A new proof of a generalization of Gerzon's bound

In this paper we give a short, new proof of a natural generalization of Gerzon's bound. This bound improves the Delsarte, Goethals and Seidel's upper bound in a special case. Our proof is a simple application of the linear algebra bound method.


Introduction
In this paper we give a linear algebraic proof of the known upper bound for the size of some special spherical s-distance sets. This result generalizes Gerzon's general upper bound for the size of equiangular spherical set.
In the following R[x 1 , . . . , x n ] = R[x] denotes the ring of polynomials in commuting variables x 1 , . . . , x n over R.
Let G ⊆ R n be an arbitrary set. Denote by d(G) the set of (non-zero) distances among points of G: d(G) := {d(p 1 , p 2 ) : p 1 , p 2 ∈ G, p 1 = p 2 }. 0 Keywords. Gerzon's bound, distance problem, linear algebra bound method An s-distance set is any subset H ⊆ R n such that |d(H)| ≤ s.
Let (x, y) stand for the standard scalar product. Let s(G) denote the set of scalar products between the distinct points of G: A spherical s-distance set means any subset G ⊆ S n−1 such that |s(G)| ≤ s.
Let n, s ≥ 1 be integers. Define Delsarte, Goethals and Seidel investigated the spherical s-distance sets. They proved a general upper bound for the size of a spherical s-distance set in [5]. Theorem 1.1 (Delsarte, Goethals and Seidel [5]) Suppose that F ⊆ S n−1 is a set satisfying |s(F)| ≤ s. Then |F| ≤ M(n, s).
Barg and Musin gave an improved upper bpund for the size of a spherical s-distance set in a special case in [1]. Their proof builds upon Delsarte's ideas (see [5]) and they used Gegenbauer polynomials in their argument. Theorem 1.2 (Barg, Musin [1]) Let n ≥ 1 be a positive integer and let s > 0 be an even integer. Let S ⊆ S n−1 denote a spherical s-distance set with inner products t 1 , . . . , t s such that We point out here the following special case of Theorem 1.1. An equiangular spherical set means a two-distance spherical set with scalar products α and −α. Let M(n) denote the maximum cardinality of an equiangular spherical set. There is a very extensive literature devoted to the determination of the precise value of M(n) (see [7], [8]). Gerzon gave the first general upper bound for M(n) in [7] Theorem 8.
Musin proved a stronger version of Gerzon's Theorem in [9] Theorem 1. He used the linear algebra bound method in his proof. Theorem 1.5 (Musin, [9] Theorem 1) Let S be a spherical two-distance set with inner products a and b. Suppose that a + b ≥ 0. Then De Caen gave a lower bound for the size of an equiangular spherical set in [3]. Theorem 1.6 (De Caen [3]) Let t > 0 be a positive integer. For each n = 3 · 2 2t−1 − 1 there exists an equiangular spherical set of 2 9 (n + 1) 2 vectors. Our main result is an alternative proof of a natural generalization of Gerzon's bound, which improves the Delsarte, Goethals and Seidel's upper bound in a special case. Our proof uses the linear algebra bound method. The following statement was proved in [6] Theorem 6.1. The original proof builds upon matrix techniques and the addition formula for Jacobi polynomials.

Preliminaries
We prove our main result using the linear algebra bound method and the Determinant Criterion (see [2] Proposition 2.7). We recall here for the reader's convenience this principle.  We use the following combinatorial statement in the proofs of our main results.

The proof
Proof of Theorem 1.7: Consider the real polynomial Let S = {v 1 , . . . , v r } denote a spherical s-distance set with inner products a 1 , . . . , a t , −a 1 , . . . , −a t . Here r = |S|. Define the polynomials Consider the set of vectors It is easy to verify that if we can expand P i as a linear combination of monomials, then we get where c α ∈ R are real coefficients for each α ∈ E(n, s) and x α denotes the monomial x α 1 1 · . . . · x αn n . Since v i ∈ S n−1 , this means that the equation is true for each v i , where 1 ≤ i ≤ r. Let Q i denote the polynomial obtained by writing P i as a linear combination of monomials and replacing, repeatedly, each occurrence of x t 1 , where t ≥ 2, by a linear combination of other monomials, using the relations (2).
Since g(v i ) = 0 for each i, hence Q i (v j ) = P i (v j ) for each 1 ≤ i = j ≤ r.
We prove that the set of polynomials {Q i : 1 ≤ i ≤ r} is linearly independent. This fact follows from the Determinant Criterion, when we define F := R, Ω = S n−1 and f i := Q i for each i. It is enough to prove that Q i (v i ) = P i (v i ) = 0 for each 1 ≤ i ≤ r and Q i (v j ) = P i (v j ) = 0 for each 1 ≤ i = j ≤ r, since then we can apply the Determinant Criterion. But . . , v r } is a spherical s-distance set with inner products a 1 , . . . , a t , −a 1 , . . . , −a t .
Then it is easy to check that we can write Q i as a linear combination of monomials in the form where d α ∈ R are the real coefficients for each α ∈ M(n, s). This follows immediately from the expansion (1) and from the relation 2.
Since the polynomials {Q i : 1 ≤ i ≤ r} are linearly independent and if we expand Q i as a linear combination of monomials, then all monomials appearing in this linear combination contained in the set of monomials

Concluding remarks
The following Conjecture is a natural strengthening of Theorem 1.7.
Conjecture 1 Let n ≥ 1 be a positive integer and let s > 0 be an even integer. Let S ⊆ S n−1 denote a spherical s-distance set with inner products t 1 , . . . , t s such that t 1 + . . . + t s ≥ 0.