Some bounds arising from a polynomial ideal associated to any t-design ∗

We consider ordered pairs (X,B) where X is a finite set of size v and B is some collection of k-element subsets of X such that every t-element subset of X is contained in exactly λ “blocks” B ∈ B for some fixed λ. We represent each block B by a zero-one vector cB of length v and explore the ideal I(B) of polynomials in v variables with complex coefficients which vanish on the set {cB | B ∈ B}. After setting up the basic theory, we investigate two parameters related to this ideal: γ1(B) is the smallest degree of a non-trivial polynomial in the ideal I(B) and γ2(B) is the smallest integer s such that I(B) is generated by a set of polynomials of degree at most s. We first prove the general bounds t/2 < γ1(B) ≤ γ2(B) ≤ k. Examining important families of examples, we find that, for symmetric 2-designs and Steiner systems, we have γ2(B) ≤ t. But we expect γ2(B) to be closer to k for less structured designs and we indicate this by constructing infinitely many triple systems satisfying γ2(B) = k. 2010 MSC: 05B05, 05E30, 13F20


Introduction
Let X be a finite set of size v and consider a k-uniform hypergraph (X, B) with vertex set X and block (hyperedge) set B. We aim to study polynomial functions on X which vanish on each element of B so that B may be viewed as the variety of some naturally defined ideal of polynomials in v variables. In order to do so, we identify each block B ∈ B with a 01-vector c B , with entries indexed by the elements of X, whose i th entry is equal to one if i ∈ B and equal to zero otherwise. In other words, c B is the B th column of the point-block incidence matrix, A, of the hypergraph. In this paper we work in characteristic zero and consider the polynomial ring R = C[x 1 , . . . , x v ]. The evaluation map given by ε(f ) (B) = f (c B ) is a ring homomorphism and its kernel is the ideal of all polynomials in v variables which evaluate to zero on each block of the hypergraph. Denoting this kernel by I, we see that I is the ideal of the finite variety {c B | B ∈ B} and we write I = I(B). Our goal in this paper is to explore connections between this ideal I and the hypergraph (X, B). Our primary question involves the identification of good generating sets G for I based on the combinatorial structure of the design (X, B). We are not concerned here with the actual size of the generating set; in fact, we prefer a set of polynomials preserved by the automorphism group of the design. By "good" here, we mean principally that polynomials in the generating set are all of low degree. But we also seek polynomials that shed light on properties of the design.

Background and related work
Algebraic geometry has a long history in the theory of error-correcting codes. We also have a theory of spherical codes and designs that involves the evaluation of multivariate polynomials at finite sets of points on the unit sphere. Möller [20] constructs good quadrature rules for spherical integration by choosing zero sets of well-chosen families of polynomials. Conder and Godsil [5] studied the symmetric group as a polynomial space. The standard module of the Johnson association scheme J(v, k) can be identified with polynomials of degree at most k in v variables in such a way that the polynomials of a given maximum degree j correspond to a sum of eigenspaces V 0 + V 1 + · · · + V j . This approach, which is implicit in [8] is worked out in more detail in a later paper of Delsarte [9] (see also [3]). These phenomena motivated Martin and Steele [17] to consider the ideal of polynomials that vanish on the shortest vectors of certain lattices. In work in progress, Martin [18] extends this to attach an ideal to any cometric association scheme by viewing the columns of the Q-polynomial generator E 1 in the Bose-Mesner algebra as a finite variety. Several important examples are connected to combinatorial t-designs and -when disentangled from the language of association schemes -the problem of determining generating sets for the ideals of those association schemes reduces to the problem we address here.
The "polynomial method" in combinatorics [24] is a powerful tool for deriving bounds on the size of combinatorial objects and for proving non-existence of extremal objects. In particular, a recent breakthrough by Croot, Lev and Pach [7] (see also [11]) has stimulated interest in collections of multivariate polynomials that vanish on some configuration in a finite vector space. Here, we work in characteristic zero and are interested in how the ideal we obtain is related to the structure of the design. By contrast, the authors of [7,11] work over fields of positive characteristic. To see the connection, we note that, since our zero set lies in Z v , the polynomial generators g ∈ G may always be chosen from Z [x]. So application of the reduction Z → Z p maps the ideal G ⊆ Z[x] into the ideal of the same finite variety considered modulo p. It will be an interesting follow-up task to determine when the image under this map is the full ideal in positive characteristic.
We have recently learned of previous Gröbner basis approaches to algebraic aliasing in the statistical design of experiments. Once a useful basis is chosen for the space of polynomial functions on the elements of the design, higher order effects can be mapped to more elementary interactions between factors. For a recent survey, see [19], with the caveat that the terminology used there differs from the terminology in this paper.

The trivial ideal
We pause to introduce our notation for the standard operations of elementary algebraic geometry. (See, e.g., [10,Chapter 15] for a basic introduction.) For a set S ⊆ C v , we let I(S) denote the ideal of all polynomials in C[x] := C[x 1 , . . . , x v ] that vanish at each point in S. And if G is any set of polynomials in C[x 1 , . . . , x v ], we denote by Z(G) the zero set of G, the collection of all points c in C v which satisfy f (c) = 0 for all f ∈ G. Note that, when S is finite, we have Z(I(S)) = S. In the opposite direction, for any ideal J of polynomials, the Nullstellensatz (see, e.g., [13, p21], [6, p179]) informs us that I(Z(J)) = Rad(J), where Rad(J) denotes the radical of ideal J, the ideal of all polynomials g such that g n ∈ J for some positive integer n. A radical ideal is an ideal which is already closed under this process: Rad(J) = J.
Our first example to consider is the complete uniform hypergraph (X, K v k ). Then Proof. One easily checks that each c B is a common zero of the polynomials in G 0 . Conversely, any point in C v which is a zero of each polynomial in G 0 must be a 01-vector with exactly k ones. In order to verify that G 0 generates the full ideal, we check that the Zariski tangent space at each point is fulldimensional. Let B ⊂ X be a k-set; evaluating the gradient ∇f of f (x) = x 2 j − x j at c B we obtain ±e j where e j is the standard basis vector with a one in position j and all other entries zero. So the Jacobian of the set G 0 of v + 1 polynomials in v variables evaluated at c B takes the form which clearly has column rank equal to v. This guarantees that each c B is a simple zero of G 0 and so the ideal is indeed radical. It now follows from the Nullstellensatz that See Section 3 for details of these last calculations.

Two parameters
In this paper, once we fix v and k, every ideal will contain the ideal we have just considered. We call this the trivial ideal and denote For any k-uniform hypergraph (X, B) on v points, the ideal I(B) := I({c B | B ∈ B}) will contain T and a polynomial f ∈ I(B) will be called non-trivial if f ∈ T and trivial otherwise.
To each C ⊆ {1, 2, . . . , v} we associate the monomial x C = j∈C x j and note that, for a block B ∈ B, the value of The following characterization of t-designs is well-known (see, for example, Godsil [14, Cor. 14.6.3]). Lemma 2.1 (Cf. Delsarte [9,Theorem 7]). Let X be a set of size v and let (X, B) be a k-uniform hypergraph defined on X with corresponding vectors c B (B ∈ B) as defined above. Then (X, B) is a t-design on X if and only if the average over B of any polynomial f (x) in v variables of total degree at most t is equal to the average of f (x) over the complete uniform hypergraph K v k defined on X.
which is exactly the average of f (x) over the block set B. So the result holds for monomials. But every polynomial in v variables of total degree at most t is a linear combination of such monomials, so the result holds for these as well by linearity.
Given (X, B), we seek combinatorially meaningful generating sets for I(B). Two polynomials f (x) and g(x) have the same value at every point c B , B ∈ B if and only if their difference belongs to the ideal I(B). For example, f (x) = x 2 j and g(x) = x j take the same value on every 01-vector so x 2 j − x j belongs to the ideal of any design. We say f (x) is a multilinear polynomial in v variables if f is linear in each x i : i.e., each monomial with non-zero coefficient in f is a product of distinct indeterminates. Modulo the trivial ideal T , each polynomial in C[x 1 , . . . , x v ] is equivalent to some (not necessarily unique) multilinear polynomial with zero constant term. With a preference for polynomials of smallest possible degree, we define two fundamental parameters.  Proof. Suppose F ∈ I(B) has degree at most t/2.
each have degree at most t/2. Since the entries of each c B are real, it is clear that f, g ∈ I(B). Then f 2 ∈ I(B) is a non-negative polynomial of degree at most t. By Lemma 2.1, its average over B is zero hence its average over K v k is also zero. Since f 2 is everywhere non-negative, it must evaluate to zero on the incidence vector c B of every k-set B. So it belongs to the ideal I(K v k ). Since this ideal is radical and contains f 2 , it also contains f . By Lemma 1.1, f must be trivial. The same argument applies to g and, hence, to F . Remark 2.4. The same sort of reasoning used in this proof shows that I(B) admits a vector space basis, even a generating set, of polynomials with integer coefficients. Let F be a polynomial which vanishes on B and let {ζ 1 . . . , ζ m } ⊆ C be a basis for the subspace of C, as a vector space over Q, that contains all the coefficients of F . Then there exist unique polynomials F 1 , . . . , F m in Q[x] with F = h ζ h F h . Since each c B is a vector with integer entries, the fact that F evaluates to zero at c B implies that each F h also vanishes at that point. So, scaling appropriately, we may assume each generator belongs to Z[x].
A standard result in the theory of designs (see; Cameron [4] and Delsarte [8,Theorem 5.21]) is the fact that a t-design with s distinct block intersection sizes satisfies t ≤ 2s. We now show that Theorem 2.3 implies a stronger result, which we believe is new.
Let C ⊆ X with characteristic vector c and suppose {|C ∩ B| : B ∈ B} = {i 1 , . . . , i s }. Then every c B for B ∈ B is a zero of the degree s zonal polynomial Of course, if |C| is sufficiently small, this polynomial belongs to the trivial ideal.
Consider the product of zonal polynomials By hypothesis, F (c B ) = 0 for every B ∈ B. Since F (χ S ) = 0, F is non-trivial. So, by Theorem 2.3, deg F > t/2.  Starting from G 0 , let us build up a meaningful generating set for I(B). The unique triple in B containing both 0 and 1 also contains 3; this combinatorial condition may be encoded as Alternatively, including the quadratic polynomial x 0 x 1 − x 0 x 3 in a generating set G for our ideal also guarantees that any vector c ∈ Z( G ) with c 0 = 1 and c 1 = 1 must have c 3 = 1 as well. Up to sign, there are 7 2 quadratic generators of this form and these, together with those in G 0 , generate the full ideal.
While we will not see further examples where the ideal is generated by G 0 and linear polynomials, we will see that the difference of two monomials appears again as a useful tool.

Radical ideals
In this section, we deal with a technicality which arises as we compute ideals of finite sets. We show here that every ideal containing the trivial ideal is radical, thereby eliminating any further need to check this property.
Given a finite set S of points in C v , it is often easy to come up with polynomials that vanish at each of those points and, with a bit of work, we might find a generating set G for some ideal J = G whose zero set is exactly S: Z( G ) = S. Hilbert's Nullstellensatz then tells us that the radical of of ideal J given by The ideal J is a radical ideal if Rad(J) = J. Our goal then is achieved in three steps: given a finite set of points S, find a nice set G of small-degree polynomials that vanish on S; verify that Z(G) = S and nothing more; verify also that G is a radical ideal. In this section, we discuss ways to achieve this last step.
/J is a finite-dimensional complex vector space and its dimension is equal to the sum of the multiplicities of all the zeros of I, The coordinate ring of a variety S ⊆ C v is defined as the quotient ring C[x]/I(S) and this is naturally identified with the ring of "polynomial functions" on the set S. If J is an ideal in C[x] with finite zero set Z(J) = S, then it is well-known (e.g., Section 5.3, Proposition 7 in [6]) that where mult(c) is the multiplicity of point c as a zero of J. This proves Proposition 3.1. With notation as above: The Jacobian of the system , the partial derivative ∂f j /∂x i evaluated at point c. Since we are dealing only with zero-dimensional varieties in this paper, we say the point c is smooth if Jac(G, c) has column rank v, so that the Zariski tangent space is zero-dimensional. A smooth point has multiplicity one. So another way to show that the ideal J is radical is to check that, at each point c of its zero set, the Jacobian Jac(G, c) has full row rank where G is some generating set for J.
be any set of polynomials such that G 0 ⊆ G (cf. Equation (1)). Then G is a radical ideal.
Proof. In the proof of Lemma 1.1, we proved that the Jacobian Jac(G 0 , c B ) of G 0 has column rank equal to v at any characteristic vector c B of any k-set B. Since Jac(G 0 , c B ) is a submatrix of Jac(G, c B ), this latter Jacobian also has full row rank and each point of the finite variety Z(G) is smooth.
We note here that this approach also provides another proof of the fundamental bound of Ray-Chaudhuri and Wilson. Proof. Assume t = |S| + 1. Since 1 + I = 1 k j x j + I we have and we may solve for x S + I.
Proof. Since the coordinate ring C[x]/I(B) has dimension |B| and the monomials {x C : |C| = s} represent linearly independent cosets in this quotient ring, we have |B| ≥ v s .
This language differs from that employed by Ray-Chaudhuri and Wilson. Consider the v s × |B| matrix A (s) with rows indexed by all C ⊆ X with |C| = s, with columns indexed by the set B of blocks of a t-(v, k, λ) design, and (C, B)-entry equal to one if C ⊆ B and equal to zero otherwise. The proof of Theorem 1 in [21] establishes that the columns c B of A (s) span the space R ( v s ) . The celebrated bound follows immediately for even t and, for odd t, one obtains |B| ≥ 2 v s whenever B is the block set of a t-(v, k, λ) design with t = 2s + 1 by applying this idea to both the derived design and the residual design. As we will use the fact later, we record it here as a lemma.

Steiner systems and partial designs
For a subset C ⊆ X and f ∈ R, define f (C) in the obvious way, by setting We want to construct small-degree generating sets G for the ideal I = I(B) where B is the block set of our design (X, B). We assume throughout that G contains G 0 so that G contains the trivial ideal T . Every zero of this latter ideal is a 01-vector with exactly k ones. As we choose the remaining generators, we need only search for multilinear polynomials: each monomial x e1 1 · · · x ev v appearing in f (x) has all e i ∈ {0, 1}. It is clear that the automorphism group of a design (X, B) acts on the ideal I(B) by permuting indeterminates; rather than minimizing |G|, we typically show a preference for sets of generators invariant under this action.
Recall, for a subset C ⊆ X, we have x C = i∈C x i . Next define, for an integer j ≤ |C|, the polynomial (This is a certain symmetric function -the elementary symmetric polynomial of degree j -in the variables Then, for every C ⊆ X, we have that g B,J (C) = 0 if and only if j ≤ |C ∩ B| < k.
Proof. If |C ∩ B| < j, then every monomial appearing in g B,J (x) evaluates to 0 on χ C . If |C ∩ B| = k, then all the monomials in g B,J (x) take nonzero values on χ C , and g B, If j ≤ |C ∩ B| < k, then some proper subset of the monomials occurring in g B,J (x) take nonzero value, and g B,J (C) cannot equal 0.
Proof. In addition to our generators G 0 for T given in (1), we will include one generator Suppose J = {j 1 , . . . , j } (possibly the empty set). Then consider the polynomial noting that g Y (x) = −x Y whenever Y is contained in no block of the hypergraph.
Let I be the ideal generated by It follows from Proposition 3.2 that I is a radical ideal. In order to prove that I = I(B) (and hence that γ 2 (B) ≤ k), we must verify Suppose that B ∈ B; then it is easy to verify from (6) that g Y (B) = 0 for any On the other hand, if Y is contained in at least one block, then g Y (C) = −1 from (6). We then have that I is a radical ideal with Z(I) = B; by Proposition 3.1(ii) we are done.
A Steiner system is a t-(v, k, λ) design with λ = 1. The question of existence of non-trivial Steiner systems with t > 5 has been recently resolved in spectacular fashion by Keevash [16].  Let (X, B) be any t-(v, k, 1) design. For a block B ∈ B and any t-element subset T contained in B, define (as in (5)) Proof. In view of Theorem 4.2, we can assume that t < k. Consider the generating set  Next, we describe a couple of variations of Theorem 4.3. A partial t-(v, k, 1)-design is a k-uniform hypergraph in which any t-subset occurs in at most one block. A partial t-(v, k, 1)-design, (X, B), is maximal if there does not exist a k-subset C ⊆ X, C ∈ B such that (X, B ∪ {C}) is a partial t-(v, k, 1)design.
Proof. As we employ the same generating set as in the proof of Theorem 4.3, we need only check that Z( G ) = B. As before, we have that g B,T (B ) = 0 for all blocks B, B ∈ B and all t-subsets T of B. Now suppose that |C| = k, C ∈ B. Because (X, B) is a maximal partial t-(v, k, 1)-design, it is possible to choose T ⊆ C, |T | = t such that there is a block B ∈ B with T ⊆ B. Then g B,T (C) = 0 as before.
By a slight extension of our construction, we do not require the partial t-(v, k, 1)-design to be maximal. Since x T evaluates to zero on every block, we still have B ⊆ Z( G ). But if C is a k-set not belonging to B, either B ∪ {C} is again a partial t-design or some t-subset T of C is contained in some block B. In the latter case, g B,T (C) = 0 as above; in the former case, every t-subset of C belongs to T so we can take any t-subset T ⊆ C and, with f (x) = x T ∈ G, we have f (C) = 0. So Z( G ) = B and the rest of the proof follows just as before.
This result gives us another upper bound on γ 2 for general t-designs.  Fisher's inequality states that, for any such 2-design, we have |B| ≥ |X| since, for t ≥ 2, the point-block incidence matrix A has rank v. A 2-design with equally many blocks and points is called a symmetric 2design. While Theorem 2.3 implies here that γ 1 (B) > 1, we may use the invertibility of A to obtain more information in this case.

Symmetric balanced incomplete block designs
k 1 lies in the left nullspace of A. So w is a scalar multiple of 1 and f is trivial. The quadratic case is more interesting. Let r i denote row i of matrix A. For any two distinct points i, j ∈ X, the entrywise product r i • r j is expressible as a linear combination of the rows of A. Say Then the polynomial f (x) given by is easily seen to belong to I(B): for a block B indexing column of matrix In fact, we may determine the coefficients w h explicitly to obtain a nice generating set for our ideal.
(ii) γ 1 (B) = γ 2 (B) = 2; Proof. Assume (X, B) is a 2-design with |B| = v and incidence matrix A. As AA = (k − λ)I + λJ, we see that the inverse of our incidence matrix is Letting r i denote the i th row of A (i ∈ X), observe that r i • r j is a 01-vector of length v with λ entries equal to one. So r i • r j = w A gives that is, I(B) contains the quadratic polynomial As x X,1 takes value k on each c B , this shows that ideal I contains f i,j (x) for every pair i, j of distinct points.   Proof. Here we have v = (q d+1 − 1)/(q − 1) and k = (q e+1 − 1)/(q − 1). Every line of PG(d, q) contains q + 1 points. Suppose B is an e-dimensional subspace of PG(d, q) and L is a line. Then |L ∩ B| ∈ {0, 1, q + 1}. In each case, g L,J (B) = 0 for any 2-element J ⊆ L by Lemma 4.1.
A subset of points of PG(d, q) that intersects any line in 0, 1 or q + 1 points is necessarily a subspace of PG(d, q). Suppose that |C| = k but C is not an e-dimensional subspace of PG(d, q). Then there exists a line L such that |L ∩ C| ∈ 0, 1, q + 1. In this case, g L,J (C) = 0 by Lemma 4.1.
We have shown that the ideal generated by G is a radical ideal with zero set B, so we are done by Proposition 3.1(ii).
Remark 5.4. Clearly we can build a much smaller generating set than our choice of G by selecting just one pair J of points in each line. We instead prefer here to choose a set G of polynomials which is invariant under the automorphism group of the design.

Triple systems
Identifying each square-free monomial x C with the set C, the multilinear polynomials with real coefficients are in bijective correspondence with the real-valued functions on the Boolean lattice. For multilinear f write Suppose that C ⊆ X and 0 ≤ s ≤ |C|. The s-incidence vector of C, denoted δ = δ s (C), is the vector of length w = s i=0 v i , whose coordinates correspond to the subsets of X of cardinality at most s, defined by where |J| ≤ s.
Now suppose that f is a multilinear polynomial in x 1 , . . . , x v of degree at most s. The vector of coefficients of f , denoted c = c(f ), is also a w-dimensional vector whose coordinates correspond to the subsets of X of cardinality at most s.
The following lemma is obvious.
For any f ∈ I(B), we have that f (B 1 ) = f (B 2 ) = · · · = f (B 7 ) = 0. It then follows from (7) and Lemma 6.1 that f (C) = 0. However, since C ∈ B, it must be the case that f (C) = 0 for some f ∈ I(B). This contradiction establishes the desired result.
Note that this linearization technique may be applied more generally. If we can find C ∈ B such that δ s (C) is a linear combination of the w-dimensional vectors {δ s (B) | B ∈ B}, then γ 2 (B) > s.
We now show one way to construct examples of 2-(v, 3, 2) designs that satisfy the hypotheses of Theorem 6.3. Our construction works for any v ≡ 1, 3 (mod 6), v ≥ 15.
Suppose we have a 2-(v, 3, 2) design that satisfies the hypotheses of Theorem 6.3. We first observe that is a set of four blocks that forms a so-called quadrilateral (or Pasch configuration). As well, is a set of three blocks that is not contained in a quadrilateral (because {4, 5, 6} is not a block).

It is shown in [22, Theorem 3.1] that the maximum number of quadrilaterals in a
, and this maximum is attained if an only if the design is isomorphic to the point-line structure of the projective geometry PG(n, 2) for some integer n ≥ 2. Take any 2-(v, 3, 1) design that is not isomorphic to the projective geometry PG(n, 2) (this can be done provided v ≡ 1, 3 (mod 6), v ≥ 9. It is easy to see that design must contain three non-collinear points that are not contained in a quadrilateral. By relabelling points in the design, we can assume that the three non-collinear points are denoted 1, 2 and 3, and they are contained in the three blocks in (9). Moreover, {4, 5, 6} is not a block in this design because the three points 1, 2, 3 are not contained in a quadrilateral.
Now we take the union of the blocks in the two 2-(v, 3, 1) designs constructed above. The result is a 2-(v, 3, 2) design that contains the seven blocks in (8) and (9). We have already noted that {4, 5, 6} is not a block in the second design. It is also not a block in the first design because the pairs {4, 5}, {4, 6} and {5, 6} occur in three different blocks in this design.
As a consequence of this discussion and Theorem 6.3, the following result is immediate. We now give a more general version of this construction, starting from an arbitrary trade. We recall some definitions from [12]. A trade is a set T of two (finite) subsets of blocks of size three, say T = {T 1 , T 2 } that satisfies the following properties: 1. each T ( = 1, 2) is a partial Steiner triple system (i.e., no pair of points occurs in more than one block) 3. the set of pairs contained in the blocks in T 1 is identical to the set of pairs contained in the blocks in T 2 .
As an example, if The volume of a trade T = {T 1 , T 2 }, which is denoted vol (T), is the number of blocks in T 1 (or, equivalently, the number of blocks in T 2 ). The foundation of T, denoted found (T), is the set of points covered by the blocks in T 1 (or, equivalently, the set of points covered by the blocks in T 2 ).
The following lemma is easy to prove. Proof. The definition of a trade T = {T 1 , T 2 } ensures that T 1 and T 2 cover the same set of pairs. So we just need to prove that T 1 and T 2 contain the same points with the same multiplicities. Suppose i ∈ found (T). Let Then it is easy to see that i is contained in |N |/2 blocks in each of T 1 and T 2 .
The following is a slight generalization of Theorem 6.3. We omit the proof, which makes use of Lemma 6.5, since it is essentially the same.  Proof. T 1 is a partial Steiner triple system on n points. The famous result of Bryant and Horsley [2] shows that T 1 can be embedded in a 2-(v, 3, 1) design for any v ≡ 1, 3 (mod 6), v ≥ 2n + 1.
is a partial Steiner triple system on n + 1 points, so (again, from [2]) it can be embedded in a 2-(v, 3, 1) design for any v ≡ 1, 3 (mod 6), v ≥ 2(n + 1) + 1. So for v ≡ 1, 3 (mod 6), v ≥ 2n + 3, we have two 2-(v, 3, 1) designs (which we can assume are defined on the same set of points), say (Y, B 1 ) and (Y, 3,2) design that contains all the blocks in We claim that B is not a block in B 1 ∪ B 2 . First, there is a unique block B ∈ B 1 that contains the pair {i, j}, and this block B is one of the blocks in T 1 . Because B ∈ T 1 and B ∈ T 2 , it follows that B = B. Therefore B ∈ B 1 . To see that B ∈ B 2 , we observe that the unique block in B 2 that contains the pair {i, j} is B * = B.
Thus we have shown that (Y, B 1 ∪ B 2 ) satisfies the hypotheses of Theorem 6.6, and the proof is complete.

Strength greater than two
We finish by addressing the ideals of non-trivial t-(v, k, λ) designs with t > 2. For Steiner systems (where λ = 1), we have using Theorem 2.3 and Theorem 4.3. When λ > 1, the lower bound still holds and we may apply Lemma 2.6: since the number of blocks is We also have Corollary 4.7 which tells us that γ 2 (B) ≤ m + 1 when m is the maximum size of the intersection of two distinct blocks.
Let (X, B) be a t-(v, k, λ) design. For i ∈ X, the derived design of (X, B) with respect to i is the ordered pair (Ẋ,Ḃ) whereẊ = X \ {i} anḋ The residual design of (X, B) with respect to i has vertex setẊ and block set {B ∈ B | i ∈ B}. If (X, B) is a t-design, then both its derived design and its residual design are (t − 1)-designs. Lemma 7.1. Let (X, B) be a non-trivial t-design with i ∈ X. With notation as above γ h (Ḃ) ≤ γ h (B) for h = 1, 2. The same inequalities hold for the residual design.
Proof. We handle the case of the derived design; the computations for the residual design are similar. To simplify the notation, we take i = 1. Let I = I(B) and define ideal J as the image of I under the ring homomorphism ϕ : x v ] mapping x 1 to 1 and mapping each x j to itself for j = 2, . . . , v. For g ∈ I writeġ := ϕ(g) ∈ J. For any (k − 1)-set C ⊆ {2, . . . , v}, we haveġ(C) = g(C ∪ {1}) and so, for C ∈Ḃ we haveġ(C) = 0 for allġ ∈ J and, for C ∈Ḃ, there existsġ ∈ J for whichġ(C) = 0. It follows that, if G is a generating set for I, then ϕ(G) is a generating set for J. This shows γ 2 (Ḃ) ≤ γ 2 (B). Next, if g is a non-trivial polynomial in I of smallest degree, thenġ has degree no larger than the degree of g and is also non-trivial since ϕ maps trivial ideal to trivial ideal.
We illustrate this and other results in this paper by recording, in the following table, the exact value of these parameters for the Witt designs and the t-designs appearing as their derived designs. Up to isomorphism, there are unique block designs with parameters 5-(24, 8, 1), 4-(23, 7, 1), 3-(22, 6, 1), 5-(12, 6, 1), 4-(11, 5, 1) and 3-(10, 4, 1). One accessible source of information on the Witt designs is the note [1] by Andries Brouwer. In the large Witt design, with parameters 5- (24,8,1), blocks intersect in 0, 2 or 4 points. So we might start with the zonal polynomials where B ∈ B. We know that the blocks of this design are the supports of the minimum weight codewords in the extended binary Golay code. We may then use the fact that this is a self-dual code to show that these, together with the generators of T , generate our ideal. But we can do better.
(ii) γ 1 (B) = γ 2 (B) = 3. satisfies c ∈ G 24 if and only if the mod 2 dot product c · c is zero for every c ∈ G 24 . Since G 24 is generated by its weight eight codewords, we may say c ∈ G 24 if and only if its inner product with these 759 codewords is zero mod two. Since we only want to recover the codewords of weight eight, we may omit integer inner product six.
Let I = G 0 ∪ {f B,i,j | i, j ∈ B ∈ B} and observe that any element of Z(I) has exactly eight entries equal to one and sixteen entries equal to zero. For c ∈ F 24 2 , let c ∈ R 24 be the corresponding 01-vector with real entries: c j = 1 if c j = 1 and c j = 0 if c j = 0. Assuming c ∈ Z(I), we have f B,i,j (c) = 0 for each B ∈ B and each i, j ∈ B which implies that either c B · c ∈ {2, 4} or c i = 1 ⇔ c j = 1 for all i, j ∈ B. This latter alternative clearly means that either c = c B or c · c B = 0. For the corresponding binary vectors, this implies c · c = 0 for each c ∈ G 24 with Hamming weight eight. As outlined above, this gives c ∈ G 24 and, in turn, c = c B for some B ∈ B. By Propositions 3.2 and 3.1(ii), we have I = I(B).
(iii) the coordinate ring C[x]/I(B) admits a basis consisting of those 23 2 cosets x i x j + I(B) represented by multilinear monomials of degree two.
Proof . Denote by B 1 , B 2 , B 3 , B 4 , B 5 the five blocks containing T := {i, j, k} and set C = B \ T . We know that two distinct blocks of our Witt design intersect in either three points or one point. We now show that h i,j,k (B) = 0 for each B ∈ B. To illustrate the simple arithmetic involved, we write where B = C ∪ T and we retain most of the parentheses here in our evaluation.

case (1)
T ⊆ B. Here, we have B = B for some ∈ {1, . . . , 5} and Here we must have |B ∩ B | = 3 for all and, as B never contains two points from the same "sub-block" C = B \ T , we have

case (3)
|T ∩ B| = 1. In this case, as there are five blocks containing T and |B| = 7, we must have |B ∩ B | = 3 for exactly three values of and, as B contains two points from the same sub-block C = B \ T in each of these three cases, we have

case (3)
T ∩ B = ∅. In this case, as there are five blocks containing T and |B| = 7, we must have |B ∩ C | = 3 for some unique sub-block C = B \ T and B must contain a unique point from each of the other four. In this case, we have On the other hand, if S is a 7-set of points which is not a block, then h i,j,k (S) = 0 for any {i, j, k} ⊆ S. For if T := {i, j, k} is contained in S and h i,j,k (S) = 0, then we have where m = |S ∩ C |. But m 1 + · · · + m 5 = 4 and we see that the only arrangement that achieves the stated equality is where some m = 4. But then S = B and we are done. So, if (ii) γ 1 (B) = γ 2 (B) = 2; For the small Witt designs, we do not have a computer-free proof of our claims. Let us instead describe some generators for the ideals.
First consider the unique Witt design on twelve points. Let (X, B) be the 5-(12, 6, 1) design. For three distinct points i, j and k, let C = X \ {i, j, k}. The twelve blocks containing i, j and k yield a 2-(9, 3, 1) design on the point set C and the four parallel classes of this affine plane may be oriented in a total of sixteen ways (each resulting in a 4-set of directed triples of blocks). We find that certain orientations yield polynomials of degree three which, together with those polynomials in G 0 , generate the ideal I(B). This shows γ 1 (B) = γ 2 (B) = 3.
To be precise, let M 12 be the subgroup of S 12 generated by vanishes on each block of the design. It follows that any image [1 π , 2 π , 3 π ], [4 π , 5 π , 9 π ], [10 π , 11 π , 8 π ] of the three triples of indices under any π ∈ M 12 yields another polynomial in the ideal. It requires a bit more computation, using the Singular computer algebra system [15], to check that the ideal I(B) is generated by these polynomials together with those in G 0 . The relative orderings within the three triples above is important and we do not have an intrinsic description of the permissible orderings that yield vanishing polynomials.
If we take the above computation as correct, we may determine γ 1 and γ 2 for the Witt design on eleven points. If (X, B) now denotes a 4-(11, 5, 1) design, we may use Theorem 2.3 to see that γ 1 (B) ≥ 3. By Lemma 7.1, we have equality, and γ 2 (B) = 3 as well.
Without proof, we note that if (X, B) is the unique 3-(10, 4, 1) design [23, Fig. 9.1], we find γ 1 (B) = γ 2 (B) = 2. In addition to the generators of the trivial ideal, we build certain quadratic generators from any pair B 1 , B 2 of disjoint blocks. In order to explain these generators, we first describe the design.
For the construction in [23], we take X = For each i ∈ B 1 , the two blocks meeting B 1 only in this point partition B 2 into two sets of size two by their intersections. We select one of these two to determine two neighbours of i along an octagon as in Figure 1. Once B 1 and B 2 have been selected and this choice of a pair of neighbours has been made, this determines a quadratic generator for our ideal. We illustrate this with B 1 = {0, 1, 2, 3} and B 2 = {4, 5, 7, 8}. The resulting polynomial is We leave it to the reader to check that the way in which any block intersects this configuration guarantees that g(c B ) = 0 for any B ∈ B. In fact, just five of these polynomials are needed -along with G 0 -to generate the ideal.

Conclusion
We have introduced an algebraic approach to the study of t-designs which builds on existing machinery tied to the space of polynomial functions on blocks. For a design (X, B), we introduced the ideal We determine the exact value of these parameters for symmetric 2-designs and the Witt designs. By constructing many triple systems with γ 2 (B) = k, we indicate that γ 2 (B) can be larger than t. While we expect the value to be more typically close to k, we leave this as an open problem.
One may also investigate the ideal vanishing on the codewords of an error-correcting code. In order to compute γ 1 and γ 2 in such a situation, we have some degree of freedom as these parameters are invariant under affine transformations (provided one is careful with the definition of the trivial ideal). Representing the codewords of a binary linear [n, k, d] code C by ±1 vectors in R n , we see that each dual codeword c = [c 1 , . . . , c n ] corresponds to an element f c (x) = −1 + j x cj j in the ideal of our code. In the linear case, γ 1 (C) is the minimum distance of C ⊥ and γ 2 (C) seems tied to the smallest g such that C ⊥ is generated by its codewords of weight g or less. So it seems interesting to classify those codes C for which γ 1 (C) = γ 2 (C) as these seem related to tight designs.