New Nonexistence Results for Spherical Designs

We obtain bounds for the smallest and largest inner products of distinct points of spherical τ -designs of relatively small cardinalities and odd strengths τ . In many cases, the restrictions obtained imply new nonexistence results. Our method works well in small dimensions as well as when the dimension tends to infinity. For τ = 3 and τ = 5, we obtain new asymptotic bounds on the minimum possible odd size of τ -designs.


Introduction
The spherical designs have been introduced in 1977 by Delsarte-Goethals-Seidel [7] on the analogy of the classical combinatorial designs. A spherical τ -design C ⊂ S n−1 is a finite subset of the unit sphere S n−1 in R n such that (µ(x) is the Lebesgue measure) holds for all polynomials f (x) = f (x 1 , . . . , x n ) of degree at most τ (i.e., the average of f over the set is equal to the average of f over S n−1 ). The number τ is called the strength of C. Let us denote by B(n, τ ) (resp., by B odd (n, τ )) the minimum possible cardinality (resp. odd cardinality) of a τ -design on S n−1 . The following Fisher-type lower bound on B(n, τ ) was obtained by Delsarte-Goethals-Seidel [ (1) In this paper we consider designs of odd strength τ = 2e − 1, e ≥ 2 is integer, and odd cardinality |C|. This case turned out to be more difficult for realization.
For τ = 2e − 1 we derive from (1) the estimate First nonexistence results for (2e − 1)-designs of odd size were proved in [5] (see also [4]). In this paper we continue this investigation by refining the approach from [5].
The following definition for spherical designs is crucial for our approach. A code C ⊂ S n−1 is a spherical τ -design if and only if for any point y ∈ S n−1 and any real polynomial f (t) of degree at most τ , the equality holds, where i (t) in terms of the Gegenbauer polynomials [1, Chapter 22]). As usual, x, y denotes the standard scalar product in R n .
We use (2) when y belongs to the design. Then (2) becomes In this paper, we propose a method for investigation of (2e − 1)-designs of odd size. As results, we derive some restrictions on the structure of such designs which are expressed as bounds on inner products of their points. Sometimes this implies nonexistence results. Interestingly, nonexistence follows already in first open cases (i.e., in small dimensions) and when the dimension n tends to infinity but the strength τ is fixed.
In Section 2 we describe our approach. We start with the assumption of the existence of certain (2e − 1)-design C ⊂ S n−1 with prescribed odd cardinality. Then we prove that some special triple of points of C do always exist. The remaining analysis is restricted to these three points with using suitable polynomials in (3).
Applications for τ = 3 and τ = 5 are shown in Sections 3 and 4. It becomes clear how this investigation could be continued for higher strengths.
Our approach is the following. First, we show for odd cardinalities |C| that some special triples (x, y, z) of points of C appear. Then we use suitable polynomials in (3) in order to derive bounds on some inner products in the sets I(x), I(y) and I(z). At the third step, we organize an iterative process by using the new bounds and (other) suitable polynomials in (3). The results are again bounds on inner products in I(x), I(y) and I(z). In many cases this implies nonexistence of the design under target.
Step 1. Our first step is based on the following simple observation. Lemma 1. Let C ⊂ S n−1 be a τ -design of odd cardinality |C|. Then there exist three distinct points x, y, z ∈ C such that t 1 (x) = t 1 (y) and t 2 (x) = t 1 (z).
Proof. Let Γ be the directed graph with vertices the points of C and edges x → y if and only if t 1 (x) = x, y . It is easy to see that cycles in Γ are possible only of length two. Since |Γ| = |C| is odd, we must have y ↔ x ← z which completes the proof.
It follows by Lemmas 1 and 2 that there exists a point x ∈ C such that t 1 (x) ≤ t 2 (x) ≤ α 0 . This observation was used in [4] to prove that ρ 0 |C| ≥ 2 is a necessary condition for existence of C. We upgrade this result by more detailed investigation of the triple (x, y, z).
Step 2. Inequalities of the type t 1 (x) = x, y ≤ t 2 (x) = x, z ≤ a may mean that the points y and z are close each other. Indeed, it is easy to see that y, z ≥ 2a 2 − 1. If 2a 2 − 1 > α e−1 , we actually have obtained new bounds on t |C|−1 (y) and t |C|−1 (z). In turn, these new bounds give better estimations t 1 (y) ≤ a < a and t 1 (z) ≤ a < a. This leads to an improvement Step 3. If 2α 2 0 − 1 > α e−1 , we can start and further organize the following iterative process, applying in fact Step 2 as many times as necessary. Set δ 0 = α 0 and let us have δ 1 = a by applying Step 2 for a = α 0 . Now 2δ 2 1 − 1 > 2δ 2 0 − 1 is a better lower bound for t |C|−1 (y) and t |C|−1 (z) and implies by a second application of Step 2 better upper bounds t 2 (x) ≤ δ 2 . We can continue this process, checking (at each iteration) the existence of C.
for some i ≥ 0, then C does not exist.
Proof. Assume that such a polynomial f exists and consider (3) for this polynomial and x ∈ C. The assertion then follows since the left-hand side of (3) is at least 2f (t 2 (x)) ≥ 2f (δ i ) for all i ≥ 0. This means that C could not exist if (4) is satisfied.
The logic of Theorem 1 is the following. If lim i→+∞ δ i = −∞ (we actually need much weaker results -see Example 1), then C does not exist. Otherwise, we have some new bounds of (C) = min{ x, y : x, y ∈ C} and s(C) = max{ x, y : x, y ∈ C, x = y}.
Bajnok [2,3] shows that 3-designs of any even size exist. He also constructs 3-designs of any odd cardinality greater than or equal to 5n/2 for n ≥ 7, 11 for n = 3, 4 and 15 for n = 5, 6. On the other hand, it was shown in [5] that k ≥ 3 in all dimensions, k ≥ 5 for n ≥ 11, k ≥ 7 for n ≥ 19, etc.
In each concrete case, the optimal value of F (a) can be found numerically by Maple. According to Step 2, we denote δ 1 = min{F (a) : a ∈ (α 0 , α 1 )}.
For the iterative process of Step 3, we use the analog of Lemma 3 by using µ 1 instead of µ 0 to obtain t 1 (y) ≤ t 1 (z) ≤ δ 2 and so on.
There were 144 open cases in dimensions 3 ≤ n ≤ 50. We rule out 50 of them.
Therefore, 2.3227 n ≤ B odd (n, 3) ≤ 2.5 n asymptotically. Our conjecture is that the upper bound gives the exact value of B odd (n, 3) either in small dimensions and as n tends to infinity.
Let k be odd and x, y, z be the points from Lemma 1. We assume that µ 0 = 2α 2 0 − 1 > α 2 as Step 2 requires. Then the new bound on t 2 (x) = t 1 (z) is given by the following lemma which can be proved as Lemma 3.  F (a, b), where and K = (2α 2 0 − 1) 2 + a(2α 2 0 − 1) + b 2 provided the denominator in the last fraction is positive.
The iterative process can be continued as in the case τ = 3 by using f (t) = (t − α 1 ) 2 (t − α 2 ) 2 in Theorem 1. The first nonexistence result shows that 33-point 5-designs on S 4 do not exist. We tested the first open cases (with ρ 0 |C| ≥ 2) in each dimension 3 ≤ n ≤ 20 until nonexistence proof is still possible. In this way, 53 designs were proved not to exist.