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Linear programming bounds for balanced arrays
Journal of Statistical Planning and Inference 137 (2007) 324 – 330 www.elsevier.com/locate/jspi
Linear programming bounds for balanced arrays夡 Peter Dukesa,∗ , Alan C.H. Lingb a Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W3P4 b Department of Computer Science, University of Vermont, Burlington, VT 05405, USA
Received 6 July 2004; received in revised form 6 June 2005; accepted 9 November 2005 Available online 18 January 2006
Abstract In this paper, we apply Delsarte’s linear programming bound to balanced arrays, a key ingredient in certain constructions of fractional factorial designs. Examples are given to illustrate the power and generality of this technique compared with previous applications of the method of moments. © 2006 Elsevier B.V. All rights reserved. MSC: Primary 05B15; secondary 05E30 Keywords: Balanced arrays; Delsarte’s inequalities; Linear programming; Fractional factorial designs
1. Introduction An N × k balanced array (BA) of strength t (over a binary alphabet) with index vector µ = (0 , . . . , t ) is an array with entriesin {0, 1} such that, in any t columns of the array, every word of weight i appears in exactly i rows. Clearly, N = ti=0 ti i . When i = for all i, such an object is called an orthogonal array (OA) of index . A BA or an OA is simple if it has no two identical rows. It should be noted that both BAs and OAs have generalizations to alphabets other than {0, 1}. While the results we discuss for BAs can be modified for larger alphabets, the binary case has attracted nearly full attention in past literature. Moreover, there are at least two natural ways to generalize the notion of weight over the alphabet Zq : the number of nonzero entries in a word and the “composition” of entries in a word. For these reasons, we focus on the binary case. Balanced arrays are of particular interest in combinatorics and statistics. For instance, applications to factorial designs are discussed in Aggarwal and Singh (1981), Kuwada and Nishii (1979), and Shirakura and Ohnishi (1985). Clearly, a BA with any index vector always exists for k = t; we simply take i copies of every vector of length t and weight i as rows. Such arrays we call trivial. More generally, a BA with k columns implies the existence of a BA with k columns for t k k. Given a binary word w1 w2 · · · wk , and indices 1i1 < i2 < · · · < is k, we say that wi1 wi2 · · · wis is a sub-word contained in w1 w2 · · · wk . Given a BA with parameters t, k, µ, we define for 0 s t the quantities s,i to be the 夡 Research of the first author is supported by an NSERC postdoctoral fellowship. Research of the second author is supported by the Army Research Office (USA) under grant number DAAD 19-01-1-0406 (Ling). ∗ Corresponding author. Tel.: +1 647 223 4338. E-mail addresses: [email protected] (P. Dukes), [email protected] (A.C.H. Ling).
0378-3758/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2005.11.007
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325
number of rows which contain a given sub-word of length s and weight i. It is clear that s,i
t−s t −s i+l = l l=0
and in particular t,i = i for all i. Note that for an OA with index , s,i = 2t−s for each i. Let Nh denote the number of rows of weight h in our BA. By counting in two ways the number of ordered pairs (R, S), where R is a row of the BA having a sub-word S of length s and weight j, we have the following moment equations. Lemma 1.1. For 0 j s t, k h k−h h=0
j
s−j
Nh =
k s s,j . s j
It is essentially these equations (in Nh ) which have been exploited in recent work (Chopra, 1990, 1991, 1996, 1998; Chopra and Dios, 1997, 2003; Dios, 2002; Dios and Chopra, 2002) to determine necessary conditions on the existence of BAs of certain strengths between 4 and 8. In the Section 2, we apply Delsarte’s inequalities (Delsarte, 1973) as a general condition for arbitrary strength t. Although this technique does not necessarily subsume the “method of moments”, we have found that in many examples it does. Comparisons are given in Section 3.
2. Delsarte’s linear programming bound We refer the reader to Chapter 30 of van Lint and Wilson (1992) for a nice introduction to the theory of association schemes. A very brief outline is given here for reference and notation. A k-class association scheme on a set X consists of k + 1 nonempty symmetric binary relations R0 , . . . , Rk which partition X × X, where R0 is the identity relation {(x, x) : x ∈ X}, and such that for any x, y ∈ X with (x, y) ∈ Rh , the number of z ∈ X such that (x, z) ∈ Ri and (z, y) ∈ Rj depends only on the indices h, i, j . These quantities are h . Let |X| = n. For h = 0, . . . , k, define the n × n adjacency matrix A , indexed by entries of X, to usually denoted pij h have ij-entry 1 if (x, y) ∈ Rh , and 0 otherwise. Since the Rh partition X × X, it follows that A0 + · · · + Ak = J , the all ones matrix. h , the matrices {A } form a basis for an algebra with respect to entrywise multiplication, called By definition of the pij h the Bose-Mesner algebra. There is also a basis E0 , . . . , Ek with respect to ordinary matrix multiplication such that (1) E0 + · · · + Ek = I ; (2) (Ei )2 = Ei for all i; and (3) Ei Ej = O for i = j . The second eigenmatrix Q is (k + 1) × (k + 1) and has entries determined by nE i = kj =0 Qij Aj . In the Hamming scheme H (k, q), the set X consists of all q-ary words of length k, and (x, y) ∈ Ri if and only if the Hamming distance (the number of disagreeing entries) between x and y is precisely i. We focus here on the binary case q = 2. For H (k, 2), the entries of the second eigenmatrix Q are given by (see Delsarte, 1973, for details) Qj i =
k h=0
(−2)h
k−h j −h
i . h
(2.1)
Note that even though the matrices Ah and Eh are symmetric for each h = 0, . . . , k, the matrix Q is not in general symmetric.
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For example, consider H (3, 2). The set X consists of all eight binary vectors of length 3, and the relations Rh contain those pairs of words at Hamming distance h. From (2.1), ⎛
1 ⎜3 Qj i = ⎝ 3 1
⎞ 1 −3 ⎟ ⎠. 3 −1
1 1 1 −1 −1 −1 −1 1
The 8 × 8 adjacency matrices A0 = I, A1 , A2 , A3 are easy to generate, and the four matrices E0 , E1 , E2 , E3 are computable from these and the entries of Q above. Given a multiset Y of points of an association scheme, the distribution vector of Y is defined to be a = (a0 , . . . , ak ), where ai = |(Y × Y ) ∩ Ri |. Note that a0 |Y |, with equality if and only if Y has no repeated elements. Delsarte’s inequalities state that aQ0 for any distribution vector a in an association scheme with second eigenmatrix Q. Since the rows of a BA with k columns are simply words in the Hamming scheme H (k, 2), we have the following specialization of Delsarte’s inequalities from (2.1). Lemma 2.1. Suppose Y is the collection of rows of a binary BA with k columns and distribution vector a. Then for j = 0, . . . , k, k
(−2)
h
h=0
k k−h i a 0. j −h h i i=0
Unfortunately, Lemma 2.1 is not very helpful until we know more about the distribution vector a. Lemma 2.2. Suppose Y is the collection of rows of a binary BA with k columns, strength t, and index vector µ = (0 , . . . , t ). Define a = (a0 , . . . , ak ) to be the distribution vector of Y. Then for s = 0, . . . , t, k k−i i=0
s
ai =
s s k 2s,j . j s
(2.2)
j =0
Proof. Fix r ∈ Y , say of weight h, and denote by i the number of rows which disagree with r in exactly i positions. By counting in two ways the number of ordered pairs (R, S) where S is a sub-word of length s of both R and r ∈ Y , we have k k−i i=0
s
i =
s h k−h j =0
j
s−j
s,j .
Now summing over all r ∈ Y , we have k k−i i=0
s
ai =
k h=0
=
s j =0
Nh
s h k−h j =0
s,j
j
s−j
s,j
k h k−h h=0
j
s−j
Nh
and the result follows by using Lemma 1.1 on the inner sum.
In the special case that Y is the collection of rows of an OA with index , the right side of (2.2) reduces to ks 2 22t−s . To decrease the number of inequalities and variables under consideration, we now substitute the equations of Lemma 2.2 into the inequalities of Lemma 2.1.
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327
Theorem 2.3. For j = t + 1, . . . , k, (−1)
j
k
(−2)
s
s=t+1
k−s j −s
k−t−1
k−i s
i=0
ai (−1)
j +1
t s k s s j 2s,i . (−2) j s i s=0
i=0
Proof. Rewriting Delsarte’s inequalities, we have for any j = 0, . . . , k, 0 = =
k h=0 k
(−2)h (−2)h
h=0 k k
k k−h i a j −h h i i=h
k−h j −h
(−2)
h
s=0 h=s
k h
(−1)s
i=0 s=0
k−s h−s
k−i s
ai
k k−h k−i s k−s (−1) ai . j −h h−s s i=0
Now split this sum into S1 + S2 , where S1 is the sum over 0 s t and S2 is the sum over t < s k. For S1 , we apply Lemma 2.2 to get S1 =
t k
h
(−2) (−1)
s
s=0 h=s s t
k−h j −h
k−s h−s
s k s 2s,i s i i=0
k s j h k 2s,i = (−1)s (−2)h i h s j s=0 i=0 h=s s t k j s 2s,i . (−2)s = (−1)j j s i s=0 i=0
The expression S2 can be simplified as S2 =
k−t−1 i=0
= (−1)
j
ai
k
s=t+1 k−t−1
ai
i=0
(−1)
s
k
k−i s
(−2)
s
k
s=t+1
h=s
k−i s
(−2)
h
k−h j −h
k−s j −s
k−s h−s
.
Remark. In simplifying sums above, we have repeatedly used the identities − − +−1 = , = (−1) − and + = . i −i i=0
In the cases k = t + 1 and t + 2, we get the following lower bounds on a0 , the sum of the squares of frequencies of each possible word. Corollary 2.4. In a BA with index vector µ and k = t + 1, s t (−1)t t +1 s 2s,i . (−2)s a0 t+1 s i 2 s=0
i=0
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Proof. This is the case k = j = t + 1 in Theorem 2.3.
Corollary 2.5. In a BA with index vector µ and k = t + 2, t s (−1)t t +1 t +2 s − 2s,i . a0 t+2 (−2)s (t + 2) s s i 2 s=0
i=0
Proof. When k = t + 2, the two inequalities in Theorem 2.3 are t s s t+1 t s t +1 2 ((t + 2)a0 + a1 ) (−1) (t + 2) 2s,i (−2) s i s=0
(2.3)
i=0
and 2
t+1
(ta 0 + a1 )(−1)
t
t s=0
(−2)
s
s t +2 s 2s,i . s i
(2.4)
i=0
Subtracting (2.4) from (2.3) gives the required bound on a0 .
For OAs with index , recall that the s,i = 2t−s are independent of i. It is an easy exercise to check that in this case Corollaries 2.4 and 2.5 reduce to 2a0 /N and 4a0 /N , respectively. When the array is simple (a0 = N ), these claims are obvious, since must be bounded by 2k−t . However, when the array is not simple, these results say more than just that is bounded by 2k−t times the maximum replication of any row. Certain other necessary conditions on the existence of BAs when k is close to t appear in Srivastava (1972). 3. Examples In general, for a fixed k, t, and index vector µ, the equations in Lemma 2.2, inequalities in Theorem 2.3, and the conditions ai 0 for i = 1, . . . , k give linear programming bounds (upper and lower) on a0 /N . There are k − t variables a0 , . . . , ak−t−1 and less than 2k constraints in this LP problem. In the examples to follow, we use these bounds in conjunction with the following easy necessary conditions on a0 /N . Lemma 3.1. In a BA with every row repeated at most r times, 1 a0 /N r. Moreover, the index vector forbids any row to be repeated too many times. Lemma 3.2. Suppose there is an r-fold row of weight h in a BA of strength t and index vector µ with k columns. Then i r for i = max(0, h − (k − t)), . . . , min(t, h). Proof. Let R be the hypothesized r-fold row. If h, k − h k − t, then there are k − t + 1 possible weights of subwords of R of length t, namely h − (k − t), . . . , h. If h < k − t then, there is a length t sub-word of R of any weight i = 0, . . . , min(t, h). Similarly, if h > t we have a length t sub-word of R of any weight i = max(0, h − (k − t)), . . . , t. Each of these subwords appears in at least r rows, so i r for these values of i. We now revisit the upper bounds on k in past examples. In most cases, linear programming problems were solved computationally. These results were instantaneous on a personal computer. • t = 4, µ = (1, 2, 5, 2, 1): k 8 shown in Chopra (1990). Let k = 7. By Lemma 3.2, any BA with these parameters must be simple. Thus a0 = N = 48. Solving the equations in Lemma 2.2, we can rewrite the condition a6 0 as 24a1 + 5a2 6440.
(3.1)
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The inequalities resulting from j = 6, 7 in Theorem 2.3 are
• •
• •
•
•
5a1 + a2 1323,
(3.2)
4a1 + a2 1084.
(3.3)
Some easy arithmetic shows a1 175 (from (3.1), (3.2)) and a2 64 (from (3.1), (3.3)). But these constraints contradict (3.1). Thus, k 6 for this index vector. t = 5, µ = (1, 0, 1, 1, 1, 1): k 6 shown in Chopra (1998). Certainly any BA with these parameters must be simple and has a0 = N = 27 rows. Suppose k = 7. The inequality (2.4) gives 5a0 + a1 94.27, which contradicts a1 0. Thus k 6 is necessary for this index vector. t = 5, µ = (2, 0, 0, 0, 1, 2): k 7 shown in Dios (2002). While Delsarte’s inequalities can be used successfully, we can actually argue directly that k 7 for this index vector. First, observe that any such array has nine rows, seven of which have at least four 1s and two of which have at least five 0s. In order to avoid a sub-word of length 5 and weight 1, 2, or 3, there must be two rows of weight 0, and the nonzero rows can contain at most one 0. Furthermore, every column must contain a 0 from some nonzero row, otherwise 4 = 1 is contradicted. So there are at most seven columns. (Index vectors with zero entries often place severe restrictions on k.) t = 6, µ = (2, 3, 3, 3, 3, 3, 2): k 9 shown in Chopra and Dios (1997). Let k = 10. The linear programming constraints cannot be satisfied, so we confirm that k 9 for this index vector. t = 8, µ = (1, 4, 3, 3, 2, 8, 4, 1, 1): k 9 shown in Dios and Chopra (2002). Let k = 9. The LP constraints give a0 /N 4.68661, so there is a 5-fold row in any such BA. But this contradicts Lemma 3.2, as there are no two consecutive values at least 5 in the index vector. Therefore, k 8; in other words, any such array is trivial. t = 8, µ = (1, 3, 6, 4, 1, 7, 5, 1, 2): k 10 shown in Chopra and Dios (2003). The LP constraints are not satisfiable for k =10. So k 9 for this index vector. With a somewhat more sophisticated argument, it is in fact possible to conclude k 8. A BA with this index vector must have N = 1029 rows. Let k = 9. Corollary 2.4 asserts that a0 /N 4.538. So by Lemma 3.1 there must be a 5-fold row. By Lemma 3.2, it follows that any 5-fold row must have weight six and there can be no 6-fold row. Now there are at most ( 96 ) = 84 different words of weight six, each occurring at most 5 times as a row. The quantity a0 is maximized when each occurs exactly 5 times, and the other 1029 − 5 · 84 = 609 rows are divided into 609 4 4-fold rows. But this implies 2 a0 84 · 5 + 609 · 4 = 4536, which is strictly less than the LP lower bound N · 4.538 4669. As in the previous example, any such array is trivial. t = 8, µ = (1, 3, 2, 2, 1, 5, 5, 2, 2): k 9 shown in Chopra and Dios (2003). Let k = 9. We use the same method as in the previous example. There are N = 701 rows. Corollary 2.4 says a0 /N 2.762, so there are 3-fold rows of weight 6. No such row can be repeated more than 5 times. So a0 5 · 84 + 4 · (701 − 5 · 84) = 1544, contradicting a0 N · 2.762 1936. Again, the only such arrays are trivial.
Acknowledgements We would like to thank John Stufken, whose suggestions improved the presentation of this paper. References Aggarwal, K.R., Singh, T.P., 1981. Methods of construction of balanced arrays with application to factorial designs. Calcutta Statist. Assoc. Bull. 30, 89–93. Chopra, D.V., 1990. Some combinatorial inequalities on the existence of balanced arrays. J. Combin. Math. Combin. Comput. 8, 9–12. Chopra, D.V., 1991. Further investigations on balanced arrays. Comput. Statist. Data Anal. 12, 231–237. Chopra, D.V., 1996. On the constraints of some balanced arrays. J. Combin. Math. Combin. Comput. 22, 129–134. Chopra, D.V., 1998. Contributions to balanced arrays of strength five. J. Combin. Math. Combin. Comput. 27, 123–128. Chopra, D.V., Dios, R., 1997. On two level balanced arrays of strength six. J. Combin. Math. Combin. Comput. 24, 249–253. Chopra, D.V., Dios, R., 2003. A note on an upper bound for the constraints of some balanced arrays. J. Combin. Math. Combin. Comput. 45, 129–135.
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Delsarte, P., 1973. An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10. Dios, R., 2002. On the existence of balanced arrays of strength five. J. Combin. Math. Combin. Comput. 41, 3–7. Dios, R., Chopra, D.V., 2002. A note on balanced arrays of strength 8. J. Combin. Math. Combin. Comput. 41, 133–138. Kuwada, M., Nishii, R., 1979. On a connection between balanced arrays and balanced fractional s m factorial designs. J. Japan Statist. Soc. 9, 93– 101. Shirakura, T., Ohnishi, T., 1985. Search designs for 2m factorials derived from balanced arrays of strength 2(l + 1) and AD-optimal search designs. J. Statist. Plann. Inference 11, 247–258. Srivastava, J.N., 1972. Some general existence conditions for balanced arrays of strength t and 2 symbols. J. Combin. Theory Ser. A 13, 198–206. van Lint, J.H., Wilson, R.M., 1992. A Course in Combinatorics. Cambridge University Press, Cambridge.
Linear programming bounds for balanced arrays夡 Peter Dukesa,∗ , Alan C.H. Lingb a Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W3P4 b Department of Computer Science, University of Vermont, Burlington, VT 05405, USA
Received 6 July 2004; received in revised form 6 June 2005; accepted 9 November 2005 Available online 18 January 2006
Abstract In this paper, we apply Delsarte’s linear programming bound to balanced arrays, a key ingredient in certain constructions of fractional factorial designs. Examples are given to illustrate the power and generality of this technique compared with previous applications of the method of moments. © 2006 Elsevier B.V. All rights reserved. MSC: Primary 05B15; secondary 05E30 Keywords: Balanced arrays; Delsarte’s inequalities; Linear programming; Fractional factorial designs
1. Introduction An N × k balanced array (BA) of strength t (over a binary alphabet) with index vector µ = (0 , . . . , t ) is an array with entriesin {0, 1} such that, in any t columns of the array, every word of weight i appears in exactly i rows. Clearly, N = ti=0 ti i . When i = for all i, such an object is called an orthogonal array (OA) of index . A BA or an OA is simple if it has no two identical rows. It should be noted that both BAs and OAs have generalizations to alphabets other than {0, 1}. While the results we discuss for BAs can be modified for larger alphabets, the binary case has attracted nearly full attention in past literature. Moreover, there are at least two natural ways to generalize the notion of weight over the alphabet Zq : the number of nonzero entries in a word and the “composition” of entries in a word. For these reasons, we focus on the binary case. Balanced arrays are of particular interest in combinatorics and statistics. For instance, applications to factorial designs are discussed in Aggarwal and Singh (1981), Kuwada and Nishii (1979), and Shirakura and Ohnishi (1985). Clearly, a BA with any index vector always exists for k = t; we simply take i copies of every vector of length t and weight i as rows. Such arrays we call trivial. More generally, a BA with k columns implies the existence of a BA with k columns for t k k. Given a binary word w1 w2 · · · wk , and indices 1i1 < i2 < · · · < is k, we say that wi1 wi2 · · · wis is a sub-word contained in w1 w2 · · · wk . Given a BA with parameters t, k, µ, we define for 0 s t the quantities s,i to be the 夡 Research of the first author is supported by an NSERC postdoctoral fellowship. Research of the second author is supported by the Army Research Office (USA) under grant number DAAD 19-01-1-0406 (Ling). ∗ Corresponding author. Tel.: +1 647 223 4338. E-mail addresses: [email protected] (P. Dukes), [email protected] (A.C.H. Ling).
0378-3758/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2005.11.007
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325
number of rows which contain a given sub-word of length s and weight i. It is clear that s,i
t−s t −s i+l = l l=0
and in particular t,i = i for all i. Note that for an OA with index , s,i = 2t−s for each i. Let Nh denote the number of rows of weight h in our BA. By counting in two ways the number of ordered pairs (R, S), where R is a row of the BA having a sub-word S of length s and weight j, we have the following moment equations. Lemma 1.1. For 0 j s t, k h k−h h=0
j
s−j
Nh =
k s s,j . s j
It is essentially these equations (in Nh ) which have been exploited in recent work (Chopra, 1990, 1991, 1996, 1998; Chopra and Dios, 1997, 2003; Dios, 2002; Dios and Chopra, 2002) to determine necessary conditions on the existence of BAs of certain strengths between 4 and 8. In the Section 2, we apply Delsarte’s inequalities (Delsarte, 1973) as a general condition for arbitrary strength t. Although this technique does not necessarily subsume the “method of moments”, we have found that in many examples it does. Comparisons are given in Section 3.
2. Delsarte’s linear programming bound We refer the reader to Chapter 30 of van Lint and Wilson (1992) for a nice introduction to the theory of association schemes. A very brief outline is given here for reference and notation. A k-class association scheme on a set X consists of k + 1 nonempty symmetric binary relations R0 , . . . , Rk which partition X × X, where R0 is the identity relation {(x, x) : x ∈ X}, and such that for any x, y ∈ X with (x, y) ∈ Rh , the number of z ∈ X such that (x, z) ∈ Ri and (z, y) ∈ Rj depends only on the indices h, i, j . These quantities are h . Let |X| = n. For h = 0, . . . , k, define the n × n adjacency matrix A , indexed by entries of X, to usually denoted pij h have ij-entry 1 if (x, y) ∈ Rh , and 0 otherwise. Since the Rh partition X × X, it follows that A0 + · · · + Ak = J , the all ones matrix. h , the matrices {A } form a basis for an algebra with respect to entrywise multiplication, called By definition of the pij h the Bose-Mesner algebra. There is also a basis E0 , . . . , Ek with respect to ordinary matrix multiplication such that (1) E0 + · · · + Ek = I ; (2) (Ei )2 = Ei for all i; and (3) Ei Ej = O for i = j . The second eigenmatrix Q is (k + 1) × (k + 1) and has entries determined by nE i = kj =0 Qij Aj . In the Hamming scheme H (k, q), the set X consists of all q-ary words of length k, and (x, y) ∈ Ri if and only if the Hamming distance (the number of disagreeing entries) between x and y is precisely i. We focus here on the binary case q = 2. For H (k, 2), the entries of the second eigenmatrix Q are given by (see Delsarte, 1973, for details) Qj i =
k h=0
(−2)h
k−h j −h
i . h
(2.1)
Note that even though the matrices Ah and Eh are symmetric for each h = 0, . . . , k, the matrix Q is not in general symmetric.
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For example, consider H (3, 2). The set X consists of all eight binary vectors of length 3, and the relations Rh contain those pairs of words at Hamming distance h. From (2.1), ⎛
1 ⎜3 Qj i = ⎝ 3 1
⎞ 1 −3 ⎟ ⎠. 3 −1
1 1 1 −1 −1 −1 −1 1
The 8 × 8 adjacency matrices A0 = I, A1 , A2 , A3 are easy to generate, and the four matrices E0 , E1 , E2 , E3 are computable from these and the entries of Q above. Given a multiset Y of points of an association scheme, the distribution vector of Y is defined to be a = (a0 , . . . , ak ), where ai = |(Y × Y ) ∩ Ri |. Note that a0 |Y |, with equality if and only if Y has no repeated elements. Delsarte’s inequalities state that aQ0 for any distribution vector a in an association scheme with second eigenmatrix Q. Since the rows of a BA with k columns are simply words in the Hamming scheme H (k, 2), we have the following specialization of Delsarte’s inequalities from (2.1). Lemma 2.1. Suppose Y is the collection of rows of a binary BA with k columns and distribution vector a. Then for j = 0, . . . , k, k
(−2)
h
h=0
k k−h i a 0. j −h h i i=0
Unfortunately, Lemma 2.1 is not very helpful until we know more about the distribution vector a. Lemma 2.2. Suppose Y is the collection of rows of a binary BA with k columns, strength t, and index vector µ = (0 , . . . , t ). Define a = (a0 , . . . , ak ) to be the distribution vector of Y. Then for s = 0, . . . , t, k k−i i=0
s
ai =
s s k 2s,j . j s
(2.2)
j =0
Proof. Fix r ∈ Y , say of weight h, and denote by i the number of rows which disagree with r in exactly i positions. By counting in two ways the number of ordered pairs (R, S) where S is a sub-word of length s of both R and r ∈ Y , we have k k−i i=0
s
i =
s h k−h j =0
j
s−j
s,j .
Now summing over all r ∈ Y , we have k k−i i=0
s
ai =
k h=0
=
s j =0
Nh
s h k−h j =0
s,j
j
s−j
s,j
k h k−h h=0
j
s−j
Nh
and the result follows by using Lemma 1.1 on the inner sum.
In the special case that Y is the collection of rows of an OA with index , the right side of (2.2) reduces to ks 2 22t−s . To decrease the number of inequalities and variables under consideration, we now substitute the equations of Lemma 2.2 into the inequalities of Lemma 2.1.
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327
Theorem 2.3. For j = t + 1, . . . , k, (−1)
j
k
(−2)
s
s=t+1
k−s j −s
k−t−1
k−i s
i=0
ai (−1)
j +1
t s k s s j 2s,i . (−2) j s i s=0
i=0
Proof. Rewriting Delsarte’s inequalities, we have for any j = 0, . . . , k, 0 = =
k h=0 k
(−2)h (−2)h
h=0 k k
k k−h i a j −h h i i=h
k−h j −h
(−2)
h
s=0 h=s
k h
(−1)s
i=0 s=0
k−s h−s
k−i s
ai
k k−h k−i s k−s (−1) ai . j −h h−s s i=0
Now split this sum into S1 + S2 , where S1 is the sum over 0 s t and S2 is the sum over t < s k. For S1 , we apply Lemma 2.2 to get S1 =
t k
h
(−2) (−1)
s
s=0 h=s s t
k−h j −h
k−s h−s
s k s 2s,i s i i=0
k s j h k 2s,i = (−1)s (−2)h i h s j s=0 i=0 h=s s t k j s 2s,i . (−2)s = (−1)j j s i s=0 i=0
The expression S2 can be simplified as S2 =
k−t−1 i=0
= (−1)
j
ai
k
s=t+1 k−t−1
ai
i=0
(−1)
s
k
k−i s
(−2)
s
k
s=t+1
h=s
k−i s
(−2)
h
k−h j −h
k−s j −s
k−s h−s
.
Remark. In simplifying sums above, we have repeatedly used the identities − − +−1 = , = (−1) − and + = . i −i i=0
In the cases k = t + 1 and t + 2, we get the following lower bounds on a0 , the sum of the squares of frequencies of each possible word. Corollary 2.4. In a BA with index vector µ and k = t + 1, s t (−1)t t +1 s 2s,i . (−2)s a0 t+1 s i 2 s=0
i=0
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P. Dukes, A.C.H. Ling / Journal of Statistical Planning and Inference 137 (2007) 324 – 330
Proof. This is the case k = j = t + 1 in Theorem 2.3.
Corollary 2.5. In a BA with index vector µ and k = t + 2, t s (−1)t t +1 t +2 s − 2s,i . a0 t+2 (−2)s (t + 2) s s i 2 s=0
i=0
Proof. When k = t + 2, the two inequalities in Theorem 2.3 are t s s t+1 t s t +1 2 ((t + 2)a0 + a1 ) (−1) (t + 2) 2s,i (−2) s i s=0
(2.3)
i=0
and 2
t+1
(ta 0 + a1 )(−1)
t
t s=0
(−2)
s
s t +2 s 2s,i . s i
(2.4)
i=0
Subtracting (2.4) from (2.3) gives the required bound on a0 .
For OAs with index , recall that the s,i = 2t−s are independent of i. It is an easy exercise to check that in this case Corollaries 2.4 and 2.5 reduce to 2a0 /N and 4a0 /N , respectively. When the array is simple (a0 = N ), these claims are obvious, since must be bounded by 2k−t . However, when the array is not simple, these results say more than just that is bounded by 2k−t times the maximum replication of any row. Certain other necessary conditions on the existence of BAs when k is close to t appear in Srivastava (1972). 3. Examples In general, for a fixed k, t, and index vector µ, the equations in Lemma 2.2, inequalities in Theorem 2.3, and the conditions ai 0 for i = 1, . . . , k give linear programming bounds (upper and lower) on a0 /N . There are k − t variables a0 , . . . , ak−t−1 and less than 2k constraints in this LP problem. In the examples to follow, we use these bounds in conjunction with the following easy necessary conditions on a0 /N . Lemma 3.1. In a BA with every row repeated at most r times, 1 a0 /N r. Moreover, the index vector forbids any row to be repeated too many times. Lemma 3.2. Suppose there is an r-fold row of weight h in a BA of strength t and index vector µ with k columns. Then i r for i = max(0, h − (k − t)), . . . , min(t, h). Proof. Let R be the hypothesized r-fold row. If h, k − h k − t, then there are k − t + 1 possible weights of subwords of R of length t, namely h − (k − t), . . . , h. If h < k − t then, there is a length t sub-word of R of any weight i = 0, . . . , min(t, h). Similarly, if h > t we have a length t sub-word of R of any weight i = max(0, h − (k − t)), . . . , t. Each of these subwords appears in at least r rows, so i r for these values of i. We now revisit the upper bounds on k in past examples. In most cases, linear programming problems were solved computationally. These results were instantaneous on a personal computer. • t = 4, µ = (1, 2, 5, 2, 1): k 8 shown in Chopra (1990). Let k = 7. By Lemma 3.2, any BA with these parameters must be simple. Thus a0 = N = 48. Solving the equations in Lemma 2.2, we can rewrite the condition a6 0 as 24a1 + 5a2 6440.
(3.1)
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329
The inequalities resulting from j = 6, 7 in Theorem 2.3 are
• •
• •
•
•
5a1 + a2 1323,
(3.2)
4a1 + a2 1084.
(3.3)
Some easy arithmetic shows a1 175 (from (3.1), (3.2)) and a2 64 (from (3.1), (3.3)). But these constraints contradict (3.1). Thus, k 6 for this index vector. t = 5, µ = (1, 0, 1, 1, 1, 1): k 6 shown in Chopra (1998). Certainly any BA with these parameters must be simple and has a0 = N = 27 rows. Suppose k = 7. The inequality (2.4) gives 5a0 + a1 94.27, which contradicts a1 0. Thus k 6 is necessary for this index vector. t = 5, µ = (2, 0, 0, 0, 1, 2): k 7 shown in Dios (2002). While Delsarte’s inequalities can be used successfully, we can actually argue directly that k 7 for this index vector. First, observe that any such array has nine rows, seven of which have at least four 1s and two of which have at least five 0s. In order to avoid a sub-word of length 5 and weight 1, 2, or 3, there must be two rows of weight 0, and the nonzero rows can contain at most one 0. Furthermore, every column must contain a 0 from some nonzero row, otherwise 4 = 1 is contradicted. So there are at most seven columns. (Index vectors with zero entries often place severe restrictions on k.) t = 6, µ = (2, 3, 3, 3, 3, 3, 2): k 9 shown in Chopra and Dios (1997). Let k = 10. The linear programming constraints cannot be satisfied, so we confirm that k 9 for this index vector. t = 8, µ = (1, 4, 3, 3, 2, 8, 4, 1, 1): k 9 shown in Dios and Chopra (2002). Let k = 9. The LP constraints give a0 /N 4.68661, so there is a 5-fold row in any such BA. But this contradicts Lemma 3.2, as there are no two consecutive values at least 5 in the index vector. Therefore, k 8; in other words, any such array is trivial. t = 8, µ = (1, 3, 6, 4, 1, 7, 5, 1, 2): k 10 shown in Chopra and Dios (2003). The LP constraints are not satisfiable for k =10. So k 9 for this index vector. With a somewhat more sophisticated argument, it is in fact possible to conclude k 8. A BA with this index vector must have N = 1029 rows. Let k = 9. Corollary 2.4 asserts that a0 /N 4.538. So by Lemma 3.1 there must be a 5-fold row. By Lemma 3.2, it follows that any 5-fold row must have weight six and there can be no 6-fold row. Now there are at most ( 96 ) = 84 different words of weight six, each occurring at most 5 times as a row. The quantity a0 is maximized when each occurs exactly 5 times, and the other 1029 − 5 · 84 = 609 rows are divided into 609 4 4-fold rows. But this implies 2 a0 84 · 5 + 609 · 4 = 4536, which is strictly less than the LP lower bound N · 4.538 4669. As in the previous example, any such array is trivial. t = 8, µ = (1, 3, 2, 2, 1, 5, 5, 2, 2): k 9 shown in Chopra and Dios (2003). Let k = 9. We use the same method as in the previous example. There are N = 701 rows. Corollary 2.4 says a0 /N 2.762, so there are 3-fold rows of weight 6. No such row can be repeated more than 5 times. So a0 5 · 84 + 4 · (701 − 5 · 84) = 1544, contradicting a0 N · 2.762 1936. Again, the only such arrays are trivial.
Acknowledgements We would like to thank John Stufken, whose suggestions improved the presentation of this paper. References Aggarwal, K.R., Singh, T.P., 1981. Methods of construction of balanced arrays with application to factorial designs. Calcutta Statist. Assoc. Bull. 30, 89–93. Chopra, D.V., 1990. Some combinatorial inequalities on the existence of balanced arrays. J. Combin. Math. Combin. Comput. 8, 9–12. Chopra, D.V., 1991. Further investigations on balanced arrays. Comput. Statist. Data Anal. 12, 231–237. Chopra, D.V., 1996. On the constraints of some balanced arrays. J. Combin. Math. Combin. Comput. 22, 129–134. Chopra, D.V., 1998. Contributions to balanced arrays of strength five. J. Combin. Math. Combin. Comput. 27, 123–128. Chopra, D.V., Dios, R., 1997. On two level balanced arrays of strength six. J. Combin. Math. Combin. Comput. 24, 249–253. Chopra, D.V., Dios, R., 2003. A note on an upper bound for the constraints of some balanced arrays. J. Combin. Math. Combin. Comput. 45, 129–135.
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