The lattice of N-run orthogonal arrays

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Abstract

If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the “expansive replacement” construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N-run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N, we investigate the height and the size of the lattice. It is shown that the height is at most ⌊c(N−1)⌋, where c=1.4039…, and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N). Using a new construction based on “mixed spreads”, all parameter sets with 64 runs are determined. Four of these 64-run orthogonal arrays appear to be new.

Introduction

Although mixed-level (or asymmetrical) orthogonal arrays have been the subject of a number of papers in recent years (see Hedayat et al., 1999, Chapter 9, for references), it seems fair to say that we know much less about them than about fixed-level orthogonal arrays (in which all factors have the same number of levels). For example, there is no analogue for mixed orthogonal arrays of one of the most powerful construction methods for fixed-level arrays, that based on linear codes (see Hedayat et al., 1999, Chapters 4 and 5).

Again, there are many instances where the linear programming bound for fixed-level orthogonal arrays gives the correct answer for the minimal number of runs needed for a specified number of factors. There is a linear programming bound for mixed arrays (Sloane and Stufken, 1996), but it is less effective than in the fixed-level case—it ignores too much of the combinatorial nature of the problem (especially when the levels involve more than one prime number), and, though generally stronger than the Rao bound, does not give correct answers as often as in the fixed-level case.

A mixed orthogonal array OA(N,s1k1s2k2svkv,t) is an array of size N×k, where k=k1+k2+⋯+kv is the total number of factors, in which the first k1 columns have symbols from {0,1,…,s1−1}, the next k2 columns have symbols from {0,1,…,s2−1}, and so on, with the property that in any N×t subarray every possible t-tuple of symbols occurs an equal number of times as a row. We usually assume 2⩽s1<s2<⋯ and all ki⩾1. Except in Section 5, only arrays of strength 2 will be considered, and we will usually omit t from the symbol for the array.

We refer to (N,s1k1s2k2⋯) as the parameter set for an OA(N,s1k1s2k2⋯). We also allow the parameter set (N,11), corresponding to the trivial array consisting of a single column of N 0's. In this paper we consider the question: if N is specified, how many different parameter sets are possible?

Given an array A=OA(N,s1k1s2k2⋯), other N-run arrays can be obtained from it by the expansive replacement method. Let S be one of the si occurring in A, and suppose B is an OA(S,t1l1t2l2⋯). The expansive replacement method replaces a single column of A at S levels by the rows of B. For example, if A=OA(16,2344) and B=OA(4,23), we obtain an OA(16,2643). If B is a trivial array OA(S,11), we are simply deleting one of the S-level factors from A. E.g. taking S=2, an OA(24,22041) trivially produces an OA(24,21941). The expansive replacement method also includes replacing a factor at s levels by a factor at s′ levels, if s′ divides s. For further details about the expansive replacement method see Hedayat et al. (1999, Chapter 9).

Let A and B be parameter sets for orthogonal arrays with N runs. We say that B is dominated by A if an orthogonal array with parameter set B can be obtained from an orthogonal array with parameter set A by a sequence of expansive replacements.

Using “dominance” as the relation, the parameter sets for orthogonal arrays with N runs form a partially ordered set, which we denote by ΛN (Hedayat et al., 1999, p. 335).

ΛN has a unique maximal element (N,N1) (corresponding to the trivial array with one factor at N levels) and a unique minimal element (N,11). It is straightforward to verify that meet (∧) and join (∨) are well defined for this relation (we omit the proof), so ΛN is in fact a lattice (cf. Welsh, 1976; Trotter, 1995).

If an OA(N,s1k1s2k2⋯) exists, then necessarily we must have:N−1⩾k1(s1−1)+k2(s2−1)+⋯,

(C5) the linear programming bound holds (see Sloane and Stufken, 1996).

These conditions are certainly not sufficient for an array to exist, and it appears to be difficult to test if an orthogonal array does exist with a putative parameter set satisfying (C1)–(C5). A further difficulty is that in order to construct ΛN it is necessary to know Λd for all proper divisors d of N.

To avoid these difficulties we define a second lattice, the idealized lattice ΛN: this has as nodes all putative parameter sets satisfying (C1)–(C4), with the dominance relation as before, except that in the expansive replacement method we may now make use of any of the nodes of any Λd′ for d dividing N.

Constructing ΛN′ is much easier than constructing ΛN, since essentially all we need to do is enumerate the solutions to (1). Of course ΛN is a sublattice of ΛN′.

To avoid having to repeat the adjective “putative”, from now on we will use “parameter set” to mean any symbol (N,s1k1s2k2⋯) satisfying conditions (C1)–(C4). The parameter sets are precisely the nodes of ΛN′. If a parameter set is also a node of ΛN then it is implied that an OA(N,s1k1s2k2⋯) does exist, i.e. that the parameter set is realized by an orthogonal array.

It is convenient to represent ΛN and ΛN by their Hasse diagrams (cf. Welsh, 1976, p. 45). These diagrams are drawn “from the bottom up”, with (N,11) as the root node at the bottom (Fig. 1 shows Λ12 and Λ12′). The height of a parameter set is the number of edges in the longest path from that node to the root. A node of height i appears on the ith level of the diagram. The height of the maximal element (N,N1) will be denoted by ht(N).

The atoms in ΛN (those nodes just above the root) are precisely the parameter sets (N,p1) for the primes p dividing N.

The dual atoms in ΛN (those nodes just below the maximal element) are especially interesting, since they dominate all other parameter sets.

We can now state our main results.

Theorem 1

(i) For all N,ht(N)⩽⌊c(N−1)⌋,wherec=i=0122i−1=1.4039….

(ii) If N=22m(m⩾0) then ht(N)=⌊c(N−1)⌋.

Let T(N) (resp. T′(N)) denote the total number of nodes in ΛN (resp. ΛN′).

Theorem 2

If N=2n,14(log2N)2(1+o(1))⩽log2T(N)⩽38(log2N)2(1+o(1)).

Theorem 3

There is a constant c1 such that for all N,lnlnT(N)⩽c1lnNlnlnN(1+o(1)).

Remark

(i) The bounds in , also apply to T′(N).

(ii) Theorem 2 shows that when N=2n, T(N) grows very roughly like Nalog2N, for some constant a between 14 and 38. This is a “superpolynomial” function of N, meaning that it grows faster than any polynomial in N.

(iii) It appears (although we have not proved this) that the upper bound in (5) can be achieved by taking N to be a certain product of powers of the first m primes, where m is about12elnNlnlnN(see Section 7). In other words, it appears that there is an infinite sequence of values of N for which T(N) grows very roughly likeexp(Nc2/lnlnN),where c2 is a constant. This is again a superpolynomial function of N, and is now close to being an exponential function, since lnlnN grows slowly.

The above discussion has shown that there is an infinite sequence of values of N for which the number of nodes in ΛN grows superpolynomially, while the height of ΛN grows at most linearly. It follows that the size of the largest antichain must also grow superpolynomially. The data in Table 3 suggest the following conjecture.

Conjecture

There is an infinite sequence of values of N for which the number of dual atoms grows superpolynomially in N.

In fact it seems likely that if N=2n, a lower bound of the form in (4) (possibly with a different constant) applies to the logarithm of the number of dual atoms, and that for some sequence of values of N a lower bound similar to the upper bound on the right-hand side of (5) will hold. However, at present these are only conjectures.

In order to construct the orthogonal arrays needed to establish the lower bound in Theorem 2 we make use of what we call “mixed spreads”, generalizing the notions of “spread” and “partial spread” from projective geometry. Arrays that can be constructed in this way we call “geometric”. Many familiar examples of orthogonal arrays, for example arrays constructed from linear codes, are geometric. The construction is not restricted to strength 2 (and is one of the few general constructions we know of for mixed arrays of strength greater than 2). The construction will be described in Section 5.

In Section 6 we use this construction to determine the lattice Λ64, and in doing so we find tight arrays with parameter sets(64,2541781),(64,41483),(64,2541084),(64,4786),which appear to be new.

When studying parameter sets of putative orthogonal arrays with N runs, it is convenient to be able to say that if the number of degrees of freedom of the parameter set (N,s1k1s2k2⋯), that is,k1(s1−1)+k2(s2−1)+⋯is small compared with N−1, then an orthogonal array certainly exists.

To make this precise, we define the threshold function B(N) to be the maximum number b such that every parameter set (satisfying conditions (C1)–(C4)) with at most b degrees of freedom is realized by an orthogonal array, but some parameter set (again satisfying (C1)–(C4)) with b+1 degrees of freedom is not realized. If every parameter set satisfying (C1)–(C4) is realized, we set B(N)=N−1.

Fig. 1 shows that B(12)=6, since there is no OA(12,2531), but every parameter set with at most six degrees of freedom is realized.

We are not aware of any earlier investigations of B(N).

Theorem 4

If N is a power of a prime thenN3/4⩽B(N).

In words, if the number of degrees of freedom in the parameter set does not exceed N3/4, then an orthogonal array exists. This is certainly weak, but is enough to establish the lower bound of Theorem 2. It would be nice to have more precise estimates for B(N).

A final remark: We could have considered the partially ordered set whose nodes are all the inequivalent orthogonal arrays with N runs, rather than just their parameter sets. However, the number of nodes then becomes unmanageably large, even for small values of N (furthermore, it appears that “meet” and “join” are no longer well defined, and so in general this partially ordered set would not be a lattice).

Consider N=28, for example. Using Kimura 1994a, Kimura 1994b enumeration of the Hadamard matrices of order 28, we have calculated1 that there are precisely 7570 inequivalent OA(28,227)'s. This would be merely a lower bound on the number of dual atoms. On the other hand we know (see Table 1) that Λ28 has precisely four dual atoms, between 47 and 55 nodes, and height 28.

Section snippets

Examples of the lattices ΛN and ΛN

There are a few general cases when we can describe ΛN explicitly (and for which ΛN′ is the same as ΛN).

If N=p is a prime then ΛN=ΛN′ has two nodes, one dual atom and height 1, as shown in Fig. 2(a) (dual atoms are circled).

If N=pq is the product of two distinct primes, ΛN=ΛN′ has five nodes, one dual atom and height 3. Λ6 is shown in Fig. 2(b).

More generally, if N is the product of u⩾2 distinct primes, it is not difficult to show that ΛN=ΛN′ has 2u−1−1 dual atoms, height 2u−1, and βu+1 nodes,

The maximum height of ΛN

In this section we give the proof of Theorem 1.

Let σ denote a specification s1k1s2k2⋯ of factors at various levels, leaving the number of runs unspecified. Given σ, there is a smallest number of runs, N0 say, for which an OA(N0,σ) exists. Let h be the height of the parameter set (N0,σ) in ΛN0. Then if the parameter set (N,σ) occurs in any other lattice ΛN, it also has height h. (E.g. the specification σ=61 has height 3 in each of Fig. 1, Fig. 2, Fig. 3.) We may therefore define ht(σ) to be h,

Upper bounds on the number of parameter sets

In this section we establish the upper bounds in Theorem 2, Theorem 3. We will bound T′(N), the number of nodes in ΛN′. Since ΛN is a sublattice of ΛN′, this is also an upper bound on the number of nodes in ΛN. Suppose first that N=22r.

We start by considering parameter sets (N,2k14k28k3⋯(2r)kr), containing no level exceeding N. From (1),N−1⩾k1+3k2+7k3+⋯+(2r−1)kr.Let γ be the number of nonnegative integer solutions (k1,k2,…,kr) to this inequality. Then γ/(N−1)r is the Riemann sum approximating

Geometric orthogonal arrays

We consider subspaces V of the vector space GF(q)n over GF(q), where q is a power of a prime. By the dimension of V, dimV, we mean the vector space dimension over GF(q) (rather than the projective dimension, which is one less). The following notion was suggested by the notions of spread and partial spread in projective geometry (cf. Thas, 1995).

Definition

A mixed spread of strength t is a collection V={V1,V2,…,Vk} of subspaces of GF(q)n such that for all choices of τt indices i1,i2,…,iτ (with 1⩽i1<i2<⋯<iτ

If the number of runs is a power of 2

In this section we consider the case N=2n, n=1,2,…. We have already discussed ht(N) in Section 3 (see Table 2). With the assistance of Michele Colgan, we used a computer to determine the number of dual atoms A′(N) and the total number of nodes T′(N) in the idealized lattice ΛN′ for n⩽9. The results are shown in the second and third columns of Table 3. Note, in particular, the extremely rapid growth from N=256 to 512. We regard this as convincing evidence that when N=2n, A′(N) (and therefore

The existence of orthogonal arrays with certain parameter sets

In this section we prove Theorem 4, the lower bound in Theorem 2, and also give some other conditions which are sufficient to guarantee that a parameter set can be realized by an orthogonal array.

Lemma 14

Suppose N=pm is a power of a prime and (N,s1k1s2k2⋯) is a parameter set with k=Σiki factors. If kp⌊(m+1)/2⌋+1 then this parameter set is realized by a geometric orthogonal array.

Proof

Suppose first that the parameter set contains a factor with s=pn>N levels. If m is even then a geometric OA(pm,(pmn)pn(pn)1

Acknowledgements

We thank Michele Colgan for computing the properties of the lattices ΛN′ shown in Table 3. The research of John Stufken was supported by NSF grant DMS-9803684.

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