Universal $p$-ary designs

We investigate $p$-ary $t$-designs which are simultaneously designs for all $t$, which we call universal $p$-ary designs. Null universal designs are well understood due to Gordon James via the representation theory of the symmetric group. We study non-null designs and determine necessary and sufficient conditions on the coefficients for such a design to exist. This allows us to classify all universal designs, up to similarity.


Introduction
We shall briefly recall some definitions and facts from the theory of designs, more details can be found in [1].
We call µ t the coefficient of the design and if µ t = 0 then we say u is a nulldesign. We say thatû is induced from u and we shall denote its restriction to sets of size j by jû .
Similarly a p-ary t-design is a function u : [v] s → F p such thatĉ is constant on sets of size t. It is well known that integral t-design is also an integral jdesign for all integers 0 ≤ j ≤ t, however this is not true over fields of positive characteristic. If a p-ary design, u, of constant block size, s, is a j design for all j < s then we say that u is a universal design for the partition (v − s, s). Graver and Jurkat [2] determined when universal integral designs exist. Theorem 2. [2] Let t ≤ s ≤ v be integers. There exists a universal integral design for (v−s, s) with coefficients µ j if and only if µ j+1 = s−j v−j µ j for 0 ≤ j < t. 1 The goal of this paper is to prove the equivalent result for p-ary designs and to describe the resulting universal designs, up to similarity. Null universal designs are well understood via a James' kernel intersection theorem, which was proved in the context of the representation theory of the symmetric group, but is re-stated in the language of designs as Theorem 9. The existence, or otherwise, of non-null universal designs depends on the partition. We shall conclude our introduction with a number of definitions required to state the main result, and also some facts on divisibility of binomial coefficients.
Definition 3. Let u and v be universal p-ary designs for (a, b) with coefficents µ j and γ j respectively . We say u and v are similar if there is some k ∈ F p such that µ j = kγ j .
We shall now state some well known results on the divisibility of binomial coefficients, as many of the results in the theory of p-ary designs involve determining whether certain binomial coefficients are 0 (mod p) or not.
Let a = α i=0 a i p i be the base p expansion of a; that is 0 ≤ a i ≤ p − 1 and a α = 0. The p-adic valuation val p (a) is the least i such that a i is non-zero, we call α the p-adic length of a and write l p (a) = α.
Lemma 5. Let p be a prime and a, b ∈ N, then val p ( a+b b ), the highest power of p that divides a+b b , is the number of carries that occurs when a and b are added in their base p expansions.
In particular, a b ≡ 0 (mod p) if and only if some a i < b i . Remark. This gives an alternative characterisation of a James partition, in particular λ = (a, b) is James if and only if a ≡ −1 (mod p lp(b) ), or equivalently l p (b) < val p (a + 1) for all i < r.
We can now state our main result: Theorem 8. Let a, b ∈ N, with a ≥ b and let u be a non-null universal p-ary design for (a, b). If (a, b) is neither pointed or James, then u is similar to the constant design. If (a, b) is James then u is unique up to similarity, while if (a, p β +b) is pointed then u = u ′ + c where u ′ is non-null only as ab-design, while c is similar to the constant design.

Uniqueness of p-ary Designs
Universal null p-ary designs are well understood, due to the work of James on the representation theory of the symmetric group. James' well-known kernel intersection theorem gives a characterisation of the Specht module S (a,b) as the collection of all null universal p-ary designs for (a, b) [4].
Then u is a null p-ary design for (v − b, b). Moreover any null p-ary design for is a linear combination of designs of this form.
Non-null designs also play an important role in the representation theory of the symmetric group, as they determine certain non-split extensions of Specht modules, investigated by the author in [5]. Theorem 2 describes the relationship between the coefficients of integral designs, and a similar analysis determines when a p-ary t-design is also a j-design.
The inclusion matrix, matrix whose rows are indexed by subsets of [v] := {1, 2, . . . , v} of size i and whose columns are indexed by subsets of [v] of size b. The entry corresponding to position X, Y is 1 if X ⊆ Y and 0 otherwise. Gottlieb showed matrix is of full rank over fields of characteristic 0 [3]. If u is an integral design of block size for (v − b, b), then considering u as a vector of length v b , we see that where 1 i is the vector of length v i consisting of 1's. It is clear that proving the necessity of the conditions in Theorem 2.
Remark. Wilson [6] showed that there are examples of t-designs which are not j designs whenever b−j t−j ≡ 0 (mod p), which is very different to the behaviour of integral designs.
In light of this result, to check a design is universal it suffices to check that it is a b − p l -design for all l ≤ l p (b).
for any j such that the sum (b − j − p l ) + p l has no carries in p-ary notation, by Lemma 5. This is precisely those j for which the coefficient of p l in the p-ary expansion of b − j, which we shall denote (b − j) l , is non zero. If j < b, then some (b − j) l = 0, and as u is a (b − p l )-design u is also a j-design by Proposition 10.
Wilson has determined when non-null p-ary t-designs exist.
Corollary 13. There are non-null p-ary If a l ≡ −1 (mod p) and b > p l+1 then setting j = b − p l we see that non-null designs can not exist. On the other hand if b ≤ p l+1 then b−j necessarily has a carry in p-ary notation, so a+b−j a+p l ≡ 0 (mod p) by Lemma 5. 4 Combining this with the relationship between coefficients, established in Proposition 10, we obtain more integers j for which a universal design for (a, b) is null.
Proposition 14 . If a universal design, u, for (a, b) is non-null as a j-design, then (b − j) m + a m < p for all m < l p (b).
Proof. Suppose u is non-null as a j-design with coefficient µ j , and let m < l p (b) be such that (b − j) m = 0. As u is non-null for j, we must have u is non-null for b − p m , by Proposition 10, as For u to be non-null as a j-design, we must have a+b−j b−p m −j = 0. Corollary 13 ensures that a m ≡ −1 (mod p) and thus (a + p m ) + (b − j − p m ) having no carries is equivalent to (a) + (b − j) having no carries. Using Lemma 5 we see that if u is non-null then (a) + (b − j) has no carries, and therefore (b − j) m + a m < p for all m < l p (b).
Our next goal is to determine what the relationship is between the non-zero coefficients of a universal design. Let u be a universal design for (a, b), and let X be the set of all j with (b − j) m + a m < p for all m < l p (b). Observe if j / ∈ X then u must be a null j-design, and so X contains all j such that u is a non-null j-design. We shall define a partial ordering on X by setting i ≥ X j if i > j and b−j i−j ≡ 0 (mod p). If i ≥ X j and µ i and µ j are the coefficients of u corresponding to i and j respectively, then we have a relationship between the coefficients appearing in the same connected component of X.
Proposition 15. If λ = (a, b) is James, then X has a single connected component.
Proposition 16. If λ = (a, b) is not James, and b = αp β +b then X has a single connected component, unless λ is pointed, in which case X has two connected components, one of which consists only of the elementb.
Proof. Observe that i, j ∈ X are comparable if and only if These may fail to be in X as it may be that (b − x) > b or b − y = 0. If, however, (b − i) m and (b − j) m are both non-zero for some m then i ∧ j ∈ X. Let x be such that (b − x) m = p − 1 − a m for m < β and (b − x) β = α − 1, and observe that x ∈ X by Proposition 14. Clearly j ∈ X with j >b is comparable to x. If j <b ∈ X, or if j =b and α = 1 then x ∧ j ∈ X.
It only remains to consider the case where j =b and α = 1, which, if b > p valp(a+1) is clearly comparable tob − p valp(a+1) , which is in the same component as x. It follows that if λ is not pointed then there is only one connected component of X.
On the other hand, when λ is pointedb is not comparable to any other element and thus is in a connected component of its own. This is as no j <b is in X as no j <b has (b − j) m = 0 for all m < l p (b) where a m ≡ −1 mod p. Similarly no j >b has (b − j) β ≥ 1, so j andb are incomparable .
Proof of uniqueness in Theorem 8. If u is a universal design for (a, b), then its coefficients are entirely determined by the connected components of X, thus an understanding of this poset allows us to determine the possible coefficients of designs. If (a, b) is not pointed, then non-null universal designs, if they exist, are unique up to similarity, while if (a, b) is pointed, then any design must be the sum of two designs, uniquely determined by its coefficients on each of the two connected components of X.

Existence of designs
In the previous section we have seen a complete characterisation of universal null p-ary designs and described, up to similarity, the uniqueness of non-null universal designs. We now move to considering the existence of non-null designs for (a, b). We first consider p-ary designs which come from the mod p reduction of integral designs. Clearly the constant design, c (a,b) , is an integral design, with coefficients µ i = a+b−i b−i , and therefore is null if and only if (a, b) is James. Proposition 17. Let (a, b) be a partition which is neither James nor pointed. Then the constant design is the unique, up to similarity, universal p-ary design for (a, b). λ = (a, b), then there exists an integral design which is not similar to the constant design if and only if λ is James.

Proposition 18. Let
Proof. Any integral design must have coefficients satisfying the conditions of Theorem 2, µ s+1 = b−s a+b−s µ s for 0 ≤ s < t. This means that To ensure that some µ i ≡ 0 (mod p) we must take µ s = k and so if p d is a unit in F p , that is if d = 0, then We have seen that if (a, b) is pointed then the constant design is non-null. We shall now construct another non-null design for (a, b) which is not similar to the constant design, completing the classification.
Proposition 20. Let (a, b) be such that b = p β and val p (a + 1) < β. Then there exists a universal p-ary design which is null as a t-design for all t > 0 and non-null as a 0-design. are all divisible by p, by Lemma 7. Then u is a universal design which is non-null only as a 0-design.
Let u be the design constructed above for the partition (a, p β ). We shall modify u to construct a design for a pointed partition (a, p β +b), whereb < p valp(a+1) < p β , which is non-zero only as ab-design.
Let u : [v] p β → F p be the design constructed above for the partition (a, p β ) ⊢ v. We shall modify u to construct a design for a pointed partition (a, p β +b), whereb < p val p (a+1) < p β , which is non-zero only as ab-design. Let Y = {a + p β + 1, . . . , a + b}, then Y is a set of sizeb.
Given a subset X ⊂ [v] we denote by δ X : [v] |X| the indicator function; that is Of course, these functions may not be designs, but we may consider the functions they induce on subsets of [v] as before. Consider j u Y : [v] j → F p , by grouping terms by the size of their intersection with Y . First, consider the case wherê b < j < b: Each of these terms is 0, by our choice of u, so jûY is 0. Similarly for j ≤b where µ 0 = 0 is the coefficient of u as a 0-design. Observe that if Y is any subset of [a+b], not necessarily {a+p β +1, . . . , a+b}, then we may define u Y as before, by first defining u on subsets of [a + b]\Y of size p β .
which is 0 when restricted to sets of size j ifb < j < b. When j ≤b, If U is a non null p-aryb-design of block size b − 1 and coefficient α then setting where the sum is over all sets X of size b − 1 and U(X) is the coefficient of X in theb-design U, we seeb and of course j u U = 0 for all j =b.
Theorem 21. Let λ = (a, b) be such that b = p β +b andb < p valp(a+1) < b. Then there is a universal design which is non-null only as ab-design.
Proof. An element of the form u U as described above is such a design, it remains to prove such an element exists; that is that there is a non null p-aryb-design Existence of p-ary designs. If (a, b) is James, then the construction of Graver and Jurkat [2] gives rise to a non-null design. If (a, b) is not James then the constant design is non-null. If (a, b) is pointed then Theorem 21 gives a nonnull universal design. Completing the proof of Theorem 8, which we state again below to conclude.
Theorem. Let a, b ∈ N, with a ≥ b and let u be a non-null universal p-ary design for (a, b). If (a, b) is neither pointed or James, then u is similar to the constant design. If (a, b) is James then u is unique up to similarity, while if (a, p β +b) is pointed then u = u ′ + c where u ′ is non-null only as ab-design, while c is similar to the constant design.