On spherical designs of some harmonic indices

A finite subset Y on the unit sphere Sn−1 ⊆ Rn is called a spherical design of harmonic index t, if the following condition is satisfied: ∑ x∈Y f(x) = 0 for all real homogeneous harmonic polynomials f(x1, . . . , xn) of degree t. Also, for a subset T of N = {1, 2, · · · }, a finite subset Y ⊆ Sn−1 is called a spherical design of harmonic index T, if ∑ x∈Y f(x) = 0 is satisfied for all real homogeneous harmonic polynomials f(x1, . . . , xn) of degree k with k ∈ T . In the present paper we first study Fisher type lower bounds for the sizes of spherical designs of harmonic index t (or for harmonic index T ). We also study ‘tight’ spherical designs of harmonic index t or index T . Here ‘tight’ means that the size of Y attains the lower bound for this Fisher type inequality. The classification problem of tight spherical designs of harmonic index t was started by Bannai-OkudaTagami (2015), and the case t = 4 was completed by Okuda-Yu (2016). In this paper we show the classification (non-existence) of tight spherical designs of harmonic index 6 and 8, as well as the asymptotic non-existence of tight spherical designs of harmonic index 2e for general e > 3. We also study the existence problem for tight spherical designs of harmonic index T for some T , in particular, including index T = {8, 4}.


Introduction and spherical designs of harmonic index t (or T )
Throughout this paper Y is assumed to be a finite non-empty set, and we denote the set of positive (resp.non-negative) integers by N (resp.N 0 ). Let be the unit sphere in the Euclidean space R n .Delsarte-Goethals-Seidel [6, Definition 5.1] (1977) gave the following definition of spherical designs.
Definition 1 (Spherical t-designs).Let t ∈ N 0 .A subset Y ⊆ S n−1 is called a spherical t-design on S n−1 , if 1 for any real polynomial f (x 1 , . . ., x n ) of degree at most t, where |S n−1 | denotes the volume (or the surface area) of the sphere S n−1 , and the integral is the surface integral on S n−1 .
The condition ( 1) is known to be equivalent to the condition: where Harm n k is the space of real homogeneous harmonic polynomials of degree k in n indeterminates.
In connection with the latter equivalent defining condition for spherical t-designs, we define a weaker concept which we call designs of harmonic index t as follows.
Definition 2 (Spherical designs of harmonic index t).A subset Y ⊆ S n−1 is called a spherical design of harmonic index t on S n−1 , if y∈Y f (y) = 0 ( * ) for all real homogeneous harmonic polynomial f (x 1 , . . ., x n ) of degree exactly t.
More generally we have the following definition.
Definition 3 (Spherical designs of harmonic index T ).Let T be a subset of N. A subset Y ⊆ S n−1 is called a spherical design of harmonic index T on S n−1 if y∈Y f (y) = 0 for all f (x 1 , . . ., x n ) ∈ Harm n k with k ∈ T.
A spherical design of harmonic index T with T = {1, 2, . . ., t} corresponds to a usual spherical t-design, and the case T = {t} corresponds to a spherical design of harmonic index t.
We should remark that the concept of spherical designs of harmonic index t (or T ) was already introduced by Delsarte-Seidel [7, Definition 4.1] (1989).However the study of this topic is started in Bannai-Okuda-Tagami [2] (2015).
The purpose of this paper is to study spherical designs of harmonic index t as well as harmonic index T for some T , and to convince the reader that these are interesting mathematical objects.Our main concerns are Fisher type lower bounds for spherical designs of harmonic index t and T , as well as the classification problems of so-called 'tight' designs.Here 'tight' means those that attain the lower bound in a Fisher type inequality.In Section 2 we provide a linear programming bound for spherical designs of harmonic index T .We also formulate Fisher type inequalities and tight spherical designs of harmonic index t or T .In Section 3, we discuss our philosophy how to find our test functions.In the subsequent sections we will study some specific problems.In Section 4 the complete non-existence results for tight spherical designs of harmonic index 6 and 8 are proved.Note that the case of t = 4 was already settled by Okuda-Yu [17] in a beautiful way by applying the SDP (semidefinite programming) to the existence problem of equiangular lines.Also note that our proofs for t = 6 and t = 8 are obtained in an elementary level without recourse to such deeper consideration as SDP.In Section 5 we show the asymptotic non-existence of harmonic index 2e case for general e 3. Then we turn our attention to the case of T = {t 1 , t 2 }.The central model is the case T = {8, 4} in Section 6.In Section 7 we study the cases T = {8, 2}, {8, 6}, {6, 2}, {6, 4}, as well as {10, 6, 2}, and {12, 8, 4}.We conclude the paper in Section 8 with some remarks.
The techniques which we used in the present paper are: (i) the linear programming method by Delsarte, (ii) the detailed information on the locations of the zeros as well as the local minimum values of Gegenbauer polynomials, (iii) the generalization by Nozaki of the Larman-Rogers-Seidel theorem on 2-distance sets to s-distance sets, (iv) the theory of elliptic diophantine equations, and (v) the semidefinite programming method of eliminating some 2-angular line systems for small dimensions.
2 Linear programming method for spherical designs of harmonic index T In this section we consider a linear programming bound for spherical designs of harmonic index T .We also introduce some terminology and notation which will be used in the subsequent sections.Let Q n,k (x) be the Gegenbauer polynomial of degree k in one variable x as introduced in [6, Definition 2.1].Recall how the polynomials Q n,k (x) are normalized [6, Theorem 2.4, Theorem 3.2]: The Gegenbauer polynomials Q n,k (x) are orthogonal polynomials on the closed interval [−1, 1] with respect to the weight function the electronic journal of combinatorics 24(2) (2017), #P2.14 where a n,k is some (normalization) constant depending on n and k, and δ k, is the Kronecker delta.From this orthogonality it is well established that to any real polynomial F (x) of degree r we can associate its Gegenbauer expansion where the Gegenbauer coefficients f k can be computed as follows: We denote by x • y the standard inner product of x and y in R n .For a subset Harm n k , then the well-known addition formula says that, for every x, y ∈ S n−1 , From the addition formula we have, for any Thus we obtain (see Definition 3 for (M2)) the following two simple observations: We introduce our main identity (see (3) below).(See [6] for the original discussion about so-called 'linear programming bounds' for spherical designs.)Suppose F (x) is a non-constant real polynomial of degree r which is of the form (2). For any Y ⊆ S n−1 , if we calculate x,y∈Y F (x • y) in two different ways, we find Now suppose Y ⊆ S n−1 is a spherical design of harmonic index T .We are interested in finding a good lower bound for then we obtain (recall (M1) and (M2)) that where the first and second inequalities are due to (LP1) and (LP2), respectively.Moreover we necessarily have f 0 > 0, since by (LP1) the integrand of the following integral 2 dx > 0. By (4) we have Therefore we want to find the following quantity: Definition 4. We call a polynomial F (x) a test function provided that both (LP1) and (LP2) hold.
Note that the multiplication of F (x) by any positive real number does not affect the consistency of (LP1) and (LP2), and does not change the value F (1)  f 0 .Thus we might adopt an equivalence relation on the set of test functions defined using the multiplication by a positive real number.
It is also harmless to restrict the range of the above supremum by only considering F (x) with (any) fixed constant term f 0 = c.If we introduce new variables g k := f k f 0 (k 0), then the computation of S in (6) becomes the following linear programming of infinite variables (g 1 , g 2 , . ..). (Notice that (6) itself is not a linear programming of the variables (f 0 , f 1 , . ..) because f 0 appears in the denominator of F (1)  f 0 so that F (1) f 0 is not actually linear in f 0 .) subject to (7) (i) All but finite number of g 1 , g 2 , . . .∈ R are zero; (iii) All g 1 , g 2 , . . .satisfy (LP2).
In this paper we consider the case where T consists of positive even integers Given n and T , what is the best choice of a test function F (x) for ( 6)?The answer is not easy: First of all, what is the definition of "goodness" for F (x)? Of course, the most obvious one is that a "better" F (x) provides a larger F (1) f 0 .However, this approach seems hopeless because the exact determination of S in (6) is too difficult.(It is not even apparent that there exists an optimal test function which attains the supremum in (6).)This means that there can be many approaches for choosing a test function with different definitions of "goodness".
In this section we show our way for choosing a test function, and explain its philosophy.We should emphasize here that our goal is not to maximize F (1)  f 0 .In this paper, for our purpose, we only deal with a test function F (x) of the following form: One reason why we only concern test functions of the form ( 8) is that it is the easiest one we can handle.More precisely, Eq. ( 8) is the easiest form for a polynomial F (x) to satisfy (LP2).Another reason is that if F (x) has the form (8), then we are able to guess when the equality in ( 5) is attained: f 0 .Since F (x) has the form (8) and M k (Y ) = 0 for all k ∈ T (Definition 3), the term k∈T f k M k (Y ) vanishes.We see from (LP1) that the equation f 0 if and only if Y is a distance set where the distances between two distinct points in Y should occur in the zero set of F (x). Now it is the time when we should explain our definition of "goodness" for F (x).Our philosophy for choosing F (x) (of the form (8)) is that we want a distance set Y with large |I(Y )| whenever |Y | attains F (1) f 0 , i.e., we hope an interesting (=large) "tight" object.Therefore we require that a test function F (x) has exactly non-negative zeros (9) because we think that is the naturally expected largest number of non-negative zeros for F (x) of the form (8). (Of course, it is still a difficult problem to determine the precise maximum number of minima for F (x) in (8).) We only deal with the case where there are only finitely many test functions satisfying our conditions (9), although the choices of test functions are possibly infinite for some index sets.Suppose that there are only m test functions F 1 (x), F 2 (x), . . ., F m (x) (up to equivalence) which satisfy our condition (9) for a given index set T .For each test function , where f (i) 0 is the constant term in the Gegenbauer expansion of F i (x).Definition 6.With the above conditions and notation, we define b n,T := max If the size of a spherical design Y of harmonic index T attains the lower bound b n,T , then Y is said to be tight.
If there exists a unique test function F (x) (up to equivalence), then Definition 6 implies that b n,T = F (1) f 0 .Remark 7. We should note that it is a very delicate problem to define "tight" designs for general T in a rigorous way.In the above argument, we required that our test function should have as many zeros as possible so that b n,T could be as large as possible.The reason to get as many zeros is that the "tight" set can be an s-distance set for larger s, otherwise the size cannot be large.
In the first section we defined the concept of spherical design of harmonic index t or more general T .This notion was already essentially defined in the literature, as "a spherical design which admits indices T ". (See Delsarte-Seidel [7], say.)On the other hand, the terminology of spherical design of harmonic index t is already defined as "a spherical design for which equality in Definition 2 ( * ) holds for any homogeneous polynomials of degree t", say in [7,14], etc.In order to avoid the confusion with these terminologies, we use the term 'spherical designs of harmonic index t (or T )'.It seems that no systematic study of spherical designs of harmonic index T has been made, before Bannai-Okuda-Tagami [2].They used the test function Q n,t (x) for T = {t}, and obtained the following theorem.(Note that this theorem is a special case of Proposition 5.) Let Y be a design of harmonic index t on S n−1 .Then the following inequality holds: where c n,t = − min Remark 9.For T = {t}, the choice of test function is unique (up to equivalence).The reason is that our test function is of form (8), i.e.In [2,17] they discussed the cases when t = 2 and 4. In the following section we discuss the existence problem of tight spherical designs of harmonic index t for t = 6 and 8.As is seen from Remark 9, if we consider T = {2e}, then there is only one positive zero of F (x), equivalently, only one positive minima of Q n,2e (x) exists.
If we take T = {8, 4}, say, there are only at most two positive zeros.With property (9) and some calculation, it is shown that our test function F (x) is determined uniquely.For T = {t 1 , t 2 , . . ., t } with t i even, it seems that we can expect that there are at most non-negative zeros although we do not know the exact answer for general T .Among the possible test functions with this property, finding the best one, namely with the largest b n,T , is not easy for general T .For example, in the discussion below, in the case of T = {12, 8, 4} the candidates of test function F (x) are not necessarily unique.Also, in some cases, no good test function exists.Thus, here we are compromising in taking the test function which seems to be the most natural one.We cannot eliminate the possibility of the existence of a better test function, in general case.For specific T , which are discussed in subsequent sections, we believe the choices of our test functions are natural and meaningful, although we do not show that rigorously at this stage.This situation may look to be an embarrassing situation, but this is even true for the definition of tight spherical t-designs, originally defined by Delsarte-Goethals-Seidel [6].In the case of ordinary tight spherical 2e-designs, the specifically chosen test function 2 satisfies the requirement that there are possible maximum e positive zeros.Moreover, it is expected to give the maximum b n,T (with T = {1, 2, . . ., 2e}), so it satisfies our criterion of 'good' test function.On the other hand, there is no easy proof that it is the best test function.Still, the concept of tight spherical t-design in this particular choice of the test function was very meaningful.Our definition of tight spherical designs of harmonic index T has the same feature, and we have to compromise that the definitions of tight designs are not completely rigorously defined in the general case of T .(To show the test function is a best one is not easy and in many cases it is still undecided.However, such a problem is an unavoidable fact in this kind of theories.)4 The non-existence of tight spherical designs of harmonic index 6 and 8 In this section we will prove the non-existence of tight spherical designs of harmonic index t = 6 and t = 8.The lower bound in Theorem 8 is obtained by the inequality (5) using following test function F (x).
the electronic journal of combinatorics 24(2) (2017), #P2.14 Throughout this section, F (x) = c n,2e + Q n,2e (x) and we say Y is a tight spherical design of harmonic index 2e if |Y | attains the lower bound b n,2e in (10).
It is known that F (x) have exactly two roots {α, −α}, which means that Y is bounded above by the cardinality of spherical 2-distance set.For any spherical 2-distance set X ⊆ S n−1 with I(X) = {α, β} and α + β 0, Musin [15] proved

The non-existence of tight spherical designs of harmonic index 6
In this subsection Y denotes a tight spherical design of harmonic index 6.The Gegenbauer polynomial Q n,6 (x) (with our normalization Q n,6 (1) = dim Harm n 6 ) is given by  10) can be obtained as well.The following are our results: In this subsection Y ⊆ S n−1 denotes a tight spherical design of harmonic index 8.The Gegenbauer polynomial Q n,8 (x) is As in the preceding subsection we can obtain α, c n,8 , and also It can be checked that if n 20 and, if n 19, the only integral value is b 2,8 = 2.By a similar argument as in Remark 11 one trivial example exists when n = 2.
Remark 13.We do not give the formulas of α, b n,8 and c n,8 explicitly, since they are extremely complicated.Here, b n,8 > n(n+1) 2 is checked from the formula of b n,8 rather than from the asymptotic form (12). 5 The asymptotic non-existence of tight spherical designs of harmonic index 2e for general e In this section we consider the existence of tight spherical designs of harmonic index 2e for e 5, since the cases e = 2, 3, 4 were already treated.Our main result in this section is the following theorem.  .On the other hand, if n is sufficiently large, then Theorem 14 implies Proof of Theorem 14. Szegő [19, p. 107] gives the asymptotic property of Gengebauer polynomial C λ t (x): lim where H t (x) is the Hermite polynomial of degree t.
−2 x) for simplicity.Then we have Take the derivative with respect to x on both sides of (13).For fixed e, since P n,e (x) uniformly converges to 2 e (2e−1)!x 2e−1 as n tends to be infinity, we get the following result: where the last equality is due to the property H t (x) = 2tH t−1 (x).Let x 1 be the largest zero of H 2e−1 (x).Then Thus the following equality can be obtained. .
the electronic journal of combinatorics 24(2) (2017), #P2.14 Remark 16.In Theorem 14 we did not give explicit evaluation of B 2e , but it is possible to give it, since the locations of the zeros of Hermite polynomials and the (local) minimum values of H 2e (x) are well studied.Also, if we want to evaluate b n,2e explicitly from below, rather than evaluating B 2e , it is also possible, although we will not discuss it in this paper.For this purpose, the following papers [4], [8], [9], [11] may be useful to do that.It seems that there are many literature on this.
6 Tight spherical designs of harmonic index {8, 4} In what follows, we assume T = {t 1 , t 2 , . . ., t } with ) even.And we investigate the case where our test function is of the form . Then as we mentioned in Section 3, it is better that F (x) have as many zeros in [−1 , 1].For this purpose we are interested in the case where L(x) attains minimum value at non-negative points α 1 , . . ., α ∈ [0, 1].Delsarte-Goethals-Seidel (1977) gave an upper bound for a spherical s-distance set Using the above theorem, we obtain the following lemma which gives the relation between the size of tight spherical design of harmonic index T and the cardinality of spherical s-distance set.
Lemma 18.If Y is a tight spherical design of harmonic index T , and takes minimum value at non-negative points, then Proof.Let c n,T,L = − min L(x) and F (x) = L(x) + c n,T,L .If e is even (resp.odd), then F (x) has 2 (resp. 2 − 1) zeros.By the assumption, F (x) has non-negative roots.
. Similarly, we can get the conclusion if e is odd.
Consider the case of T = {t 1 , t 2 } with t 2 = even and t 1 = 2e > t 2 .In this case we use the test function In this case we are interested in the cases when L(x) has the minimum value −c n,T at exactly two non-negative points α, β (with α > β) and choose f 0 = c n,T .For this purpose we take and we want to determine f t 2 such that for some α, β and c n,T .Suppose t 1 = 8 and t 2 = 4. Then the problem is to find f 4 such that The Gegenbauer polynomial Q n,4 (x) is By comparing the coefficients in ( 14), we obtain the following equations: Therefore, Hence α 2 and β 2 are the solutions of the following quadratic equation in the variable u: Finally, we obtain the electronic journal of combinatorics 24(2) (2017), #P2.14 where b n,T is the lower bound in (5).
In the following we shall prove the non-existence of tight spherical designs of harmonic index {8, 4} stated in Theorem 23 below.If Y is a tight spherical design of harmonic index {8, 4}, then I(Y ) ⊆ {±α, ±β}.We define must be an integer with |k i | U (N ).
2 , then the following two numbers are integers: ( 3 + 1, then the following two numbers are integers: .
Theorem 23.There exists no tight spherical design of harmonic index {8, 4} on S n−1 for all n.
We follow the argument in [10, p. 79] to obtain rigorous bound of our semidefinite programming problems.We independently solve the dual problem to avoid numerical issue in CVX and guarantee that our bounds are justified.We checked that the error did not affect our results.i.e. our computational SDP bounds plus the error still strictly less than linear programming bound of tight spherical designs of harmonic index T .Then, such tight designs do not exist.
In our problem, the solved value and the error are 1 + 1 i=1 x i i is very small and bounded above by 0.00011592.)Hence our computational SDP bound plus error is still strictly less than the LP bound (=65) for spherical design of harmonic index {8, 4}.We can conclude there exists no tight spherical design of harmonic index {8, 4}.
Theorem 25.Let Y be a spherical 4-distance set with inner products a, b, c and d.Let p be a positive integer.The cardinality |Y | is bounded above by the solution of the following semidefinite programming problem: subject to the electronic journal of combinatorics 24(2) (2017), #P2.14 where ) matrices with entries related to Gegenbauer polynomial and explicit definition can be found in [3] equation (2)(3).
In this theorem the variables x 1 , . . ., x 4 refer to the number of ordered pairs of vectors in Y with inner product a, b, c and d respectively; for instance ( The variables x(u, v, t) refer to the number of triples in Y such that inner products are (u, v, t), where (u, v, t) ∈ {a, b, c, d}.The number of (u, v, t) is the combinations with repetition for choosing three times out of 4 elements and it is = 20.We use CVX solver in MATLAB to solve the above optimization problems.We set up p = 8 i.e. S n 0 matrix is an 9 × 9 matrix.The key inequalities to set up SDP bounds for spherical few distance sets are: Therefore, for spherical s-distance sets with s 5, it is not hard to set up the SDP upper bounds for the cardinality and clarify the feasibility of our SDP bounds.

7.1
The non-existence of tight spherical designs of harmonic index {6, 4} Find f 4 such that By comparing the coefficients in (15), we get the following results: Solving for f 4 and α 2 from these two equations gives: (n + 4)(n + 6) .
If n 5, then f 4 is a complex number.So we cannot find ) satisfying our assumption.It is easy to check that b 2,T = 2, b 3,T = 3 (or 0) and b 4,T = 2.By Remark 11, there exists no tight spherical design of harmonic index {6, 4} when n = 2 and |Y | = 2.When n = 3 and n = 4, observe that the lower bounds for spherical designs of harmonic index 6 are about 3.41 and 5.29, respectively, which are strictly larger than b 3,T and b 4,T .Note that spherical design of harmonic index {6, 4} should also satisfy the condition for harmonic index 6.From the discussion above, there exists no tight spherical design of harmonic index {6, 4} for any n.

The non-existence of tight spherical designs of harmonic index {6, 2}
The Gegenbauer polynomial Then with similar calculation we have the following results.Then we can give a weaker condition for α with 3-distance set as follows.
Proof.Let X be a set of unit vectors whose mutual inner product set is {0, ±α}, and let G be the Gram matrix of such vectors.Then, .Note A cannot have more than one eigenvalue of multiplicity m because A is a |X| × |X| matrix.Therefore −1/α is rational, since it is also an algebraic integer, hence −1/α is an integer.
There is an example of X in S n−1 with I(X) = {0, ±α} and |X| > 2n so that 1/α is an even integer.
Example 28.The E 8 root system consists of 240 points in S 7 with inner products 0, ±1/2, −1.An example of the above is a half of E 8 -roots, which consists of 120 points in S 7 by choosing one of antipodal pairs.
Therefore Eq. ( 18) and the above two relations imply that We denote the LHS of (20) as g(ξ), then g(ξ) must have at least one non-negative root since we define ξ = α 2 β 2 .However we can prove that g(ξ) only has one real root and that root is negative.g(ξ) is a degree 3 polynomial with positive leading coefficient and has two critical points: .
From the calculation, we obtain the lower bound for |Y | is If 2 n 4, it is easy to check that b 4,T = 2 is the unique case when b n,T ∈ Z.However, the lower bound for spherical design of harmonic index 6 is about 5.29, which is strictly larger than b 4,T .Therefore, there is no tight spherical design of harmonic index {8, 6}.

7.4
The non-existence of tight spherical designs of harmonic index {8, 2} We want to determine f 2 such that From calculation we have the following results: If n = 4, the SDP upper bound for 4-distance set is 8.9981 (< 9).
For n = 9, we assume X is a spherical 4-distance set with I(X) = {±α, ±β} such that an integer.By Lemma 22, the case where n = 9 is also impossible.
Consider the case where L ) takes minimum value −c n,T,L at three non-negative points {0, α, β}, i.e., We can solve for f 6 and f 2 as follows. .
Then we can get α 2 , β 2 and the lower bound b n,T . .
The following theorem is very useful to consider the existence of spherical design of harmonic index {10, 6, 2}.
In this subsection, we assume X is an antipodal spherical 6-distance set with with E | 2 × 3 × 5 × 7. We can obtain the integral solutions of these equations.
the electronic journal of combinatorics 24(2) (2017), #P2.14 Let Comparing the coefficients in F (x) and R(x), we can obtain the following results.
We should remark that there is another test function 0 , where

Concluding remarks
In this paper we considered mainly spherical designs of harmonic index T = {t}, or T = {t 1 , t 2 }.For some T = {t 1 , t 2 , . . ., t } (with t 1 = 2e > t 2 > • • • > t and all t i are even), it seems that the general interesting case is where L(x) = Q n,2e (x) + f t 2 Q n,t 2 (x) + f t 3 Q n,t 3 (x) + • • • + f t Q n,t (x) and the minimum value of L(x) is at non-negative points α 1 , α 2 , . . ., α .Thus, further studies along this line would be interesting.
As we have shown in Section 5 as well as in previous sections, it seems remarkable that a spherical design of harmonic index t = 2e has a Fisher type lower bound |Y | (constant)•n e , which is the same order as for spherical 2e-design.So, all harmonic index T -designs are between harmonic index 2e-designs and spherical 2e-designs.It seems that considering tight T -designs have some meaning, although it seems tight harmonic index T -designs rarely exist.
As it is discussed in Bannai-Okuda-Tagami [2, Proposition 4.1] that to some extent, is also studied.The result is explained in terms of Bessel functions.As some special cases are mentioned in [2, p. 10], b n becomes greater than n(n + 1)/2 if n = 7, 8, 9, 10.This implies that tight spherical designs of harmonic index 2e do not exist, if t = 2e becomes large, say for n = 7, 8, 9, 10.On the other hand if n 6, it seems possible to determine the non-existence of tight designs in these cases, but it is not clear how we can show the non-existence of such harmonic index 2e-designs whose sizes are close to the Fisher type lower bound.It seems that this remains as an interesting open problem.
In concluding this paper, we remark that the theory as well as the concept of harmonic index T -designs in Q-polynomial association schemes exactly go parallel with the spherical case.The concept of T -design for an arbitrary subset T of the index set of nontrivial relations {1, 2, . . ., d} is already defined in Delsarte [5, Section 3.4] (1973).On the other hand, it seems that any systematic study on some specific choices of T , beyond the case T = {1, 2, . . .t} has not begun, even for the case T = {t}.We hope to discuss more on this topic in a separate paper.
x) is uniquely determined, from the monotonicity of the local minima of the Gegenbauer polynomial.Hence b n,{t} = 1 + Qn,t(1)   cn,t and we write b n,t = b n,{t} for simplicity.

2 2 .
if n 37.Moreover, b 2,6 = 2 and b 24,6 = 231 are the only two cases for which b n,6 ∈ Z when n 36.Remark 11.We should remark that not all the roots of F (x) in(11) will necessarily appear in I(Y ) when n is small.Consider the case n = 2. Recall that b 2,2e = 2 is proved for general e in[2, p. 6].Let y 1 , y 2 be two unit vectors in R 2 with angle θ = jπ/2e for odd j.Then, by the argument in [2, p. 2], Y = {y 1 , y 2 } is a tight spherical design of harmonic index 2e on S 1 .Larman-Rogers-Seidel (1977) proved the following fact.Theorem 12 ([12, Theorem 2]).Let X be a 2-distance set in R n with Euclidean distances c and d (c < d).If |X| > 2n + 3, then we have c 2 d 2 = (k − 1) k for some integer k with 2 k 1+ √ 2n the electronic journal of combinatorics 24(2) (2017), #P2.14 Suppose n = 24.Then Y is an at most 2-distance set in S 23 with I(Y ) ⊆ {±α}.Assume X is a spherical 2-distance set with I(X) = {±α} such that |Y | |X|.(There exists such X, otherwise |Y | is strictly larger than the cardinality of any 2-distance set.)If we put c = √ 2 − 2α and d = √ 2 + 2α, then c and d become the Euclidean distances between two distinct vectors in X.Note that |X| |Y | = b 24,6 = 231 > 2 × 24 + 3 = 51.However, in this case, we obtain c 2 /d 2 = 1/3 from easy calculation, contrary to Theorem 12. Hence there exists no tight spherical design of harmonic index 6 when n = 24.4.2 The non-existence of tight spherical designs of harmonic index 8

Theorem 14 .
Let e 2 be fixed.Then there exist positive constants A 2e and B 2e such that lim n→∞ c n,2e n e = A 2e and lim n→∞ b n,2e n e = B 2e , where A 2e and B 2e depend only on e.Therefore, c n,2e = A 2e n e (1 + o(1)) and b n,2e = B 2e n e (1 + o(1)).

Corollary 15 .
Let e 3 be fixed.If n is sufficiently large, then there exist no tight spherical designs of harmonic index 2e.Proof.If Y is a tight spherical design of harmonic index 2e, then I(Y ) ⊆ {±α} for some α > 0, and it follows from Lemma 10 that |Y | n(n+1) 2

Case ( 3 ): If 2 n 8 ,
b 8,T = 65 is the unique case with b n,T ∈ Z.We set up the semidefinite programming (SDP) method on the upper bounds for spherical 4-distance sets with the indicated inner product values.Theorem 25 is what we set up to estimate the upper bounds of spherical 4-distance sets.Such an SDP formula can be obtained from special setting of or generalization of Barg-Yu [3, Theorem 3.1] for spherical 2-distance sets.We choose the positive semidefinite matrices S n k with size (9 − k) × (9 − k) and linear constraints
where x ∈ {0, ±α} and x ∈ {0, ±1}.G is a symmetric and positive semidefinite matrix of order |X|.It has the smallest eigenvalue 0 of multiplicity m |X| − n.Therefore, A has the smallest eigenvalue −1/α of multiplicity m |X| − n.Moreover, −1/α is an algebraic integer since A is an integer matrix, and every algebraic conjugate of −1/α is also an eigenvalue of A with multiplicity m.If |X| > 2n, then m > |X| 2
2and get the following table for some integers y 1 , y 2 , y 3 .
* ) They are often displayed as a list [a 1 , a 2 , a 3 , a 4 , a 6 ].More information about the database of elliptic curve is available from: If 9 n 75, then |X| |Y | 2 n+1 2 .Using the first statement in Lemma 22, we see that both 1−α 2 β 2 −α 2 and 1−β 2 α 2 −β 2 are integers.It is easy to check that neither of them is an integer for 9 n 75.