S E X C Cm
A :E
E x A x :A E Ei e x with ex A A 0,AU,0,1,0,AU0 A Ea C X and 1 is in the x-th coordinate.
If X is primitive, then A is injective.
Moreover, x, y A qi A j A A p j Ai A 1 Krein Parameters Since the Bose-Mesner algebra is also closed under the Hadamard product, then we may write Ei A E j A Euq E k A0 for some real numbers qijk , called Krein parameters.
These parameters are uniquely determined by the eigenmatrices P and Q :
Theorem 2.
2 (Krein conditio.
For all i, j , k Ea A0,1,AU, d A we have, qijk A Eu k q Al A q Al A q Al A C 0 mk X l A0 where kl A pl A 0 A and mk A qk A 0 A .
Let X A A X .
ARi A0Ci C d A be a symmetric association scheme.
Let also P and Q be the eigenmatrices of the scheme.
The association scheme X is called P- On Tight Euclidean 6-Designs: An Experimental Result polynomial .
Q-polynomia.
scheme with respect to the ordering R0 .
R1 ,AU Rd .
E0 .
E1 ,AU Ed ] if there exist some polynomials vi A x A .
vi* A x A ] of degree i A 0 C i C d A such that Ai A vi A A1 A .
Ei A vi* A E1 A under the Hadamard produc.
Regarding the Q-polynomial scheme.
Delsarte gives necessary and sufficient conditions for any symmetric association scheme to become a Q-polynomial scheme (.
Theorem 5.
), which together with the Krein condition can be restated as follows.
Theorem 2.
3 A symmetric association scheme is Q-polynomial if and only if the Krein parameters qijk satisfy the following two conditions: q1iiA1 A 0 and q1ki A 0 for k A i A 1 A 0 C i C d A 1A .
In particular, for the case Q-polynomial scheme of class 3, the following corollary is immediate.
Corollary 2.
4 A symmetric association scheme is Q-polynomial of class 3 if and only if the Krein parameters qijk satisfy the conditions: q113 A 0 , q112 A 0 , and q12 A 0 .
Distance Sets.
Spherical and Euclidean Designs 1 Distance Sets A finite subset X Es C n is called antipodal if A x Ea X , for any x Ea X .
For a finite subset X Es C n , we define A A X A A A x A y : x, y Ea X , x C yA.
X is called an s-distance set if A A X A A s .
For A Ea A A X A , we define vA A x A A A y Ea X : x A y A A A .
Any subset X is called distance invariant if vA A x A does not depend on the choice of x Ea X , and depend only on A , for any fixed A Ea A A X A .
The upper bound for the cardinality of any s-distance set X in S n A1 is given by (.
Theorem 4.
Djoko Suprijanto
E n A1A s E E n A 2 A s E
EAE
E n A1 E E n A1 E
X CE
2 Spherical t-designs Here is the exact definition of spherical t-designs as introduced by Delsarte.
Goethals, and Seidel in 1977 .
Definition 2.
5 Let t be a positive integer.
A finite nonempty subset X Es S n A1 is called spherical t-design if the following condition holds:
n A1 E f A x A dA A x A A X Eu f A x A n A1 .
xEa X for any polynomial f A x A Ea C Au x1 , x2 ,AU, xn Ay of degree at most t, where A A x A is the O A n A -invariant measure on S n A1 and S n A1 is the area of the sphere S n A1 .
The maximum value of t for which X is a spherical t-design is called the strength of X .
The lower bound for the cardinality of any spherical t-design X in S n A1 is given by (.
Theorem 5.
11 and Theorem 5.
E n A 1 A Au 2t Ay E E n A 2 A Au t A21 Ay E
EAE
E n A1 E E n A1 E
X CE
An s-distance set .
spherical t-desig.
X is called tight if the bound .
] is attained.
Again, the theorem below was also proved by Delsarte.
Goethal, and Seidel .
Theorem 2.
Theorem 7.
Let X be a spherical t-design as well as an sdistance set in S n A1 .
If t C 2s A 2 , then X carries an s-class Q-polynomial Remark 2.
7 In fact .
Theorem 7.
, neither stated nor proved that the association scheme is Q-polynomial.
The detail proof is given, e.
, in .
Theorem 7.
, .
Theorem 9.
) but Bannai and Bannai never claim the above theorem to be from them, instead they always refer the theorem to Delsarte.
Goethal, and Seidel .
ee, e.
, .
Theorem 3.
, .
On Tight Euclidean 6-Designs: An Experimental Result Now, let us turn to Euclidean designs.
3 Euclidean t-designs Let X be a finite set in C n , n C 2 .
Let Ar1 , r2 ,AU rp A A A x : x Ea X A , where x is a norm of x defined by standard inner product in C n and ri is possibly 0.
For each i, we define Si A x Ea Cn : x A ri , the sphere of radius ri centered at We say that X is supported by the p concentric spheres S1 .
S2 ,AU.
S p .
ri A 0 , then Si A A0A .
Let X i A X AO Si , for 1 C i C p .
Let A A x A be the O A n A - invariant measure on the unit sphere S n A1 Es Cn .
We consider the measure A i A x A on each Si so that Si A ri nA1 S nA1 , with Si is the surface area of Si , namely Si A 2ri n A1A AN A n2 A We associate a positive real valued function w on X , which is called weight of X .
We define w A X i A A Eu xEaX w A x A .
Here if ri A 0 , then we define S1 E f A x A dA i A x A A f A 0 A , for any function f A x A defined on C .
Let S A Ai Si .
Let Au S Ea A0,1A be defined by i A1 E1, 0 Ea S AuS A E E0, 0 Ea S We give some more notation we use.
Let Pol A C n A A C Au x1 , x2 .
AU xn Ay be the A A be the vector space of polynomials in n variables x1 , x2 ,AU xn .
Let Homl C n subspace of Pol A C n A spanned by homogeneous polynomials of degree l.
Let A A be the subspace of Pol A C A consisting of all harmonic Let Harml A C A A Harm A C A AO Homl A C A .
Then we have Poll A C A A EIi A0 Homi A C A .
Let Pol A S A .
Poll A S A .
Homl A S A .
Harm A S A , and Harml C Harml A S A be the sets of corresponding polynomials restricted to the union S of A A AA .
p concentric spheres.
For example Pol A S A A f |S : f Ea Pol Cn With the notation mentioned above, we define a Euclidean t-design as follows.
Djoko Suprijanto Definition 2.
8 Let X be a finite set with a weight function w and let t be a positive integer.
Then A X , w A is called Euclidean t-design in C n if the following condition holds:
wA Xi A i A1 Eu E f A x A dA A x A A Eu w A x A f A x A , xEa X A A of degree at most t.
The maximum value for any polynomial f A x A Ea Pol C n of t for which X is a Euclidean t-design is called the strength of X .
The following theorem gives a condition which is equivalent to the definition of Euclidean t-designs.
Theorem 2.
9 (Neumaier-Seide.
Let X be a finite nonempty subset in C n with weight function w.
Then the following .
are equivalent:
X is a Euclidean t-design.
Eu w A u A u uEa X A A u A A 0 , for any polynomial A Ea Harml A C n A with 1 C l C t and 0 C j C Au t A2l Ay .
Let X be a Euclidean 2e-design in C n .
Then it is known that (.
Theorem .
, .
Theorem 5.
X C dim A Pole A S A A .
Following .
, we define the tightness for the Euclidean designs as given below.
Definition 2.
10 Let X be a Euclidean 2e-design supported by S .
X A dim A Pole A S A A holds we call X tight Euclidean 2e-design on S .
Moreover, if A A AA dim A Pole A S A A A dim Pole Cn holds, then X is called tight Euclidean 2e-design.
The following lemma is crucial in our study of Euclidean designs.
On Tight Euclidean 6-Designs: An Experimental Result Lemma 2.
Lemma 1.
Let X be a tight Euclidean 2e-design on p concentric spheres.
Then the following hold:
The weight function w is constant on each X i , for 1 C i C p .
X i is at most an e-distance set 1 C i C p .
If the weight function w is constant on X \ A0A , then p A Au S C e .
As an application of the above lemma.
Et.
Bannai .
proved the following The theorem gives a certain connection between spherical designs and Euclidean designs.
Theorem 2.
Theorem 1.
Let X be a tight Euclidean 2e-design on p concentric spheres.
If p A Au S C e , then each X i is .
imilar t.
a spherical A 2e A 2 p A 2Au S A 2A -design.
Moreover, if p C EE eA 2Au2 A3 EE , then each X i is a distance invariant set.
Let A X , w A be a finite weighted subset in C n .
Let S1 .
S2 ,AU S p be the p concentric spheres supporting X and let S A Ai ipA1 Si .
A A
For any A ,A Ea Harm C n , we define the following inner-product
A ,A A
S n A1 E A A x AA A x A dA A x A .
n A1 Then we have the following .
, .
Lemma 2.
13 The following three statements hold:
A A is a positive definite inner-product space under A.
A and has the orthogonal decomposition Harm A C A AA Harm A C A .
Harm C n A A .
Pole C n A EI 0Ci A 2 j C e Cu i A0 nAe A A with dim A Pol A C A A A EE e EE .
Harmi C n E Djoko Suprijanto Pol A S A A x 2 j : 0 C j C min A p A 1.
Au e AyA EI Harm 1C i C e E e A1 E 0 C j C minA p A Au A1.
EE 2 EEA
If p C EE 2 EE then e AAu S dim A Pole A S A A A Au S A 2A p AAu S A A1
E n A e A i A 1E e E n A e A i A 1E E n A e E
E AEu E
EAE
E n A 1 E i A0 E n A 1 E E e E Eu E i A0 where e is a non-negative integer.
If p C EE 2 EE A 1 , then e AAu S En A eE E e E dim A Pole A S A A A E where e is a non-negative integer.
Remark 2.
14 Definition 2.
10 and Lemma 2.
13 show that a tight Euclidean 2edesign is the same as a tight Euclidean 2e-design on p concentric spheres with p C EE 2 EE A 1 .
e AAu S The next lemma is stated in Bannai and Bannai .
Lemma 2.
Proposition 1.
Let X Ea Cn be a tight Euclidean 2e-design.
If 0 Ea X , then e is even, p A 2e A 1 , and X \ A0A is a tight Euclidean 2e-design on 2e concentric spheres.
A AA and A ,AU.
A Let h1 A dim Harml C n l ,1 l , hl be an orthonormal basis of A A with respect to the inner-product defined above.
Then, by Lemma Harml C n E 2j E e E EE E x : 0 C j C min E p A 1.
E E EE Ai
E 2 E EE
E 2j E e A l E EE E x Al ,i A x A :1 C l C e,1 C i C hl , 0 C j C min E p A Au S A 1.
E 2 EE EE
gives a basis of Pole A S A .
On Tight Euclidean 6-Designs: An Experimental Result
Now, we are going to construct a more convenient basis of Pole A S A for our Let
A A
AN Cn Pole A S A A x : 0 C j C p A 1A .
Let AN A X A A Ag | : g Ea A C AA .
Then A x : 0 C j C p A 1A is a basis of AN A X A .
We define an inner-product A.
A l on AN A X A by f , g l A Eu wA xA x f A x A g A x A , for 1 C l C e .
xEaX We apply the Gram-Schmidt method to the basis A x : 0 C j C p A 1A to construct an orthonormal basis Ag A x A , g A x A ,AU, g A x AA l ,0 l , p A1 l ,1 of AN A X A with respect to the inner-product A.
A l .
We can construct them so that for any l the following holds:
gl , j A x A 1, x ,AU, x deg A gl , j A A 2 j for 0 C j C p A 1 .
As an example, for p A 2 , we can express g l , j in the following way:
gl ,0 A x A C , gl ,1 A with al A Eu xEaX w A x A x al al A 2 A a l A1 E 2 al A1 E E x A E Now we are ready to give a new basis for Pole A S A .
Let us consider the following sets:
E e E EE AO0 A E g 0, j : 0 C j C min E p A 1.
E E EE .
E 2 E EE
E e A l EE E ,1 C i C hl E , for 1 C l C e.
E 2 EE
AO1 A E gl , jAl ,i : 0 C j C min E p A Au S A 1.
Then AO A AielA0 AOl is a basis of Pole A S A .
Djoko Suprijanto We close this section by the following lemma .
for the proof.
Lemma 2.
If A X , w A is a tight Euclidean 2e-design on S , then the following .
The weight function of X satisfies Eu 1C l C e .
A l, j A u A Ql A1A A A A min p A1.
EE 2e EE g 0,2 j A u A A j A0 w A uA , for all u Ea X .
0 C j C min p A Au S A1.
EE e2Al EE .
For any distinct points u, v Ea X , we have Eu 1C l C e .
A E u, v E minA p A1.
EE EEA u v gl , j A u A gl , j A v A Ql E E A Eu g 0, j A u A g 0, j A v A A 0.
j A0 E u v E 0 C j C min p A Au S A1.
EE e2Al EE .
Here u, v is the standard inner-product in Euclidean space C n and Ql AA A is the Gegenbauer polynomial of degree l.
Proof of Main Theorem We prove Theorem 1.
2 by contradiction.
The general idea is to show that the assumption of the existence of tight Euclidean 6-design of certain given parameters does not carry a Q-polynomial scheme of class 3.
Hence we get a The detail follows.
Let X A X 1 Ai X 2 be a tight Euclidean 6-design in C n , for 2 C n C 8 .
Theorem 2.
12, we know that X i is .
imilar t.
a spherical 4-design.
We also know, by Lemma 2.
11, that a tight spherical 4-design X i is also a 2-distance set, while the non-tight one is also a 3-distance set.
Therefore.
Theorem 2.
guarantees that the non-tight spherical 4-design X i should carry a 3-class Qpolynomial scheme.
On Tight Euclidean 6-Designs: An Experimental Result On the other hand, van Dam .
gives all feasible character tables of the 3class symmetric association schemes on points up to 100.
By the help of Lemma 2.
1, we know that the symmetric association scheme on X i can be embedded into a unit sphere, which also give us the feasible 3-inner product set.
Hence, keeping in mind that any distance set in the unit sphere has one-to-one correspondence with an inner product set, we can investigate whether the finite set X i carries a 3-class Q-polynomial scheme, by comparing the numerical 3inner product sets .
btained from Lemma 2.
with the feasible ones .
iven by van DamAos character table.
Let us consider first some special cases.
Some Special Cases We begin with some elementary facts.
Let N .
Ni denotes the cardinality of X .
X i , for i A 1, 2 , respectively.
Suppose N1 C N 2 .
Then the lower bound Lb and upper bound U b of N1 is given by [Lb,U.
We notice that there are three kind 3-class symmetric association schemes of AydegenerateAy case .
o follow van Dam .
Dam-.
They are: .
the schemes generated by n disjoint union of strongly regular graphs SRG A v, k .
A .
A A .
the schemes generated by SRG A v, k .
A .
A A EE J n , and .
the rectangular scheme R A m, n A .
The character tables of each case are .
, p.
v A1A k A n A 1A v E v A1A k A1 A r A1 A s E 1 nk E , .
E 1 nr
E1 0
E 1 ns
n A1 n Av A1A k A E
n A1
n A1
n A A1 A r A E
n A A1 A s A E
Djoko Suprijanto
A m A 1AA n A 1A
n A 1 m A 1E
A1 E
1A m
m A 1E
1A n n A1
A1 E
By direct calculation it is easy to see that some values of Krein parameters are .
q113 A 0, q112 A 0 , .
q112 A 0, q123 A 0 .
q113 A A n A 1AA m A 2 A , q112 A A n A 2 AA m A 1A , q123 A n A 1 .
Hence for the first two cases, the schemes are not Q-polynomial, while for the last case, the scheme is Q-polynomial if and only if n A 2 or m A 2 , that is if cardinality of the finite set carrying the scheme is even.
Furthermore, the only feasible 3-inner product set given by this scheme is AC mA1 1 .
A1A , for m C 2 .
will include this feasible set in our observation below.
Remark 3.
1 We checked the above degenerate cases for all possible ordering of the primitive idempotent basis E0.
E1.
E2.
E3.
Here we consider the character table PC obtained from P by applying a permutation to the set of its rows but the first .
here are six possibilitie.
and we have: two of them give q113 A 0 , q112 A 0 and the others q113 C 0 .
or type .
and two of them give q113 A A n A 1AA m A 2 A , q112 A A n A 2 AA m A 1A , q123 A n A 1 and the others q113 A 0 , q112 A 0 .
or type .
Next, let us consider the following cases.
Case ( n.
N1 ) = .
, .
, .
, .
, .
, .
, .
, .
, .
, .
These are the cases where X 1 is a tight spherical 4-design.
By BannaiDamarellAos criteria .
, c.
Remark .
), it is known that if a tight spherical 4-design X 1 Es S n A1 exists, then n A A 2m A 1A A 3 holds, for some integer m.
Since there is no m satisfying A 2m A 1A A 6 , .
7, 8, 10, and .
, then there does not exist tight spherical 4-design on S2, .
S3.
S4.
S6, and S.
On Tight Euclidean 6-Designs: An Experimental Result Hence there is no tight Euclidean 6-design in C 3 .
C 4 .
C 5 .
C 7 , and C 8 ] supported by two concentric spheres with such parameters.
Remark 3.
2 Results for the cases .
= .
, .
, .
, .
, and .
, .
are also a direct consequence of a work of Boyvalenkov and Nikova .
, where they improved the lower bound of tight spherical 4-designs on S2.
S3, and S4, from 9, 14, and 20 to 10, 15, and 21, respectively.
Now, let us turn to the general treatment.
We begin with the constant weight Case 1: Constant Weight For n = 2, then N i = 5, i.
X i are regular pentagon .
-go.
We may assume that x A 1 , for x Ea X 1 and w A x A A 1 , for x Ea X .
Proposition 2.
, c.
Theorem 2.
From equation .
we have that the weight function for any point sitting on the second sphere is w A x A A 1 .
Our assumption implies x A 1 , for any x Ea X 2 , which is impossible.
Hence tight Euclidean 6-design in C 2 supported by two concentric spheres does not exist.
Next, let us consider the other cases.
We mention first the procedure we use in our observation.
Let X A X 1 Ai X 2 be a tight Euclidean 6-design with constant weight in C n A 2 C n C 8A , let X 1 A N1 , and that x A 1 , for x Ea X 1 , and w A x A A 1 , for x Ea X .
X 2 A N 2 .
We may assume Step 1: Given n.
N1, and N2.
Step 2: If there exist 3-class symmetric association schemes on N1 and N2 points simultaneously, then further check if Xi carries a Q-polynomial scheme .
y Corollary 2.
If such a polynomial scheme exists then Ae Ae Calculate the radius of the second sphere x A R , for x Ea X 2 by equation .
in Lemma 2.
If R C 1 , then substitute R to equation .
in Lemma 2.
16 to get the 3-inner product set AA1 .
A 2 .
A 3 A .
Step 3: Compare AA1 .
A 2 .
A 3 A with the entries of corresponding character table A A A .
A A .
A A A , for 2 C i C 4 .
(See the Appendix for example of calculation results.
Djoko Suprijanto Case 2: Non-constant Weight For n=2, then Ni = 5, i.
Xi are regular 5-gons.
Bajnok .
has constructed an example of such a design:
E 2 j A k E E 2 j A k EE A E , rk E A E E :1 C j C 5,1 C k C 2 E .
E 5
E E 5
X A Ebkj A E rk cos E The weight function is given by w A bkj A A r1 , for k =1,2.
Hence there exists a tight Euclidean 6-design in C 2 supported by two concentric spheres.
Moreover, it is easy to show that in fact it is the only tight Euclidean 6-design in C 2 .
The argument is as follows.
Let X C A X 1C Ai X 2C be a tight Euclidean 6-design in C 2 .
Then X 1C and X 2C should be tight spherical 4-designs, namely regular pentagons.
Hence, up to the action of orthogonal group O A 2 A , we can write X 1C A X 1 and X 2C A A A X 2 A , for some rotation A Ea O A 2 A .
Using the Neumaier-Seidel's Theorem above it can be shown that A A I , the identity, namely X 2C A X 2 .
Next, let us consider the other cases.
We mention first the procedure we use in our observation.
Let X A X 1 Ai X 2 be a tight Euclidean 6-design with nonconstant weight in Cn A 3 C n C 8A , let X1 A N1 and X 2 A N 2 .
Again, we may assume that x A 1 , for x Ea X 1 and w( .
A 1 , for x Ea X 1 .
Step 1: Given n.
N1, and N2.
Step 2: If there exist 3-class symmetric association schemes on N1 and N2 points simultaneously, then further check if Xi carries a Q-polynomial scheme .
y Corollary 2.
If such a polynomial scheme exists, then Ae Calculate the weight function w by equation .
in Lemma 2.
Ae Substitute the weight function w to equation .
in Lemma 2.
16 to get the 3-inner product set {A1 .
A 2 .
A 3 } .
(Here A i .
C i C .
are functions of the second radius R).
Step 3: Compare {A1 .
A 2 .
A 3 } with the entries of corresponding character table A p1 .
) A , p2k2.
) , p3k3.
) , for 2 C i C 4 .
(Here we consider at most 3 C 9 A 27 equations for one corresponding character tabl.
On Tight Euclidean 6-Designs: An Experimental Result Ae If, say.
A1 A 1k1 implies the positive real value of radius R0 (C .
, p .
) then substitute R0 to A 2 and A 3 .
Ae Check whether {A1 ( R0 ).
A 2 ( R0 ).
A 3 ( R0 )} A A p1 .
) A , p2k2.
) , p3k3.
) , for some i Ea.
,3, .
(See the Appendix for example of calculation results.
In summary, our assumption of the existence of tight Euclidean 6-designs with certain given parameters implies:
the non-existence of tight spherical 4-design in a Euclidean space of given dimension .
y Bannai-Damarell's criteri.
, or the non-existence of 3-class symmetric association scheme on N1 or N2 points .
y checking on van Dam's tabl.
, or the non-existence of 3-class Q-polynomial scheme on N1 or N2 points .
y Krein conditio.
, or the 3-class symmetric association scheme on N1 points does not provide the 3-inner product set .
y looking at van Dam's tabl.
, or the numerical 3-inner product set does not appear, or the numerical 3-inner product set does not coincide with the 3-inner product set provided by the character table of Q-polynomial scheme on N1 points.
All of these lead to a contradiction.
Hence, we have proved the main theorem.
Concluding Remarks As we have seen, there is no tight Euclidean 6-design supported by two concentric spheres in Euclidean spaces of small dimensions, namely in Ae n A 2 C n C 8A , for almost all feasible parameters.
The only exception is tight Euclidean 6-designs of non-constant weight in Ae 2 .
The designs was constructed by Bajnok (.
Theorem .
Our effort here might be regarded as a continuation of the work on giving classifications of tight Euclidean designs in Ae n .
Hence, the current status of this work is as follows:
tight Euclidean 2-designs in Ae n , for n C 2 , supported by all feasible concentric spheres .
tight Euclidean 3-designs in Ae n , for n C 2 , supported by all feasible concentric spheres .
Djoko Suprijanto tight Euclidean 4-designs in Ae n , for n C 2 , with constant weight, supported by two concentric spheres .
tight Gaussian 4-designs, i.
a special kind of tight Euclidean designs, in Ae n , for n C 2 supported by two concentric spheres .
tight Euclidean 5-designs in Ae n , for n C 2 supported by two concentric spheres .
tight Euclidean 6-designs in Ae n , for 2 C n C 8 , supported by two concentric Recently Et.
Bannai .
has constructed some examples of .
on-constant weigh.
tight Euclidean 4-designs supported by two concentric spheres, but the problem of classification of such designs still far of being solved.
Besides.
Bajnok .
, .
has also constructed some other sporadic examples of Euclidean Very recently.
Bannai.
Bannai.
Hirao, and Sawa .
announced that they have classified tight Euclidean 7-designs supported by two concentric spheres completely, but the complete proof is now in preparation by Bannai and Bannai.
Regarding the tight Euclidean 6-designs on two concentric spheres, all existing phenomena mentioned above lead us to a conjecture on the complete classification of such designs.
We end this paper by the conjecture:
Conjecture 4.
1 The only tight Euclidean 6-designs in Ae n , for n C 2 supported by two concentric spheres are:
E2jAk E E 2 j A k EE A E , rk sin E A E E :1 C j C 5,1 C k C 2 E .
E 5
E 5
X A Ebkj A E rk cos E The weight function of these designs are w.
kj ) A r15 for k A 1, 2 .
Remark 4.
2 One of the referee.
Eiichi Bannai, sent me the paper .
, where they give a new example of tight Euclidean 6-design in Ae 22 .
It becomes the first example of tight Euclidean 6-design in n-dimensional Euclidean space, for 3 C n C 438 , supported by two concentric spheres.
It also gives a counter-example of the above Conjecture.
However.
Bannai et.
seem to believe that it is the only existing tight Euclidean 6-design in Ae n , for n C 3 .
Hence, we modify the conjecture to the following question.
On Tight Euclidean 6-Designs: An Experimental Result Question 4.
2 Find another example of tight Euclidean 6-design in Ae n , for n C 3 , on two concentric spheres, or prove that there exist no such tight Euclidean 6-design except the ones given in the above conjecture and in .
Acknowledgement The author would like to thank Prof.
Eiichi Bannai and Prof.
Etsuko Bannai for introducing the problem and many valuable suggestions.
The author would also like to thank the referees for careful reading of the manuscript and many helpful References