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Balanced Arrays and Orthogonal Arrays
J. N. Srivastava et al., eds., A Survey of Combinatorial Theory © North-Holland Publishing Company, 1973 CHAPTER 35
Balanced Arrays and Orthogonal Arrays J. N. SRIVASTAVA t Colorado State University, Fort Collins, Colo., U.S.A. and D. V. CHOPRA Wichita State University, Wichita, Kansas 67208, U.S.A. Summary We point out the relation between the theory of balanced arrays and various com binatorial areas of design of experiment. Recalling some combinatorial theorems from Srivastava [1971], we apply these to prove a class of new and simple but rather stringent results on the existence of balanced arrays. Applications to the special case of orthogonal arrays are also considered. 1. Introduction Although orthogonal arrays have been known for almost a quarter of a century, the balanced arrays are much more recent. It will therefore be useful to first define the latter. Definition 1.1. A balanced array (B-array) T with s symbols, m rows, N columns, and strength t, is an m x N matrix T with elements belonging to a set S containing s symbols, such that for every (/ x N) submatrix T0 of T, and for every vector v of size (t x 1) with elements from S, we have A(v, T0) = A(P(v), T0)9
for every P.
(1.1)
Here, λ(\9 T0) stands for the number of columns of T0 which are identical with v, and P(v) is any vector obtained by permuting the elements of v. Thus P stands for permutation. Definition 1.2. The matrix T considered in Definition 1.1 is called an orthogonal array with the same parameters, if for every (t x N) submatrix T0 of T9 and for every pair of vectors v and v* of size (t x 1) each, we have A(v, Γ 0 ) = λ(ν*, To).
(1.2)
Clearly, an orthogonal array is a special case of a balanced array. Also, t The work of this author was partly supported by NSF Grant No. GP 30598 X. 411
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clearly, N must be a multiple of s*. The positive integer μ = Ns~* is called the index of the orthogonal array. Definition 1.3. Let one of the elements of S be denoted by zero (0), so that the remaining s— 1 elements of S may be called non-zero elements. Let v be any vector with elements from S. Then the weight of\, denoted by w(v), is defined to be the number of non-zero elements in v. Under Definition 1.1, consider the special case s = 2, with S having the elements 0 and 1. From (1.1), it is clear that λ(\, Τ0) will be a constant for all vectors v whose weight is the same. Thus, in this case, there exist nonnegative integers μ{ (k = 0, 1,. . ., t) such that if \(t x 1) is of weight k, then λ(\, Τ0) = μ\ for every (t x N) sub-matrix T0 of T. The vector (μ'0, μ\9.. ., μ[) is called the index set of the array. We shall now discuss the connection of balanced arrays with other older and more well known branches of the combinatorial theory of design of experiments and coding theory. Consider first the special case of orthogonal arrays. When t = 2 and s = 2, we get an important class of arrays which are equivalent to Hadamard matrices. On the other hand, the case t = 2 and N = s2, corresponds to a set of (m —2) mutually orthogonal Latin squares of order s. General orthogonal arrays can be classified into three categories A, B, and C. Arrays of type A are those whose columns form the set of solutions of an equation of the form Hx — c, where H(k x m) is a matrix over GF(s) of rank k, and c(k x 1) is a vector over GF(.s). Obviously, in this case, s must be a prime power and N = sm~k. It can be shown (Rao [1946]) that the sm~k solutions to the above equation will form an orthogonal array of strength /, if the matrix H has the property Qt which states that every vector in the row space of H must be of weight greater than or equal to t+l. Arrays of type A may also be called "vector space" arrays, since the set of columns of such an array obviously forms a vector space over GF(s·). Type A arrays are also equivalent to group codes. Indeed, the null space of the columns of H can be shown to be a terror correcting code if t = 2d, and a rf-error correcting and (d+ l)-error detecting code if t = 2d+\. Also, type A arrays are important from the point of view of the classical theory of confounding and fractional replication as well. Thus this class of arrays forms the famous link (Bose [1961]) between the theory of factorial designs and coding theory. Both from the point of view of obtaining a code with minimum redundancy, and a fractionally replicated design having the smallest number of treatment combinations, one is interested in obtaining matrices H with property Qt which maximize k (for given m) or maximize m (for given r, where r = m — k). It is well known that the latter problem is equivalent to Bose's "packing problem": Find the maximum number m
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413
of points in PG(r— 1, s) such that no subset of t points lies on a hyperplane of dimension less than t— 1. The solution of the packing problem is therefore important both from the coding and the design point of view. It may be useful to remark that the cases t = 2, and t = 3, s = 2, are solved (Bose [1947]). Also, the problem has been solved for many other combinations of s, t and r, with the result that almost all cases of practical interest from the design viewpoint are taken care of. However, from the coding point of view, the problem is far from solved. Here the interest lies in low values of t, particularly t = 4, and relatively large values of r (say between 20 to 200). Incidentally, an interesting link between the packing problem and some aspects of abstract combinatorial geometry has recently been established by Dowling [1971]. For some references to work on the packing problem, see Barlotti [1965]. We now discuss orthogonal arrays of type B. Consider the equations Hx = Ci (i = 1 , . . . , / ) , where H is (kxm), cf is (kx 1), and rank(i/) = k < m. We assume that H does not have the property Qt. Let Tfyn x sm~k) be the array obtained by solving Hx = cf. Let T = [T± T2 · · · Tf], and let C = [CJL, c 2 , . . . , Cf]. Then the following (unpublished) result (Srivastava [1967]) holds: The matrix T(mxf-sm~k) forms an orthogonal array of strength t if and only if for every nonnull vector u'(l x k) such that weight(u'/0 < t, the vector (u'C) is an orthogonal array of (size (1 xf) and) strengthj. The arrays of type B are obviously of a different structure than those of type A. For type B arrays, N does not have to be a power of s. Incidentally, there exist examples of type A and type B arrays with the same m and N9 and m being the maximum possible for the given N. Arrays which are not of types A or B are said to be of type C. A useful array which is not of type A is given in Bose and Bush [1953]. The parameters are s = 3, m = 7, N = 18, t = 2, μ = 2. Coming back to balanced arrays, let us first consider the special case s = 2, t = 2. In this case it can be shown that the array is the incidence matrix of a balanced incomplete block design (BIBD) with possibly unequal block sizes, written BIBDU for short. The ordinary BIBD's form a special case of these when the block sizes are constrained to be all equal, and thus correspond to arrays with the further restriction that each column be of equal weight. This shows that certain techniques of constructing BIB designs, such as the method of difference sets, could be generalized to help in the construction of balanced arrays. The case s = 2, t = 3 also is important. The so-called inversive planes form a sub-class of such arrays when the column weights are constant. Balanced arrays in general are useful in the theory of factorial designs. Indeed, balanced arrays of strength 2, 3 and 4 are identical with balanced fractional factorial designs of resolutions 3, 4 and 5, respectively. We shall
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not discuss here the detailed properties of balanced arrays considered as fractional factorial designs. The interested reader may refer to Srivastava and Chopra [1971a,b]. From the foregoing discussion, it may appear that the theory of partially balanced designs is not related to the theory of balanced arrays. However, this is not true. As is shown in Bose and Srivastava [1964], certain important principal submatrices of the "information matrix" corresponding to a balanced fractional factorial design (i.e., a balanced array) belong to the linear associative algebras generated by certain well known partially balanced association schemes. (Incidentally, association schemes are merely a different name for what are known as strongly regular graphs.) These algebras proved very helpful in certain statistical studies by Srivastava and Chopra [1971a]. Apart from the above, there are other more direct connections between PBIB designs and B-arrays. We in fact believe that most of the association schemes can be defined using certain balanced arrays. For some work in this field, see Srivastava and Anderson [1971]. It might be important to stress here that balanced arrays not only provide a mathematically challenging field of research which unites various branches of the combinatorial theory of design of experiments, they are also urgently needed for practical problems arising in factorial experimentation. For some literature on B-arrays in general, the reader is referred to Chakravarti [1956], Srivastava [1971] and Rafter [1971]. In some of the earlier work, B-arrays have been called "partially balanced" arrays. However, since the B-arrays are related directly to BIBD's rather than the PBIBD's, we have adopted the new terminology. 2. Preliminaries In this section, we present without proof some results from Srivastava [1971] for later use. Theorem 2.1. A set of necessary and sufficient conditions for the existence of a 5-rowed B-array T, of strength 4 and index set μ' = (μ0, μί9 μ2, μζ, μ*) is that there exists an integer d such that d > *AiiOO = max (0, μ*-μ3, d< Ψι2(μ) = ΐΤίίη(μ4,μ4-μ3
μ^-μζ+μ2-μι)9 + μ29μ^-μ3 +
~ „ μ2-μ1+μ0\
Theorem 2.2. A set of necessary and sufficient conditions for the existence of a 6-rowed array T of strength 4 and index set μ' is that there exists an integer d0 such that d0 > ΦιΜ = max (0, ^ . 2 + 01, rfw + 02, ^ι·6 + 03), d0 < ^2 2 (μ) = min (d69 d^-θ^ rf2.6-05),
~
2)
CH.
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where d^ = dt + di+1+ · · · +dj9j > i;for each i9 d = dt satisfies (2.1); and dx ^ d2 > * · · ^ d6. Also, the 0\s in (2.2) are given in terms of the tfs by 0i = - / U , 02 = - μ 2 + 2 μ 3 - 3 μ 4 , 03 = - μ 0 + 2 μ 1 - 3 μ 2 + 4 μ 3 - 5 μ 4 , (2.3) 04 = - μ 3 + 2μ4, 0 5 = - μ ι + 2 μ 2 - 3 μ 3 + 4μ 4 . Equivalently, (2.2) can be rewritten as "0 (a) 'de —di—d2+d6 (b) -μ4 - μ 2 + 2μ 3 --3μ 4 —d1—d2—d3 — d4+d6 (c) - μ 0 + 2μ1-3μ2+4μ3-5μ4 —d1—d2—d3—d4.—d5 (d) - μ 3 + 2μ4 dA+ds+d6 (e) (2.2) —d1—d2+d4.+ds+d6 μ4-μ3 (f) > 5^ —d1—d2—d3+d5+d6 -μ2+μ3-μ4 (g) - μ0 + 2μ! - 3μ2 + 3μ3 - 3μ4 —d — d —d 1 2 3 00 - μ ι + 2 μ 2 - 3 μ 3 + 4μ 4 d2+d3+d4.+d5+d6 Ö) - μ ι + 2 μ 2 - 3 μ 3 + 3μ4 -di+ds + d^+ds+de (j) -di+ds+de -μι+μ2-μ3+μ4 (k) _-μ0 + μ ι - μ 2 + μ 3 - μ 4 -di j (1) Theorem 2.3. Consider an array T(m9 N9 4; μ'). Let xt be the number of columns of T of weight i (i = 0, 1,. .., m)9 so that YJ=O Xi = N. Then the following single diophantine equations (SDE) must hold:
Σ 0(Z-j)*i = Οφμ,·,
J = 0, 1, 2, 3, 4.
(2.4)
Definition 2.1. An array T with 2 symbols 0 and 1, and m rows is called^ "trim" if x0 = xm = 0. Theorem 2.4. Let T be a trim array with parameters (m9 N9 4; μ'), and let xt be defined as in Theorem 2.3. Let there be g distinct solutions (dr0, drl9. . . dr6)for the vector (d09 dl9.. ., d6) under the inequalities (2.2). Note that each of these solutions would correspond to a distinct 6-rowed array; let these be denoted by Tr(r = 1 , . . . , g). Let yr denote the number of times the 6-rowed array TT occurs as a subarray* of T9 so that Yfr= ί yr = (g). Define 5rj (r = 1, .9g;j = 0, 1 , . . . , 6) by S0 = — θ3 — 6cl+d09 δι = -e5+5a-d09 δ2 = -92-4ä'+d09 <53 = -04+33-4» (2.5) <54= -e^ll+do, <55 = - 2 -d09 δ6 = d09 δ' = (δθ9 δΐ9 . . ·,δ6)9 3 = (i)(rf1+---+rf6), * Here, by a 6-rowed subarray of T we mean an array which would be left after cutting out any set of m—6 rows of T. The rows of the 6-rowed array which is left are not to be permuted.
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where the Fs are defined at (2.3). Let π,. = £ * = 1 SrJyr; j = 0, 1,. . ., 6. Then the following triple diophantine equations (TDE) must be satisfied for the case m = 8: (a) lx1+x2 = π 0 , (b) 21JC1 + 12X 2 + 3JC3 = 6πΐ9 (c) 15x 2 + 15*3+ 6x4 = 15π2, (d) 10*3+ 16Χ 4 +10Λ:5 = 20π 3 , (e) 6*4+15*5 + 15x6 = 15π4, (f) 3*5 + 12* 6 +21* 7 = 6π 5 , (g) * 6 + 7*7 = π 6 , (h) Yfr=l yr = 28. The equations (2.6) are termed TDE because there are three sets of variables involved, namely, the *'s, the d's and the y9s. Corollary 2.1. When g = 1, the equations (2.6) reduce to the following double diophantine equations (DDE): (a) 7*x+* 2 = 28(50, (b) 21* 1 + 12*2 + 3* 3 = 168^, (c) 15*2 +15*3 + 6*4 = 420(52, (d) 10*3 +16* 4 +10* 5 = 560<53 (e) 6*4+15*5 + 15*6 = 420<54, (f) 3* 5 + 12*6 + 21* 7 = 168i 5 , ^ ' ' (g) * 6 + 7* 7 = 28(56. In eqs. (2.7), we have dropped the subscript 1 in the dn(j = 0 , . . . , 6). Corollary 2.2. Suppose that for some j9 we have Sj = Sj+2 = 0 and öj+ι Φ 0. Then the DDE are contradicted. We also recall the following from Srivastava and Chopra [1971a]: Theorem 2.5. Let T be a B-array with m ^ 8, and t = 4. Then ΑΊ+μ 3 >$μ2* 4μ! < μι^ι+μ^
+ όμ^.
(2.8) (2.9)
3. Analysis of the SDE For m = 8, the SDE (2.4) reduce to 70*o + 35* 1 + 15*2 + 5* 3 +*4 = 70μ0> 35*!+40* 2 + 30*3+16*4 + 5*5 = 280μ1? 15*2 + 30*3 + 36*4 + 30*5 + 15*6 = 420μ2, 5*3 + 16*4 + 30*5 + 40* 6 + 35* 7 = 280μ3, * 4 + 5*5 + 15*6 + 35* 7 + 70* 8 = 70μ4·
(3.1) (3.2) (3.3) (3.4) (3.5)
The above equations involve nine variables. Among these, * 0 and * 8 can, in most cases, be gotten rid of by using Theorem 3.1 below. Let A = Α(μΐ9 μ29 μ 3 ) be the class of all B-arrays of strength 4 and m = 8 with fixed values of μΐ9 μ2, μ$. Let A0 = Α0(μΐ9 μ2, μ 3 ) be a subclass of A with the further restriction that if an array T belongs to A0 then T is *'trim", i.e., T does not contain any column of weight 0 or 8. Thus, if Te A0 then the value of (x0+x8)9 for T9 equals zero.
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BALANCED ARRAYS AND ORTHOGONAL ARRAYS
Theorem 3.1. (a) If for given μΐ9 μ2 and μ3, the subclass Α0(μί9 μ2, μ3) is empty, then the whole class Α(μΐ9 μ2, μ3) is empty, (b) If A0 is nonempty, Te A0, and T has index set (μ0, μί9 μ29 μ3, μ^)9 then there exists an infinite number of arrays in the class A n Ä0 (i.e.9 with x0+xs Φ 0) with the index set of the form (μ*9 μΐ9 μ29 μ3, μ*), with μ* ^ μθ9 μ* > μ 4 AU) by adding (μ* —μ0) vectors of weight 0 and (μ* —μ4) vectors of weight 8. Henceforth we shall assume x0 = x8 = 0 in the equations (3.1)—(3.5). In (3.1)—(3.5), we now make the following transformations: x = x 4 , u = x2+x69 v = x3+x59 w = χί+χΊ9 u' = x2—x6, v' = x3—xS9 H>' = xx—xl9 (3.6)
μ" = μο+μ*, μ' = μι+μ 3 , μ'ό = μο-μ*> μΌ = μι-μ* = 0. In the above, and throughout this paper, the symbol = means that the corresponding inequality holds for arrays with μχ = μ2 = μ3. From (3.1-3.6), we have, by simple additions and subtractions, 5w+10t;+12;c = 140μ2,
(3.7)
?>5w+\5u + 5v + 2x = 70μ",
(3.8)
= 280μ' = 560μ2, lw' + 3u' + v' = 14μ{)',
35Η>+40Μ+35Ι; + 32Λ;
7w' + Su' + 5v' = 56μ£ = 0, 5u + 6v + 6x = 14(4μ'-μ") = 14(8μ2-μ"), 2ι> + 3* = 7(10μ 2 -4μ'+μ") = 7(2μ2+μ"), \4w + 9u-4x = 28(4μ'-7μ 2 ) = 28μ2.
(3.9) (3.10) (3.11) (3.12) (3.13) (3.14)
.7), it is clear that 12* < 140μ2, and 5\x9 so that x = Sk9
0 < k < %μ2.
(3.15)
where A: is a non-negative integer. From (3.13), we have that 7\(2v+k). Let 2v + k = lkl9 where k1 is a non-negative integer. Then v = i(7k1-k)9 (3.16) which implies kt > jk9 and k+kx is even. (3.17)
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From (3.13), we obtain 2k+kx
= 10μ2 - 4 μ ' + μ " = 2μ2+μ".
(3.18)
μ" = 4 μ ' - 1 0 μ 2 + (2£ + &1) = -2μ2 + (2/<+/€1),
(3.19)
(3.17) and (3.18) give
\k < ikj. = ( 1 0 μ 2 - 4 μ ' + μ " ) - 2 £ = (2μ 2 +μ")-2£. Substituting from (3.15) and (3.16) into (3.9), we have, 0 ^ u = 28μ 2 -7£ 1 -11&. Substituting from (3.21) and (3.15) into (3.14), we get
(3.20) (3.21)
(3.22) 0 < w = 2(V-16/i 2 )+i(17Jk + 9ik1) = - 1 6 μ 2 + Κ17Λ + 9Λχ) Combining (3.21) with (3.17), we have \k^k, <[4μ2-^], (3.23) where [x] denotes the largest integer less than or equal to x. Subtracting (3.11) from (3.10), we have -5U'-4O'
= 140IS-4/I'O) =
14^.
Therefore 14|(5w' + 4t;')> and u' is even. Thus we have (3.24) u' = 2/ 2 , l2 being any integer. Also, from (3.11), we have 7|(w' — 2υ'). Therefore u'-2v' = 7/ lf (3.25) /x is any integer. From (3.24) and (3.25), we have v' = h-ih = /a-7/3, (3.26) where /! = 2/ 3 . Hence w', 1/ are known as soon as we choose l2 and / 3 . Substituting from (3.26) and (3.24) into (3.11), we have w ' = 8 ^ - 3 / 2 + 5/ 3 = 5 / 3 - 3 / 2 . Similarly, (3.27), (3.26), (3.24) and (3.10) give
(3.27)
h = -μο + 4μ'ο + 21* = 21*-μΌ· (3-28) The above equations give the value of (w, v, w, x; u\ v\ w') in terms of μ', k, ku l2 and / 3 . Knowing the values of the former, the equations (3.1)(3.5) can be solved for the x f 's using (3.6). Furthermore, from (3.6), it is obvious that u > |ii'|,
v ^ \v'\, w > |w'|, μ' > \μ'0\, μ" ^ | ^ Ι ,
(3.29)
and (u±u')9 (v±v')9 (νν + νν'), (μ'±μ'0) and (μ"±μ'ό) are even. Now, (3.3) together with (3.16), (3.26), (3.18) and (3.28) implies that i(k + k1)±(l2-l3) and kx±(l2-2l3) are both even.
(3.30) (3.31)
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Furthermore, v > \v'\ and u ^ |w'| give us respectively #H7A:i-*) + /J > h > \l-Wki-k) + hl (3.32) \u > l2 > -\u. (3.33) The x/s, or more precisely (pcl9..., x7), will be called the primary weight parameters of an array, and the set (u, v, w; u', v', w') will be called the secondary weight parameters. In the above analysis, we have expressed the secondary weight parameters in terms of (k, μ2, kl9 μ'; ll9 /3, μό)> which will be called "tertiary" weight parameters. We have not included μ" and μβ in the last set, since by (3.19) and (3.28) they get fixed when the other tertiary parameters are fixed. Next, from these results we obtain a lower bound on the number of assemblies in a B-array of strength four with two symbols, and m ^ 8. Theorem 3.2. For a trim B-array with 2 symbols, t = 4 andm > 8, we have N ^ ^J. μ2. (As will be shown later, the bound is attainable for μ2 = 6). Proof. Clearly it is enough to prove the theorem when m = 8. We first observe that (3.34) N = μ0 + 4μ1 + 6μ2 + 4μ 3 +μ 4 = 6μ 2 +μ" + 4μ', where μ', μ" have been defined in (3.6). Substituting from (3.19) and (3.22) in the above, we have N > 28μ2-Κ13Α: + 7Α:1) = 2&μ2-(χ3 + χ5 + ΙχΑ). (3.35) Using (3.15) and (3.21) in (3.35), we obtain the result. Corollary 3.1. Ifk = 0,k1 = 2 and μ2 = 1, then μ' ^ 3. Proof. This follows from the inequality (3.22). Theorem 3.3. Let The a trim B-array with (a) μ2 = 1 and Xt = 5, then N ^ 23; (b) μ2 > 0, then μ± αηάμ$ are not both zero, αηάμ0 αηάμ^ are not both zero. Proof, (a) From (3.15), (3.17) and (3.21) we have k = kt = 1. Then (3.22) and (3.19) give respectively μ' > 3 and μ" ^ 5. The result follows from (3.34). (b) ΤΐΊ^μχ = μ3 = 0 is not possible follows from (3.2)-(3.4) since μ2 = 1. Similarly for μ0 and μ4. Theorem 3.4. (a) If no B-array exists corresponding to a given set (x, u,v,w; u', v1, w') of weight parameters and a given value of μ2, then so is the case with the weight parameters (x, u, v, w; —u', —v', —w')for that value of μ2. (b) If no array exists corresponding to a given set (k, μ2, ku μ'; l2, /3, μ'0) then so is the case for the set (k, μ2, kl9 μ'; —12, —/3, — μό). Proof, (a) This follows by observing that if there exists a B-array T with
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the secondary parameters (x, u, v, w; — u', — v', — w'), then by interchanging 0 and 1 in T we shall get the complementary array T with parameters (x, u, v, w; u', v', w'). (b) Assume, as in (a), that an array T exists corresponding to the para meters (k, μ2, ku μ', μ"; —/2, —/3, —μ'θ9 — μ'ό). Then it is clear that for T the values of k, μ2, kl9 μ', μ" will remain unchanged. Since the parameters (—u', —i?', — w') of T change to (u\ v', w') in T, it can be checked using successively equations (3.24), (3.26), (3.27) and (3.28) that the values {-l2, — h> —μ'ο) f ° r ^change to (l2, / 3 , μ'0) for T. This completes the proof. Theorem 3.5. Let Tbe a B-array with parameters (m, N,1; μ'). Suppose that for some i (0 < i'^ m) we have Xi + xm_i > 0, where xt is the number of columns of weight i. Then there exist two non-trim arrays 7\ and T2 with m — i and i rows, respectively, and having other parameters the same as those ofT. Proof. There exists a column v in T such that w(y) = /, say. Then 7\ (or T2) are obtained by omitting all rows of T which have 1 (or 0) in v. Similarly when w(v) = m — i. 4. Arrays with μι = 1 First we note a result for later use. Theorem 4.1. (a) There does not exist any trim B-array with m = 7, μ2 = 1 and N < 28, except when N = 21. The array with N = 21 has index set (0, 0, 1,3, 3), and is unique except for an interchange ofO and 1. (b) Let there be an array with 21 ^ N *ζ 27, m = Ί and μ2 = 1. Then, apart from an interchange ofO and 1, its index set is of the form (μ0, 0, 1,3, μ 4 ) with μ0 ^ 0, μ 4 ^ 3. Proof, (a) The nonexistence of arrays with 21 < TV < 28 follows from Theorems 2.1 and 2.6, and Corollary 2.1 of Chopra and Srivastava [1971]. Using these same theorems, it is easily observed that N = 21 implies μ' = (0, 0, 1, 3, 3), apart from an interchange of 0 or 1. By using the SDE for N = 21, one then obtains x2 = 21, from which uniqueness of the array is easily shown. (The array is obtained by taking all 7-vectors of weight 2). (b) This follows from (a). Theorem 4.2. Let T(m xN) be a B-array with two symbols, t = 4, m ^ 8 and μ2 = 1. Then N ^ 28. Proof. Clearly, it is sufficient to prove the result for trim arrays T. From (3.15), we get k = 0, 1, 2, and hence x = 0, 5, 10. When k = 2, (3.21) gives lkx < 6, and hence kx = 0. But then (3.20) gives k = 0, a contradiction. Hence x = 0 or 5.
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Now, from (2.1), we have 0 < μ " - μ ' + 1 , or μ" ^ μ ' - Ι . Hence N = μ"+4μ' + 6μ2 S* 5(μ' + 1). Then N < 28 implies μ' < 4. Also, (2.9) gives μ' > 2. Hence 2 < μ' < 4. From (3.9), we have 8μ' = w+fy+v+^x = N+±u—gsX. From (3.7), we get u+2v+^x = 28. Hence, since 7|(2t;+A:), the permissible values (for N < 28) of (x, u) are (0, 14), (0, 0) and (5, 10). These values of (x, u) correspond respectively to 8μ' = N+2, N and (N+1). Hence N < 28 gives μ' < 4. When μ' = 2, (JC, w) = (0, 14), we get N = 14 < 5μ' + 5. Hence, since lkx = 2v+k and Ι / + 2 Ι ; + ^ Λ : = 28, and in view of Theorem 3.3(a), the possible values (for N < 28) of (k, kl9 μ', Ν) are (0, 4, 2, 16), (0, 2, 3, 22), (1, 1, 3, 23) and (0, 4, 3, 24). We consider these one by one. Let N = 16. Then μ' = 2 = μ". Hence μ' is of the form (μ0, μΐ9 1, 2—μ1? 2—μ0). If an array with this index set exists, then by interchanging 0 and 1 in this array, we can obtain an array with index set (2—μ0, 2—μί9 1, μΐ5 μ0). By adjoining the two arrays, we shall get an array with m = 8, N = 32, and index set (2, 2, 2, 2, 2). Such an array is, however, known to be nonexistent (Seiden and Zemach [1966]). Hence N Φ 16. When 22 < N < 24, we get μ' = 3 and w > 0. By Theorem 3.4, μχμ3 Φ 0. But, using Theorem 3.5 with m = 8, / = 1, and also Theorem 4.1(b), it follows that we must have μ1μ3 = 0, a contradiction. This completes the proof. Theorem 4.3. There exists a unique {apart from (0, 1) interchange) B-array of strength 4, m = 8, μ2 = 1, and 28 assemblies. Its index set is (0, 0, 1,4, 6), and it is obtained by taking all the distinct %-vectors of weight 2. Proof. From Theorem 2.17 of Chopra and Srivastava [1971], it follows that the arrays with m = 7, iV = 28 and μ2 = 1 must have index sets of the forms (3,1,1,3,3) or (a, 0, 1, 4, 3+j8),wherea > Ο,β ^ Ο^ΐιβημ' = (3, 1, 1,3,3) and m = 8, (3.18) gives 2k+kx = 0, so that k = kx = 0. Then, from (3.21), u = 28, and hence x = v = w = 0. Then (3.10) and (3.11) give respectively 3«' = 0, 8w' = 112, a contradiction. From Theorem 4.1, we find that m = 8, μ2 = 1, N = 28 implies that the array must be non-trim, so that in the SDE we have x0 = x8 = 0. If μ' = (α,Ο, 1,4, 3 + β), (3.11) and (3.10) give a = 0, and hence β = 3. Again, from (3.18), k = kt = 0. Hence u = 28, and (3.10) gives Zu' = 84, or u' = 28. This completes the proof. 5. The intermediate diophantine equations (IDE) We now derive a new set of diophantine equations, the IDE. These will be useful for later developments. Though these equations are new, the basic idea is the same as that for deriving the TDE (Srivastava [1971]) at (2.6).
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Indeed, the basic idea is simple. Suppose we are considering the existence of an (m x N) array T. Suppose further that the (k x N) subarrays T0 of T may be of g' different types Tou . . . , Tog'9 and that for each i and j (/ = 0, 1, . . . , k;j = 1,. .., g), we know the number of columns of Toj with weight /. Now, assuming that there are xi columns of T whose weight is /, we can compute the number of k-vectors of weight i which are contained in the columns of T. Also, we can compute this same quantity by assuming that out of the (?) subarrays of T there are Zj arrays of type T0J. Equating, we shall get k+1 equations, one for each value of /. The unknowns in these equations will be the xh the zp and probably some other quantities. The SDE and the TDE are obtained by developing this idea for the cases k = t and (t + 2), respectively, where t denotes the strength of the array. The IDE correspond to k = t+1. Consider a B-array T of size (m x N), strength t, and index set (μ'0, μ\9. . ., μ}). Let T0 be any ((t+1) x TV) subarray of T. Let d O 0) be the number of columns of T0 which equal (1, 1,. . ., 1)'. Then, in equation (2.4) of Srivastava [1971], it is shown that if φ·ν denotes the number of times any particular (t+ l)-vector of weight i occurs as a column of T0, then every (t + l)-vector of weight / occurs φι times in T0, and ^• = ^ - ^ + i + ^ + 2 - - ; - + ( - i y - i + 2 ^ + ( - i y + 1 " ^
0 < ί < f, (5.1a)
φΐ+1 = ά.
(5.1b)
The following is helpful in the calculation of the 0's: 4>i = Vt-i+u /' = 0, 1 , . . . , f. (5.2) Since each φι is nonnegative, (5.1a,b) show that d is bounded both above and below by linear functions of the JU'S. Suppose Jean take h different values. (Notice that when t = 4, h = Ι+φα — φιι^ where the ^'s are given by (2.1), then g' — h). Clearly, any (t + l)-rowed array is completely determined by the value of d. Thus, for simplicity, the (i+l)-rowed arrays will be denoted by Tod (instead of T0j), where d ranges over its possible set of values. Now, the number of (t+ l)-veotors of weight / (0 < i ^ t+1) which are contained in the columns of T (without row permutations) equals £ d *<*[idCI1)], where φίά is the value of φι for the array T0d. Hence we obtain the IDE: m
Σ (ΙΧ,ΪΓ',-Κ = CV)E^«], o < i ^ t+i, γζά = GTO. (5.3)
9= 0
d
d
These equations contain two sets of unknowns, the x's and the z's. Since φΜ are easy to compute (relative to the ö's needed for the TDE or DDE), the IDE are (relatively) easier to use. Still, however, in many applications, one needs to go to the TDE. For the case m = 8, t = 4, (5.3) can be written in full as 56*0+ 21*! + 6x2 + *3 = 0* >
(5.4a)
CH.
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BALANCED ARRAYS AND ORTHOGONAL ARRAYS
35*! + 30*2 +15*3+4* 4 = 5φ*9 20* 2 + 30Λ:3 + 24*4 +10* 5 = 1005, 10*3 + 24*4 + 30*5 + 20* 6 = 100*, 4*4 +15*5 + 30* 6 + 35* 7 = 50J, * 5 + 6 * 6 + 21* 7 + 56*8 = φ%,
423
(5.4b) ( 5 - 4c ) (5.4d) (5.4e) (5.4f)
where
# = Σ*Αι>
« = 0,1
5.
(5.5)
6. Arrays with μι = 2, 3 In Srivastava and Chopra [1971b], arrays with μ2 = 2 and N *ξ 41 have been studied. Other arrays with μ2 = 2 shall be studied in similar papers elsewhere. Here we consider only the case μ2 = 3. Henceforth, if P denotes any statement (such as the IDE, DDE, etc., or any inequality such as (2.2c)), then R(P) shall mean "the rejection of P ' \ Thus "R(TDE)" will mean "a contradiction of the TDE". Also, let
m
α = μ 1 - μ 0 , β = μ3-μ^ (6-1) Theorem 6.1. Consider a B-array T with parameters (m, N, ί;μ'), where ^ 8, t = 4, μ2 = 3, andoc + ß = 0. Then a = - ß = 0, 1 or - 1 .
Proof. It is enough to show that if a < 0, then a = — 1. If possible, let - a = ß > 1. Then (2.1) gives us ß = 3 or 2. Case (a), ß = 3. Here (2.1) gives if/u = \j/12 = 0, and hence d1 = d6 = 0. Also, μ0 > 3. From (2.2c,e,h), we get μ 4 = 3, and 3 < μ0 < 6. Hence Φ21 = Φ22 = 0, and d0 = cl = 1, δ 3 = δ5 = 0, and <54 = 3, leading to R(DDE). Case (b). β = 2. Here ^ i r = 0, φ12 = 1. Hence, recalling the IDE at (5.4), we get h = 2, and φ* = z l9 φ* = 3z1 + 2z0, and <£* = z 0 . Hence, 56 = Zo + Zi = ^f + #J = 2*2 + 3*3+ (2.4)*4 +2*5+ 6*6+ 21*7 + 56* 8 . Now the SDE (3.4), (3.5) give 280jS = 560 = 5*3 + 12*4+10* 5 -20*6-105* 7 -280* 8 . Multiplying the first equation by 10 and subtracting the second, we get Xi = 0(i = 2 , . . . , 8). But this contradicts (5.4d). Hence the proof is complete. Theorem 6.2. Lei i k ö possible B-array with μ2 = 3, m ^ 8, a = 0 Ö«U? j? > 0. Then ß = 1. (Similarly, ß = 0, a > 0 wip/fey a = 1.) Proof. Let j8 > 1; then (2.1) shows that ß = 3 or 2. Case (a). /? = 3 implies d^ = d6 = 0 and μ' = (0, 0, 3, 6, 3). Here d0 = 0, and this leads to R(DDE). Case (b). β = 2. Here 0 < ^r n < ψί2 < 1. In the IDE,- take A = 2. Then 05d = d, φ4ά = μ 4 - ί / , Φζά = 2 + rf, 02d = l-^> 0id = μ ο - Ι + ^ » Φο<ι =*
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l-d, a n d 0 ? = ζΐ9φ% = 5βμΑΓ-ζΐ9φ% = 112+ζΐ9φ% = ζθ9 φ* = 56μ0 + ζθ9 φ* = ζ0. Hence, from (5.4), we have 0 = φ* + φ* + 3φ*—φ* = 56χ0 + 2lx1 + Sx2 + 3x3+x5 + \6x6 + 63x1 + l6Sxs. Hence, xf = 0, for / Φ 4. Hence 56 = z0+zt = φ* + φ* = 0, a contradiction. This completes the proof. Theorem 6.3. Let T be a B-array of strength 4 with m ^ 8, μ2 = 3. Then a and β cannot be both positive. Proof. Suppose a > 0, β > 0. As before, we find /? < 3 and x + ß *ζ 3, so that (α, β) = (1,2), (2,1) or (1,1). Case (i). (a, ß) = (1, 2). Here (^ l l 5 ^ 1 2 ) = (0, 0), dt = 0 (all i), and using (2.2c,e), we find that μ' = (μ0, μ 0 +1> 3, μ4 + 2, μ 4 ), with μ 4 = 1 or 2. Then R(DDE) follows from δ' = (0, μ0, μ 4 - 1 , 2 - μ 4 , μ4, 0, 0). Proceed similarly for (α, /?) = (2, 1). Co«? (ii). (α,]8) = (1,1) with (ψ1ΐ9 φ12) = (0, 0). We get μ' = (μ0 ,μ 0 + h 3, 1, 0). Here d0 = 0, δ' = ( 3 - μ 0 , μ0-2,1,1,0, 0, 0), and R(DDE) follows. Case (iii). (α, β) = (1, 1) with (φ1ΐ9 φί2) = (0, 1). Here μ' = (μ0, μ0+ U 3, 1 + μ4, μ 4 ) and then 0 < rf6 < rfx < 1. Because of (2.2f,g) all values of d' with d2 φ d5 are rejected, and we consider the remaining four values one by one. Take d1 = d6 = 1. Using (2.2c,e,h,k) and (2.2 d,i), the possible values of (μο> μ*) are (0, 2), (0, 3), (1, 3) and (1, 4), and d0 = 3 + μ 0 - μ 4 . Then δ' = (0, 0, μ0, 1 - μ 0 , 1 + μο> ^4 — ^0 —2, 3 + μ 0 - μ 4 ) , and we have R(DDE). Next, take di = d5 = 1, 0. 77ze« e/fAer a = 0, ß > 0 or a > 0, ß = 0. Proof. The proof follows from Theorem 6.3 if we show that a ^ 0, ß ^ 0. Suppose a < 0. Then, as before, /? = 3 or 2. C&se (a), ß = 3. Here dt = 0 for each /, and μ' = (μ0, μ0 + α, 3, 6, 3), d0 = 0, leading to R(DDE). Case (b). j8 = 2. Then a = - l a n d 0 ^ ^ n < ^ 1 2 < 1. For the IDE, we then obtain φ% = ζθ9 φ% = l\2 + zl9 φ% = ζί9 ζ0 + ζ1 = 56. Then 0 = 2φ% + 3φ%-φ% = 4 χ 2 + 5Χ 3 + (2.4)Λ;4 + 2Χ5 + 16Χ 6 + 63Χ 7 + 168Λ:8. Hence xf = 0 (/ = 2 , . . ., 8), a contradiction. Theorem 6.5. Consider a B-array T of strength four, with m ^ 8, μ2 = 3, μί ^ 2, a«rf α + jß < 0. 77te/? a ^ 0, /? < 0, H>#A strict inequality in at least one case.
CH.
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BALANCED ARRAYS AND ORTHOGONAL ARRAYS
425
Proof· Let a > 0. Now a and — ß are positive. It is obvious that μχ and μ 4 are both nonzero. (2.1) implies a < 3, a < μχ and —ß>oc. Case (i). a = 2 = μ±. When μ1 = 2, we get ψ^ = i/f12 = 1-/J, so that μ' = (0, 2, 3, μ 4 + β, μ 4 ). But this contradicts (2.2). When μι > 2, we have Φιι = Άΐ2 = —ß> a n d μ' = (μ0, μο + 2, 3,0, —β). From (2.2), we observe μ0 = 4, 5, so that μ' = (4, 6, 3, 0, — jß) and (5,7, 3,0, — ß). Finally (using the DDE), the nonexistence of the corresponding arrays follows from (d0;cl) = (-j8; - / 0 , «i = <53 = 0, δ2 = 3; and (d0;3) = ( - / ? ; -j3), <50 = <53 = 0, δχ=δ2
= 3.
Core (ii). Next, we take a = 3. Then μι ^ 3, and i^n = ψ21 = —ß> a n d μ' = (μ0, μο + 3, 3, μ 4 + β, μ 4 ). From (2.2h,j,e), we have μ0 = 3, and - j ? < μ 4 ^ -2jS. Hence μ' = (3, 6, 3, μ 4 + 0, μ 4 ), ^ 2 ι = Ά22 = -μ 4 -2.}> and <5j = <53, leading to R(DDE). Case (iii). a = 1 implies (ψ1ί9 φ12) = (2-jS, 2-j8), ( l - j 8 , 2-j8) or (-j?, 1 — jß) according as μ! = 1, 2 or > 2. These are considered separately. (a) For a = μχ = 1, we have dx = d6 = 2 — β9 and μ' = (0, 1, 3, μ 4 + β, μ 4 ). From (2.2c,i), we get 3-j? < μ 4 ^ 5-β, leading to ^ 2 i = ^22 = 5 - 2 0 - μ 4 and δ' = (0,0,0, 1, 1 , μ 4 - 3 + 0, 5-2β-μ4). In the DDE, this implies x6 < 0, a contradiction. (b) When a = 1, μί = 2, we get (^ l l 5 ^ 1 2 ) = (1 - / ? , 2-jß) and μ' equals (1, 2, 3, μ 4 + β, μ 4 ). Because of (2.2f,g,j), we obtain (i) dx = >2 and 30x5 = — 20y2 —120j>!. This is possible only when y1 = y2 = 0, a contradiction. 7. Arrays with μ2 = 4 , 5 , 6
·
Theorem 7.1. Consider an (m x TV) B-array T of strength 4 H>/7A /Wex .sei μ', and μ2 = 4, m ^ 8. 77ze« TV ^ 56. Proof. Using (2.9), we get μ' S* 6. Moreover, if μ' ^ 8,theniV = μ" + 4μ' + 6μ2 ^ 56. Hence μ' = 6, 7. Let us consider these separately. Case (a). If μ' = 6, then (3.22), (3.15), (3.17) and (3.21) give (a) 9ki + llk ^ 160, (b) 0 < k < 9, (c) k^lk,^ 28μ 2 -11£. (7.1)
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Now, using (3.17), (3.18), we get (k, k^ = (7, 5). From (3.18), (3.22), we have μ" = 3 and w = 2. Now (3.21) gives u = 0, and hence * 2 = * 6 = 0; and (3.15), (3.16) give x = 35 and v = 14. Therefore, the SDE reduce to (a) 35* χ + 5* 3 + 35 = 70μ0, (b) 35*! +30JC 3 + 560+ 5*5 = 280μΐ9 (c) 30*3 + 1260 + 30x5 = 1680, (7.2) (d) 5*3 + 560 + 30*5 + 35* 7 = 280μ3, (e) 35 + 5* 5 + 35* 7 = 70μ4. From (7.2a,c,d,e) we find that 7|* 3 , 2|(* 1 +* 3 + l), 7|* 5 , 2|(* 1 +* 5 ), 2|(* 3 + * 7 ) and 2|(* 5 +* 7 + l). Also, v = 14, w = 2, together with the above conditions, give us (*3, * 5 ) = (0, 14), (7, 7) or (14, 0), and (xl9 * 7 ) = (0, 2), (1, 1) or (2, 0). Now (xl9 * 7 ) = (0, 2) and (*3, * 5 ) = (0, 14) contradict (7.2b), and (xl9 * 7 ) = (2, 0) and (*3, * 5 ) = (14, 0) contradict (6.4d). Finally, (*l5 * 7 ) = (1,1) and (*3, * 5 ) = (7, 7) is rejected by (7.2). Thus there does not exist a B-array having * 4 = 35, u = 0, v = 14 and w = 2. Case (b). When μ' = 7, we have 9k± + 17A: ^ 144. It can be easily checked that the only possible (k, kx) in this case are (9, 1), (8, 2), (7, 3), (7, 5), (6, 6), (0, 16). Of these, the pair (9, 1) is rejected because of (7.1c), and for the remaining possible values of (k, kx) we find that N ^ 56. This completes the proof. Theorem 7.2. There does exist a B-array Γ(8, N, 4; μ') having N = 56, μ2 = 4, and index set (4, 6, 4, 1, 0). The array is obtained by writing the 56 ^-vectors of weight 3 each. This array is unique, apart from an interchange of 0 and 1. Furthermore, this is the only array with N = 56. Proof. Using the result in Section 4, check that N = 56 implies (k, k^ = (0, 16). Also, then, μ' = 7, μ" = 4, * = u = w = 0, v = 56. Using the SDE, 5|(μ3 —1). Hence |t/| = 56. This gives the result. Theorem 7.3. For a B-array T of strength 4, with m ^ 8 and μ2 = 5, we have N ^ 66. (However, it is not known whether a B-array of strength 4 with m = 8, μ2 = 5 and N = 66 exists.) Proof. We have TV = 6μ2 + 4μ' + μ" = 30 + 4μ' + μ", which implies that μ' and μ" have to be minimum for N to be the smallest possible. Moreover, from (3.22), μ' is such that 16μ' ^ 64μ2 —[17^ + 9^!], where [*] denotes the greatest integer less than or equal to *, and therefore we take that value of μ' which satisfies
* - Η.~~ϊ6~-\|·
(7 3)
·
It is obvious from (7.3) that 17k+9k1 has to be as large as possible if μ' is to be minimum. From (3.15), we obtain 0 < k ^ 11. Now, using (3.17) and (3.23), we find that k φ 11, and that the possible values of (k, kx) (with k1 a maximum) are (0, 20), (1, 17), (2, 16), (3, 15), (4, 12), (5, 11), (6, 10), (7, 9),
CH. 35
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BALANCED ARRAYS AND ORTHOGONAL ARRAYS
(8,6), (9,5) and (10,4). Furthermore, we observe that Ws(\7k+9k1)] assumes the largest value 12 when (k, k^) = (6, 10), (7, 9), (9, 5) and (10,4). Of these four competing pairs, the minimum value of N is given by that pair for which μ" is the least. It can be checked from (3.18) that (k, kx) = (6,10) gives the least value of μ", which equals 4. Hence the minimum number of assemblies required is 66. Theorem 7.4. Let T(m xN) be a B-array of strength 4, with μ2 = 6, and m > 8. Then Tmust have N ^ 70. Proof. From (2.9), we have μ' ^ 8. If μ' > 8, then N ^ 4μ' + 6μ2 ^ 72. When μ' = 8, the only values of μχ and μ 3 satisfying (2.9) are μί = μ 3 = 4. For these cases, we have N ^ 68, and the non-existence of arrays with N = 68 or 69 will complete the proof of the theorem. Possible μ' with N = 68 or 69 are (0,4,6,4,0), (1,4,6,4,0) and (0,4,6,4,1). For μ' = (0, 4, 6, 4, 0), from the SDE (3.1), (3.3), we have x5 = 224, which contradicts N < 70. Similar contradictions are obtained for the other two values of μ'. 8. Orthogonal arrays Before closing, we make a few remarks on orthogonal arrays of strength 4 and index μ. The following result throws light on how some of the preceding development can be brought to bear on orthogonal arrays. Theorem 8.1. An orthogonal array of strength 4, m rows and index μ exists if and only if there exists a trim balanced array with the same parameters, and index set (μ0, μ, μ, μ, μ4) with μ0 < μ, and μ 4 =ζ μ. J*
It was in view of this result that in Section 3, we used the symbol" — " to indicate the expressions one obtains when μ± = μ2 = μ3. Theorems 2.1 and 2.2 can be used to get existence conditions for orthogonal arrays with 5 and 6 rows, respectively. Thus, for example, for an orthogonal array of index μ and m = 5, we have ψ11 = 0, ψ12 = μ, so that there are 1 + μ non-isomorphic arrays. For six-rowed arrays, we have the following interesting result. Theorem 8.2. A necessary and sufficient condition that there exists a B-array with m — 6, t = 4, and index set μ', is that there exist integers d*(^ 0) and i (0 < i < 5) such that {d^ = dt = rf* + 1 , di+1 = d6 = d*) is a solution of (2.2), where for i = 0 the equality (dt = dt = d* +1) is to be ignored. Proof. The result is established by observing that the coefficient matrix (say A) of the vector d' = (dl9..., d6) on the left side of (2.2) is such that each element in any column of A is larger than or equal to the corresponding element in the preceding column of A. Notice that the above theorem greatly simplifies the inequalities (2.2), since the left hand side of (2.2) now uses only one variable d* instead of six
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J.N.SRIVASTAVA AND D.V.CHOPRA
CH. 35
variables dl9..., d6. For orthogonal arrays with (a general) index μ, there is further simplification since now the right hand side of (2.2) involves just μ (instead of μθ9. .., μ4). Thus it becomes a simple matter to check the existence of orthogonal arrays with m = 6. Indeed, using the above, we would easily find that orthogonal arrays with m = 6, t = 4 exist for all μ, except for μ = 1 and 3 (Seiden and Zemach [1966]). The considerable simplification occurring for m = 6 holds out promise for the study of existence of orthogonal arrays with m ^ 7. References A. Barlotti, 1965, Some topics infinitegeometrical structures, University of North Carolina, Institute of Statistics, Mimeo Series No. 439 (Chapel Hill, N. Car.). R. C. Bose, 1947, Mathematical theory of the symmetrical factorial design, Sankhya 8, 249-256. R. C. Bose, 1961, On some connections between the design of experiments and information theory, Bull. Intern. Statist. Inst. 38, part IV. R. C. Bose and K. A. Bush, 1952, Orthogonal arrays of strength two and three, Ann. Math. Statist. 23, 508-524. R. C. Bose and J. N. Srivastava, 1964, Multidimensional partially balanced designs and their analysis, with applications to partially balanced factorial fractions, Sankhya Ser. A 26, 145-168. I. M. Chakravarti, 1956, Fractional replication in asymmetrical factorial designs and parti ally balanced arrays, Sankhya 17,143-164. D. V. Chopra and J. N. Srivastava, 1971, Optimal balanced 27 fractional factorial designs of resolution V, N < 41 (submitted for publication). P. Dembowski, 1968, Finite Geometries (Springer, New York). T. A. Dowling, 1971, Codes, packings and the critical problem, Proc. Conf. on Combinatorial Geometry and Its Applications (A. Barlotti, ed.; Perugia, Italy). J. A. Rafter, 1971, Contributions to the theory and construction of partially balanced arrays, Ph.D. dissertation (under Prof. E. Seiden), Michigan State University, East Lansing, Mich. C. R. Rao, 1947, Factorial arrangements derivable from combinatorial arrangements of arrays, Suppl. J. Roy. Statist. Soc. 9,123-139. E. Seiden and R. Zemach, 1966, On orthogonal arrays, Ann. Math. Statist. 37, 1355-1370. J. N. Srivastava, 1967, Investigations on the basic theory of 2™3η fractional factorial designs of resolution V and related orthogonal arrays (Abstract), Ann. Math. Statist. 38, 637. J. N. Srivastava, 1971, Some general existence conditions for balanced arrays of strength t and 2 symbols, /. Combin. Theory, to appear. J. N. Srivastava and D. A. Anderson, 1971, Factorial subassembly association schemes with application to the construction of multi-dimensional partially balanced designs, Ann. Math. Statist. 42, 1167-1181. J. N. Srivastava and D. V. Chopra, 1971a, On the characteristic roots of the information matrix of balanced fractional 2m factorial designs of resolution V, with applications, Ann. Math. Statist. 42, 722-734. J. N. Srivastava and D. V. Chopra, 1971b, Balanced optimal 2m fractional factorial designs of resolution V, m < 6, Technometrics 13, 257-269.
Balanced Arrays and Orthogonal Arrays J. N. SRIVASTAVA t Colorado State University, Fort Collins, Colo., U.S.A. and D. V. CHOPRA Wichita State University, Wichita, Kansas 67208, U.S.A. Summary We point out the relation between the theory of balanced arrays and various com binatorial areas of design of experiment. Recalling some combinatorial theorems from Srivastava [1971], we apply these to prove a class of new and simple but rather stringent results on the existence of balanced arrays. Applications to the special case of orthogonal arrays are also considered. 1. Introduction Although orthogonal arrays have been known for almost a quarter of a century, the balanced arrays are much more recent. It will therefore be useful to first define the latter. Definition 1.1. A balanced array (B-array) T with s symbols, m rows, N columns, and strength t, is an m x N matrix T with elements belonging to a set S containing s symbols, such that for every (/ x N) submatrix T0 of T, and for every vector v of size (t x 1) with elements from S, we have A(v, T0) = A(P(v), T0)9
for every P.
(1.1)
Here, λ(\9 T0) stands for the number of columns of T0 which are identical with v, and P(v) is any vector obtained by permuting the elements of v. Thus P stands for permutation. Definition 1.2. The matrix T considered in Definition 1.1 is called an orthogonal array with the same parameters, if for every (t x N) submatrix T0 of T9 and for every pair of vectors v and v* of size (t x 1) each, we have A(v, Γ 0 ) = λ(ν*, To).
(1.2)
Clearly, an orthogonal array is a special case of a balanced array. Also, t The work of this author was partly supported by NSF Grant No. GP 30598 X. 411
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clearly, N must be a multiple of s*. The positive integer μ = Ns~* is called the index of the orthogonal array. Definition 1.3. Let one of the elements of S be denoted by zero (0), so that the remaining s— 1 elements of S may be called non-zero elements. Let v be any vector with elements from S. Then the weight of\, denoted by w(v), is defined to be the number of non-zero elements in v. Under Definition 1.1, consider the special case s = 2, with S having the elements 0 and 1. From (1.1), it is clear that λ(\, Τ0) will be a constant for all vectors v whose weight is the same. Thus, in this case, there exist nonnegative integers μ{ (k = 0, 1,. . ., t) such that if \(t x 1) is of weight k, then λ(\, Τ0) = μ\ for every (t x N) sub-matrix T0 of T. The vector (μ'0, μ\9.. ., μ[) is called the index set of the array. We shall now discuss the connection of balanced arrays with other older and more well known branches of the combinatorial theory of design of experiments and coding theory. Consider first the special case of orthogonal arrays. When t = 2 and s = 2, we get an important class of arrays which are equivalent to Hadamard matrices. On the other hand, the case t = 2 and N = s2, corresponds to a set of (m —2) mutually orthogonal Latin squares of order s. General orthogonal arrays can be classified into three categories A, B, and C. Arrays of type A are those whose columns form the set of solutions of an equation of the form Hx — c, where H(k x m) is a matrix over GF(s) of rank k, and c(k x 1) is a vector over GF(.s). Obviously, in this case, s must be a prime power and N = sm~k. It can be shown (Rao [1946]) that the sm~k solutions to the above equation will form an orthogonal array of strength /, if the matrix H has the property Qt which states that every vector in the row space of H must be of weight greater than or equal to t+l. Arrays of type A may also be called "vector space" arrays, since the set of columns of such an array obviously forms a vector space over GF(s·). Type A arrays are also equivalent to group codes. Indeed, the null space of the columns of H can be shown to be a terror correcting code if t = 2d, and a rf-error correcting and (d+ l)-error detecting code if t = 2d+\. Also, type A arrays are important from the point of view of the classical theory of confounding and fractional replication as well. Thus this class of arrays forms the famous link (Bose [1961]) between the theory of factorial designs and coding theory. Both from the point of view of obtaining a code with minimum redundancy, and a fractionally replicated design having the smallest number of treatment combinations, one is interested in obtaining matrices H with property Qt which maximize k (for given m) or maximize m (for given r, where r = m — k). It is well known that the latter problem is equivalent to Bose's "packing problem": Find the maximum number m
CH. 35
BALANCED ARRAYS AND ORTHOGONAL ARRAYS
413
of points in PG(r— 1, s) such that no subset of t points lies on a hyperplane of dimension less than t— 1. The solution of the packing problem is therefore important both from the coding and the design point of view. It may be useful to remark that the cases t = 2, and t = 3, s = 2, are solved (Bose [1947]). Also, the problem has been solved for many other combinations of s, t and r, with the result that almost all cases of practical interest from the design viewpoint are taken care of. However, from the coding point of view, the problem is far from solved. Here the interest lies in low values of t, particularly t = 4, and relatively large values of r (say between 20 to 200). Incidentally, an interesting link between the packing problem and some aspects of abstract combinatorial geometry has recently been established by Dowling [1971]. For some references to work on the packing problem, see Barlotti [1965]. We now discuss orthogonal arrays of type B. Consider the equations Hx = Ci (i = 1 , . . . , / ) , where H is (kxm), cf is (kx 1), and rank(i/) = k < m. We assume that H does not have the property Qt. Let Tfyn x sm~k) be the array obtained by solving Hx = cf. Let T = [T± T2 · · · Tf], and let C = [CJL, c 2 , . . . , Cf]. Then the following (unpublished) result (Srivastava [1967]) holds: The matrix T(mxf-sm~k) forms an orthogonal array of strength t if and only if for every nonnull vector u'(l x k) such that weight(u'/0 < t, the vector (u'C) is an orthogonal array of (size (1 xf) and) strengthj. The arrays of type B are obviously of a different structure than those of type A. For type B arrays, N does not have to be a power of s. Incidentally, there exist examples of type A and type B arrays with the same m and N9 and m being the maximum possible for the given N. Arrays which are not of types A or B are said to be of type C. A useful array which is not of type A is given in Bose and Bush [1953]. The parameters are s = 3, m = 7, N = 18, t = 2, μ = 2. Coming back to balanced arrays, let us first consider the special case s = 2, t = 2. In this case it can be shown that the array is the incidence matrix of a balanced incomplete block design (BIBD) with possibly unequal block sizes, written BIBDU for short. The ordinary BIBD's form a special case of these when the block sizes are constrained to be all equal, and thus correspond to arrays with the further restriction that each column be of equal weight. This shows that certain techniques of constructing BIB designs, such as the method of difference sets, could be generalized to help in the construction of balanced arrays. The case s = 2, t = 3 also is important. The so-called inversive planes form a sub-class of such arrays when the column weights are constant. Balanced arrays in general are useful in the theory of factorial designs. Indeed, balanced arrays of strength 2, 3 and 4 are identical with balanced fractional factorial designs of resolutions 3, 4 and 5, respectively. We shall
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CH. 35
not discuss here the detailed properties of balanced arrays considered as fractional factorial designs. The interested reader may refer to Srivastava and Chopra [1971a,b]. From the foregoing discussion, it may appear that the theory of partially balanced designs is not related to the theory of balanced arrays. However, this is not true. As is shown in Bose and Srivastava [1964], certain important principal submatrices of the "information matrix" corresponding to a balanced fractional factorial design (i.e., a balanced array) belong to the linear associative algebras generated by certain well known partially balanced association schemes. (Incidentally, association schemes are merely a different name for what are known as strongly regular graphs.) These algebras proved very helpful in certain statistical studies by Srivastava and Chopra [1971a]. Apart from the above, there are other more direct connections between PBIB designs and B-arrays. We in fact believe that most of the association schemes can be defined using certain balanced arrays. For some work in this field, see Srivastava and Anderson [1971]. It might be important to stress here that balanced arrays not only provide a mathematically challenging field of research which unites various branches of the combinatorial theory of design of experiments, they are also urgently needed for practical problems arising in factorial experimentation. For some literature on B-arrays in general, the reader is referred to Chakravarti [1956], Srivastava [1971] and Rafter [1971]. In some of the earlier work, B-arrays have been called "partially balanced" arrays. However, since the B-arrays are related directly to BIBD's rather than the PBIBD's, we have adopted the new terminology. 2. Preliminaries In this section, we present without proof some results from Srivastava [1971] for later use. Theorem 2.1. A set of necessary and sufficient conditions for the existence of a 5-rowed B-array T, of strength 4 and index set μ' = (μ0, μί9 μ2, μζ, μ*) is that there exists an integer d such that d > *AiiOO = max (0, μ*-μ3, d< Ψι2(μ) = ΐΤίίη(μ4,μ4-μ3
μ^-μζ+μ2-μι)9 + μ29μ^-μ3 +
~ „ μ2-μ1+μ0\
Theorem 2.2. A set of necessary and sufficient conditions for the existence of a 6-rowed array T of strength 4 and index set μ' is that there exists an integer d0 such that d0 > ΦιΜ = max (0, ^ . 2 + 01, rfw + 02, ^ι·6 + 03), d0 < ^2 2 (μ) = min (d69 d^-θ^ rf2.6-05),
~
2)
CH.
35
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415
where d^ = dt + di+1+ · · · +dj9j > i;for each i9 d = dt satisfies (2.1); and dx ^ d2 > * · · ^ d6. Also, the 0\s in (2.2) are given in terms of the tfs by 0i = - / U , 02 = - μ 2 + 2 μ 3 - 3 μ 4 , 03 = - μ 0 + 2 μ 1 - 3 μ 2 + 4 μ 3 - 5 μ 4 , (2.3) 04 = - μ 3 + 2μ4, 0 5 = - μ ι + 2 μ 2 - 3 μ 3 + 4μ 4 . Equivalently, (2.2) can be rewritten as "0 (a) 'de —di—d2+d6 (b) -μ4 - μ 2 + 2μ 3 --3μ 4 —d1—d2—d3 — d4+d6 (c) - μ 0 + 2μ1-3μ2+4μ3-5μ4 —d1—d2—d3—d4.—d5 (d) - μ 3 + 2μ4 dA+ds+d6 (e) (2.2) —d1—d2+d4.+ds+d6 μ4-μ3 (f) > 5^ —d1—d2—d3+d5+d6 -μ2+μ3-μ4 (g) - μ0 + 2μ! - 3μ2 + 3μ3 - 3μ4 —d — d —d 1 2 3 00 - μ ι + 2 μ 2 - 3 μ 3 + 4μ 4 d2+d3+d4.+d5+d6 Ö) - μ ι + 2 μ 2 - 3 μ 3 + 3μ4 -di+ds + d^+ds+de (j) -di+ds+de -μι+μ2-μ3+μ4 (k) _-μ0 + μ ι - μ 2 + μ 3 - μ 4 -di j (1) Theorem 2.3. Consider an array T(m9 N9 4; μ'). Let xt be the number of columns of T of weight i (i = 0, 1,. .., m)9 so that YJ=O Xi = N. Then the following single diophantine equations (SDE) must hold:
Σ 0(Z-j)*i = Οφμ,·,
J = 0, 1, 2, 3, 4.
(2.4)
Definition 2.1. An array T with 2 symbols 0 and 1, and m rows is called^ "trim" if x0 = xm = 0. Theorem 2.4. Let T be a trim array with parameters (m9 N9 4; μ'), and let xt be defined as in Theorem 2.3. Let there be g distinct solutions (dr0, drl9. . . dr6)for the vector (d09 dl9.. ., d6) under the inequalities (2.2). Note that each of these solutions would correspond to a distinct 6-rowed array; let these be denoted by Tr(r = 1 , . . . , g). Let yr denote the number of times the 6-rowed array TT occurs as a subarray* of T9 so that Yfr= ί yr = (g). Define 5rj (r = 1, .9g;j = 0, 1 , . . . , 6) by S0 = — θ3 — 6cl+d09 δι = -e5+5a-d09 δ2 = -92-4ä'+d09 <53 = -04+33-4» (2.5) <54= -e^ll+do, <55 = - 2 -d09 δ6 = d09 δ' = (δθ9 δΐ9 . . ·,δ6)9 3 = (i)(rf1+---+rf6), * Here, by a 6-rowed subarray of T we mean an array which would be left after cutting out any set of m—6 rows of T. The rows of the 6-rowed array which is left are not to be permuted.
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where the Fs are defined at (2.3). Let π,. = £ * = 1 SrJyr; j = 0, 1,. . ., 6. Then the following triple diophantine equations (TDE) must be satisfied for the case m = 8: (a) lx1+x2 = π 0 , (b) 21JC1 + 12X 2 + 3JC3 = 6πΐ9 (c) 15x 2 + 15*3+ 6x4 = 15π2, (d) 10*3+ 16Χ 4 +10Λ:5 = 20π 3 , (e) 6*4+15*5 + 15x6 = 15π4, (f) 3*5 + 12* 6 +21* 7 = 6π 5 , (g) * 6 + 7*7 = π 6 , (h) Yfr=l yr = 28. The equations (2.6) are termed TDE because there are three sets of variables involved, namely, the *'s, the d's and the y9s. Corollary 2.1. When g = 1, the equations (2.6) reduce to the following double diophantine equations (DDE): (a) 7*x+* 2 = 28(50, (b) 21* 1 + 12*2 + 3* 3 = 168^, (c) 15*2 +15*3 + 6*4 = 420(52, (d) 10*3 +16* 4 +10* 5 = 560<53 (e) 6*4+15*5 + 15*6 = 420<54, (f) 3* 5 + 12*6 + 21* 7 = 168i 5 , ^ ' ' (g) * 6 + 7* 7 = 28(56. In eqs. (2.7), we have dropped the subscript 1 in the dn(j = 0 , . . . , 6). Corollary 2.2. Suppose that for some j9 we have Sj = Sj+2 = 0 and öj+ι Φ 0. Then the DDE are contradicted. We also recall the following from Srivastava and Chopra [1971a]: Theorem 2.5. Let T be a B-array with m ^ 8, and t = 4. Then ΑΊ+μ 3 >$μ2* 4μ! < μι^ι+μ^
+ όμ^.
(2.8) (2.9)
3. Analysis of the SDE For m = 8, the SDE (2.4) reduce to 70*o + 35* 1 + 15*2 + 5* 3 +*4 = 70μ0> 35*!+40* 2 + 30*3+16*4 + 5*5 = 280μ1? 15*2 + 30*3 + 36*4 + 30*5 + 15*6 = 420μ2, 5*3 + 16*4 + 30*5 + 40* 6 + 35* 7 = 280μ3, * 4 + 5*5 + 15*6 + 35* 7 + 70* 8 = 70μ4·
(3.1) (3.2) (3.3) (3.4) (3.5)
The above equations involve nine variables. Among these, * 0 and * 8 can, in most cases, be gotten rid of by using Theorem 3.1 below. Let A = Α(μΐ9 μ29 μ 3 ) be the class of all B-arrays of strength 4 and m = 8 with fixed values of μΐ9 μ2, μ$. Let A0 = Α0(μΐ9 μ2, μ 3 ) be a subclass of A with the further restriction that if an array T belongs to A0 then T is *'trim", i.e., T does not contain any column of weight 0 or 8. Thus, if Te A0 then the value of (x0+x8)9 for T9 equals zero.
CH. 35
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BALANCED ARRAYS AND ORTHOGONAL ARRAYS
Theorem 3.1. (a) If for given μΐ9 μ2 and μ3, the subclass Α0(μί9 μ2, μ3) is empty, then the whole class Α(μΐ9 μ2, μ3) is empty, (b) If A0 is nonempty, Te A0, and T has index set (μ0, μί9 μ29 μ3, μ^)9 then there exists an infinite number of arrays in the class A n Ä0 (i.e.9 with x0+xs Φ 0) with the index set of the form (μ*9 μΐ9 μ29 μ3, μ*), with μ* ^ μθ9 μ* > μ 4
μ" = μο+μ*, μ' = μι+μ 3 , μ'ό = μο-μ*> μΌ = μι-μ* = 0. In the above, and throughout this paper, the symbol = means that the corresponding inequality holds for arrays with μχ = μ2 = μ3. From (3.1-3.6), we have, by simple additions and subtractions, 5w+10t;+12;c = 140μ2,
(3.7)
?>5w+\5u + 5v + 2x = 70μ",
(3.8)
= 280μ' = 560μ2, lw' + 3u' + v' = 14μ{)',
35Η>+40Μ+35Ι; + 32Λ;
7w' + Su' + 5v' = 56μ£ = 0, 5u + 6v + 6x = 14(4μ'-μ") = 14(8μ2-μ"), 2ι> + 3* = 7(10μ 2 -4μ'+μ") = 7(2μ2+μ"), \4w + 9u-4x = 28(4μ'-7μ 2 ) = 28μ2.
(3.9) (3.10) (3.11) (3.12) (3.13) (3.14)
.7), it is clear that 12* < 140μ2, and 5\x9 so that x = Sk9
0 < k < %μ2.
(3.15)
where A: is a non-negative integer. From (3.13), we have that 7\(2v+k). Let 2v + k = lkl9 where k1 is a non-negative integer. Then v = i(7k1-k)9 (3.16) which implies kt > jk9 and k+kx is even. (3.17)
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From (3.13), we obtain 2k+kx
= 10μ2 - 4 μ ' + μ " = 2μ2+μ".
(3.18)
μ" = 4 μ ' - 1 0 μ 2 + (2£ + &1) = -2μ2 + (2/<+/€1),
(3.19)
(3.17) and (3.18) give
\k < ikj. = ( 1 0 μ 2 - 4 μ ' + μ " ) - 2 £ = (2μ 2 +μ")-2£. Substituting from (3.15) and (3.16) into (3.9), we have, 0 ^ u = 28μ 2 -7£ 1 -11&. Substituting from (3.21) and (3.15) into (3.14), we get
(3.20) (3.21)
(3.22) 0 < w = 2(V-16/i 2 )+i(17Jk + 9ik1) = - 1 6 μ 2 + Κ17Λ + 9Λχ) Combining (3.21) with (3.17), we have \k^k, <[4μ2-^], (3.23) where [x] denotes the largest integer less than or equal to x. Subtracting (3.11) from (3.10), we have -5U'-4O'
= 140IS-4/I'O) =
14^.
Therefore 14|(5w' + 4t;')> and u' is even. Thus we have (3.24) u' = 2/ 2 , l2 being any integer. Also, from (3.11), we have 7|(w' — 2υ'). Therefore u'-2v' = 7/ lf (3.25) /x is any integer. From (3.24) and (3.25), we have v' = h-ih = /a-7/3, (3.26) where /! = 2/ 3 . Hence w', 1/ are known as soon as we choose l2 and / 3 . Substituting from (3.26) and (3.24) into (3.11), we have w ' = 8 ^ - 3 / 2 + 5/ 3 = 5 / 3 - 3 / 2 . Similarly, (3.27), (3.26), (3.24) and (3.10) give
(3.27)
h = -μο + 4μ'ο + 21* = 21*-μΌ· (3-28) The above equations give the value of (w, v, w, x; u\ v\ w') in terms of μ', k, ku l2 and / 3 . Knowing the values of the former, the equations (3.1)(3.5) can be solved for the x f 's using (3.6). Furthermore, from (3.6), it is obvious that u > |ii'|,
v ^ \v'\, w > |w'|, μ' > \μ'0\, μ" ^ | ^ Ι ,
(3.29)
and (u±u')9 (v±v')9 (νν + νν'), (μ'±μ'0) and (μ"±μ'ό) are even. Now, (3.3) together with (3.16), (3.26), (3.18) and (3.28) implies that i(k + k1)±(l2-l3) and kx±(l2-2l3) are both even.
(3.30) (3.31)
CH.
35
BALANCED ARRAYS AND ORTHOGONAL ARRAYS
419
Furthermore, v > \v'\ and u ^ |w'| give us respectively #H7A:i-*) + /J > h > \l-Wki-k) + hl (3.32) \u > l2 > -\u. (3.33) The x/s, or more precisely (pcl9..., x7), will be called the primary weight parameters of an array, and the set (u, v, w; u', v', w') will be called the secondary weight parameters. In the above analysis, we have expressed the secondary weight parameters in terms of (k, μ2, kl9 μ'; ll9 /3, μό)> which will be called "tertiary" weight parameters. We have not included μ" and μβ in the last set, since by (3.19) and (3.28) they get fixed when the other tertiary parameters are fixed. Next, from these results we obtain a lower bound on the number of assemblies in a B-array of strength four with two symbols, and m ^ 8. Theorem 3.2. For a trim B-array with 2 symbols, t = 4 andm > 8, we have N ^ ^J. μ2. (As will be shown later, the bound is attainable for μ2 = 6). Proof. Clearly it is enough to prove the theorem when m = 8. We first observe that (3.34) N = μ0 + 4μ1 + 6μ2 + 4μ 3 +μ 4 = 6μ 2 +μ" + 4μ', where μ', μ" have been defined in (3.6). Substituting from (3.19) and (3.22) in the above, we have N > 28μ2-Κ13Α: + 7Α:1) = 2&μ2-(χ3 + χ5 + ΙχΑ). (3.35) Using (3.15) and (3.21) in (3.35), we obtain the result. Corollary 3.1. Ifk = 0,k1 = 2 and μ2 = 1, then μ' ^ 3. Proof. This follows from the inequality (3.22). Theorem 3.3. Let The a trim B-array with (a) μ2 = 1 and Xt = 5, then N ^ 23; (b) μ2 > 0, then μ± αηάμ$ are not both zero, αηάμ0 αηάμ^ are not both zero. Proof, (a) From (3.15), (3.17) and (3.21) we have k = kt = 1. Then (3.22) and (3.19) give respectively μ' > 3 and μ" ^ 5. The result follows from (3.34). (b) ΤΐΊ^μχ = μ3 = 0 is not possible follows from (3.2)-(3.4) since μ2 = 1. Similarly for μ0 and μ4. Theorem 3.4. (a) If no B-array exists corresponding to a given set (x, u,v,w; u', v1, w') of weight parameters and a given value of μ2, then so is the case with the weight parameters (x, u, v, w; —u', —v', —w')for that value of μ2. (b) If no array exists corresponding to a given set (k, μ2, ku μ'; l2, /3, μ'0) then so is the case for the set (k, μ2, kl9 μ'; —12, —/3, — μό). Proof, (a) This follows by observing that if there exists a B-array T with
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the secondary parameters (x, u, v, w; — u', — v', — w'), then by interchanging 0 and 1 in T we shall get the complementary array T with parameters (x, u, v, w; u', v', w'). (b) Assume, as in (a), that an array T exists corresponding to the para meters (k, μ2, ku μ', μ"; —/2, —/3, —μ'θ9 — μ'ό). Then it is clear that for T the values of k, μ2, kl9 μ', μ" will remain unchanged. Since the parameters (—u', —i?', — w') of T change to (u\ v', w') in T, it can be checked using successively equations (3.24), (3.26), (3.27) and (3.28) that the values {-l2, — h> —μ'ο) f ° r ^change to (l2, / 3 , μ'0) for T. This completes the proof. Theorem 3.5. Let Tbe a B-array with parameters (m, N,1; μ'). Suppose that for some i (0 < i'^ m) we have Xi + xm_i > 0, where xt is the number of columns of weight i. Then there exist two non-trim arrays 7\ and T2 with m — i and i rows, respectively, and having other parameters the same as those ofT. Proof. There exists a column v in T such that w(y) = /, say. Then 7\ (or T2) are obtained by omitting all rows of T which have 1 (or 0) in v. Similarly when w(v) = m — i. 4. Arrays with μι = 1 First we note a result for later use. Theorem 4.1. (a) There does not exist any trim B-array with m = 7, μ2 = 1 and N < 28, except when N = 21. The array with N = 21 has index set (0, 0, 1,3, 3), and is unique except for an interchange ofO and 1. (b) Let there be an array with 21 ^ N *ζ 27, m = Ί and μ2 = 1. Then, apart from an interchange ofO and 1, its index set is of the form (μ0, 0, 1,3, μ 4 ) with μ0 ^ 0, μ 4 ^ 3. Proof, (a) The nonexistence of arrays with 21 < TV < 28 follows from Theorems 2.1 and 2.6, and Corollary 2.1 of Chopra and Srivastava [1971]. Using these same theorems, it is easily observed that N = 21 implies μ' = (0, 0, 1, 3, 3), apart from an interchange of 0 or 1. By using the SDE for N = 21, one then obtains x2 = 21, from which uniqueness of the array is easily shown. (The array is obtained by taking all 7-vectors of weight 2). (b) This follows from (a). Theorem 4.2. Let T(m xN) be a B-array with two symbols, t = 4, m ^ 8 and μ2 = 1. Then N ^ 28. Proof. Clearly, it is sufficient to prove the result for trim arrays T. From (3.15), we get k = 0, 1, 2, and hence x = 0, 5, 10. When k = 2, (3.21) gives lkx < 6, and hence kx = 0. But then (3.20) gives k = 0, a contradiction. Hence x = 0 or 5.
CH. 35
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421
Now, from (2.1), we have 0 < μ " - μ ' + 1 , or μ" ^ μ ' - Ι . Hence N = μ"+4μ' + 6μ2 S* 5(μ' + 1). Then N < 28 implies μ' < 4. Also, (2.9) gives μ' > 2. Hence 2 < μ' < 4. From (3.9), we have 8μ' = w+fy+v+^x = N+±u—gsX. From (3.7), we get u+2v+^x = 28. Hence, since 7|(2t;+A:), the permissible values (for N < 28) of (x, u) are (0, 14), (0, 0) and (5, 10). These values of (x, u) correspond respectively to 8μ' = N+2, N and (N+1). Hence N < 28 gives μ' < 4. When μ' = 2, (JC, w) = (0, 14), we get N = 14 < 5μ' + 5. Hence, since lkx = 2v+k and Ι / + 2 Ι ; + ^ Λ : = 28, and in view of Theorem 3.3(a), the possible values (for N < 28) of (k, kl9 μ', Ν) are (0, 4, 2, 16), (0, 2, 3, 22), (1, 1, 3, 23) and (0, 4, 3, 24). We consider these one by one. Let N = 16. Then μ' = 2 = μ". Hence μ' is of the form (μ0, μΐ9 1, 2—μ1? 2—μ0). If an array with this index set exists, then by interchanging 0 and 1 in this array, we can obtain an array with index set (2—μ0, 2—μί9 1, μΐ5 μ0). By adjoining the two arrays, we shall get an array with m = 8, N = 32, and index set (2, 2, 2, 2, 2). Such an array is, however, known to be nonexistent (Seiden and Zemach [1966]). Hence N Φ 16. When 22 < N < 24, we get μ' = 3 and w > 0. By Theorem 3.4, μχμ3 Φ 0. But, using Theorem 3.5 with m = 8, / = 1, and also Theorem 4.1(b), it follows that we must have μ1μ3 = 0, a contradiction. This completes the proof. Theorem 4.3. There exists a unique {apart from (0, 1) interchange) B-array of strength 4, m = 8, μ2 = 1, and 28 assemblies. Its index set is (0, 0, 1,4, 6), and it is obtained by taking all the distinct %-vectors of weight 2. Proof. From Theorem 2.17 of Chopra and Srivastava [1971], it follows that the arrays with m = 7, iV = 28 and μ2 = 1 must have index sets of the forms (3,1,1,3,3) or (a, 0, 1, 4, 3+j8),wherea > Ο,β ^ Ο^ΐιβημ' = (3, 1, 1,3,3) and m = 8, (3.18) gives 2k+kx = 0, so that k = kx = 0. Then, from (3.21), u = 28, and hence x = v = w = 0. Then (3.10) and (3.11) give respectively 3«' = 0, 8w' = 112, a contradiction. From Theorem 4.1, we find that m = 8, μ2 = 1, N = 28 implies that the array must be non-trim, so that in the SDE we have x0 = x8 = 0. If μ' = (α,Ο, 1,4, 3 + β), (3.11) and (3.10) give a = 0, and hence β = 3. Again, from (3.18), k = kt = 0. Hence u = 28, and (3.10) gives Zu' = 84, or u' = 28. This completes the proof. 5. The intermediate diophantine equations (IDE) We now derive a new set of diophantine equations, the IDE. These will be useful for later developments. Though these equations are new, the basic idea is the same as that for deriving the TDE (Srivastava [1971]) at (2.6).
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CH. 35
J.N.SRIVASTAVA AND D.V.CHOPRA
Indeed, the basic idea is simple. Suppose we are considering the existence of an (m x N) array T. Suppose further that the (k x N) subarrays T0 of T may be of g' different types Tou . . . , Tog'9 and that for each i and j (/ = 0, 1, . . . , k;j = 1,. .., g), we know the number of columns of Toj with weight /. Now, assuming that there are xi columns of T whose weight is /, we can compute the number of k-vectors of weight i which are contained in the columns of T. Also, we can compute this same quantity by assuming that out of the (?) subarrays of T there are Zj arrays of type T0J. Equating, we shall get k+1 equations, one for each value of /. The unknowns in these equations will be the xh the zp and probably some other quantities. The SDE and the TDE are obtained by developing this idea for the cases k = t and (t + 2), respectively, where t denotes the strength of the array. The IDE correspond to k = t+1. Consider a B-array T of size (m x N), strength t, and index set (μ'0, μ\9. . ., μ}). Let T0 be any ((t+1) x TV) subarray of T. Let d O 0) be the number of columns of T0 which equal (1, 1,. . ., 1)'. Then, in equation (2.4) of Srivastava [1971], it is shown that if φ·ν denotes the number of times any particular (t+ l)-vector of weight i occurs as a column of T0, then every (t + l)-vector of weight / occurs φι times in T0, and ^• = ^ - ^ + i + ^ + 2 - - ; - + ( - i y - i + 2 ^ + ( - i y + 1 " ^
0 < ί < f, (5.1a)
φΐ+1 = ά.
(5.1b)
The following is helpful in the calculation of the 0's: 4>i = Vt-
Σ (ΙΧ,ΪΓ',-Κ = CV)E^«], o < i ^ t+i, γζά = GTO. (5.3)
9= 0
d
d
These equations contain two sets of unknowns, the x's and the z's. Since φΜ are easy to compute (relative to the ö's needed for the TDE or DDE), the IDE are (relatively) easier to use. Still, however, in many applications, one needs to go to the TDE. For the case m = 8, t = 4, (5.3) can be written in full as 56*0+ 21*! + 6x2 + *3 = 0* >
(5.4a)
CH.
35
BALANCED ARRAYS AND ORTHOGONAL ARRAYS
35*! + 30*2 +15*3+4* 4 = 5φ*9 20* 2 + 30Λ:3 + 24*4 +10* 5 = 1005, 10*3 + 24*4 + 30*5 + 20* 6 = 100*, 4*4 +15*5 + 30* 6 + 35* 7 = 50J, * 5 + 6 * 6 + 21* 7 + 56*8 = φ%,
423
(5.4b) ( 5 - 4c ) (5.4d) (5.4e) (5.4f)
where
# = Σ*Αι>
« = 0,1
5.
(5.5)
6. Arrays with μι = 2, 3 In Srivastava and Chopra [1971b], arrays with μ2 = 2 and N *ξ 41 have been studied. Other arrays with μ2 = 2 shall be studied in similar papers elsewhere. Here we consider only the case μ2 = 3. Henceforth, if P denotes any statement (such as the IDE, DDE, etc., or any inequality such as (2.2c)), then R(P) shall mean "the rejection of P ' \ Thus "R(TDE)" will mean "a contradiction of the TDE". Also, let
m
α = μ 1 - μ 0 , β = μ3-μ^ (6-1) Theorem 6.1. Consider a B-array T with parameters (m, N, ί;μ'), where ^ 8, t = 4, μ2 = 3, andoc + ß = 0. Then a = - ß = 0, 1 or - 1 .
Proof. It is enough to show that if a < 0, then a = — 1. If possible, let - a = ß > 1. Then (2.1) gives us ß = 3 or 2. Case (a), ß = 3. Here (2.1) gives if/u = \j/12 = 0, and hence d1 = d6 = 0. Also, μ0 > 3. From (2.2c,e,h), we get μ 4 = 3, and 3 < μ0 < 6. Hence Φ21 = Φ22 = 0, and d0 = cl = 1, δ 3 = δ5 = 0, and <54 = 3, leading to R(DDE). Case (b). β = 2. Here ^ i r = 0, φ12 = 1. Hence, recalling the IDE at (5.4), we get h = 2, and φ* = z l9 φ* = 3z1 + 2z0, and <£* = z 0 . Hence, 56 = Zo + Zi = ^f + #J = 2*2 + 3*3+ (2.4)*4 +2*5+ 6*6+ 21*7 + 56* 8 . Now the SDE (3.4), (3.5) give 280jS = 560 = 5*3 + 12*4+10* 5 -20*6-105* 7 -280* 8 . Multiplying the first equation by 10 and subtracting the second, we get Xi = 0(i = 2 , . . . , 8). But this contradicts (5.4d). Hence the proof is complete. Theorem 6.2. Lei i k ö possible B-array with μ2 = 3, m ^ 8, a = 0 Ö«U? j? > 0. Then ß = 1. (Similarly, ß = 0, a > 0 wip/fey a = 1.) Proof. Let j8 > 1; then (2.1) shows that ß = 3 or 2. Case (a). /? = 3 implies d^ = d6 = 0 and μ' = (0, 0, 3, 6, 3). Here d0 = 0, and this leads to R(DDE). Case (b). β = 2. Here 0 < ^r n < ψί2 < 1. In the IDE,- take A = 2. Then 05d = d, φ4ά = μ 4 - ί / , Φζά = 2 + rf, 02d = l-^> 0id = μ ο - Ι + ^ » Φο<ι =*
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J. N.SRIVASTAVA AND D.V.CHOPRA
CH. 35
l-d, a n d 0 ? = ζΐ9φ% = 5βμΑΓ-ζΐ9φ% = 112+ζΐ9φ% = ζθ9 φ* = 56μ0 + ζθ9 φ* = ζ0. Hence, from (5.4), we have 0 = φ* + φ* + 3φ*—φ* = 56χ0 + 2lx1 + Sx2 + 3x3+x5 + \6x6 + 63x1 + l6Sxs. Hence, xf = 0, for / Φ 4. Hence 56 = z0+zt = φ* + φ* = 0, a contradiction. This completes the proof. Theorem 6.3. Let T be a B-array of strength 4 with m ^ 8, μ2 = 3. Then a and β cannot be both positive. Proof. Suppose a > 0, β > 0. As before, we find /? < 3 and x + ß *ζ 3, so that (α, β) = (1,2), (2,1) or (1,1). Case (i). (a, ß) = (1, 2). Here (^ l l 5 ^ 1 2 ) = (0, 0), dt = 0 (all i), and using (2.2c,e), we find that μ' = (μ0, μ 0 +1> 3, μ4 + 2, μ 4 ), with μ 4 = 1 or 2. Then R(DDE) follows from δ' = (0, μ0, μ 4 - 1 , 2 - μ 4 , μ4, 0, 0). Proceed similarly for (α, /?) = (2, 1). Co«? (ii). (α,]8) = (1,1) with (ψ1ΐ9 φ12) = (0, 0). We get μ' = (μ0 ,μ 0 + h 3, 1, 0). Here d0 = 0, δ' = ( 3 - μ 0 , μ0-2,1,1,0, 0, 0), and R(DDE) follows. Case (iii). (α, β) = (1, 1) with (φ1ΐ9 φί2) = (0, 1). Here μ' = (μ0, μ0+ U 3, 1 + μ4, μ 4 ) and then 0 < rf6 < rfx < 1. Because of (2.2f,g) all values of d' with d2 φ d5 are rejected, and we consider the remaining four values one by one. Take d1 = d6 = 1. Using (2.2c,e,h,k) and (2.2 d,i), the possible values of (μο> μ*) are (0, 2), (0, 3), (1, 3) and (1, 4), and d0 = 3 + μ 0 - μ 4 . Then δ' = (0, 0, μ0, 1 - μ 0 , 1 + μο> ^4 — ^0 —2, 3 + μ 0 - μ 4 ) , and we have R(DDE). Next, take di = d5 = 1, 0. 77ze« e/fAer a = 0, ß > 0 or a > 0, ß = 0. Proof. The proof follows from Theorem 6.3 if we show that a ^ 0, ß ^ 0. Suppose a < 0. Then, as before, /? = 3 or 2. C&se (a), ß = 3. Here dt = 0 for each /, and μ' = (μ0, μ0 + α, 3, 6, 3), d0 = 0, leading to R(DDE). Case (b). j8 = 2. Then a = - l a n d 0 ^ ^ n < ^ 1 2 < 1. For the IDE, we then obtain φ% = ζθ9 φ% = l\2 + zl9 φ% = ζί9 ζ0 + ζ1 = 56. Then 0 = 2φ% + 3φ%-φ% = 4 χ 2 + 5Χ 3 + (2.4)Λ;4 + 2Χ5 + 16Χ 6 + 63Χ 7 + 168Λ:8. Hence xf = 0 (/ = 2 , . . ., 8), a contradiction. Theorem 6.5. Consider a B-array T of strength four, with m ^ 8, μ2 = 3, μί ^ 2, a«rf α + jß < 0. 77te/? a ^ 0, /? < 0, H>#A strict inequality in at least one case.
CH.
35
BALANCED ARRAYS AND ORTHOGONAL ARRAYS
425
Proof· Let a > 0. Now a and — ß are positive. It is obvious that μχ and μ 4 are both nonzero. (2.1) implies a < 3, a < μχ and —ß>oc. Case (i). a = 2 = μ±. When μ1 = 2, we get ψ^ = i/f12 = 1-/J, so that μ' = (0, 2, 3, μ 4 + β, μ 4 ). But this contradicts (2.2). When μι > 2, we have Φιι = Άΐ2 = —ß> a n d μ' = (μ0, μο + 2, 3,0, —β). From (2.2), we observe μ0 = 4, 5, so that μ' = (4, 6, 3, 0, — jß) and (5,7, 3,0, — ß). Finally (using the DDE), the nonexistence of the corresponding arrays follows from (d0;cl) = (-j8; - / 0 , «i = <53 = 0, δ2 = 3; and (d0;3) = ( - / ? ; -j3), <50 = <53 = 0, δχ=δ2
= 3.
Core (ii). Next, we take a = 3. Then μι ^ 3, and i^n = ψ21 = —ß> a n d μ' = (μ0, μο + 3, 3, μ 4 + β, μ 4 ). From (2.2h,j,e), we have μ0 = 3, and - j ? < μ 4 ^ -2jS. Hence μ' = (3, 6, 3, μ 4 + 0, μ 4 ), ^ 2 ι = Ά22 = -μ 4 -2.}> and <5j = <53, leading to R(DDE). Case (iii). a = 1 implies (ψ1ί9 φ12) = (2-jS, 2-j8), ( l - j 8 , 2-j8) or (-j?, 1 — jß) according as μ! = 1, 2 or > 2. These are considered separately. (a) For a = μχ = 1, we have dx = d6 = 2 — β9 and μ' = (0, 1, 3, μ 4 + β, μ 4 ). From (2.2c,i), we get 3-j? < μ 4 ^ 5-β, leading to ^ 2 i = ^22 = 5 - 2 0 - μ 4 and δ' = (0,0,0, 1, 1 , μ 4 - 3 + 0, 5-2β-μ4). In the DDE, this implies x6 < 0, a contradiction. (b) When a = 1, μί = 2, we get (^ l l 5 ^ 1 2 ) = (1 - / ? , 2-jß) and μ' equals (1, 2, 3, μ 4 + β, μ 4 ). Because of (2.2f,g,j), we obtain (i) dx = >2 and 30x5 = — 20y2 —120j>!. This is possible only when y1 = y2 = 0, a contradiction. 7. Arrays with μ2 = 4 , 5 , 6
·
Theorem 7.1. Consider an (m x TV) B-array T of strength 4 H>/7A /Wex .sei μ', and μ2 = 4, m ^ 8. 77ze« TV ^ 56. Proof. Using (2.9), we get μ' S* 6. Moreover, if μ' ^ 8,theniV = μ" + 4μ' + 6μ2 ^ 56. Hence μ' = 6, 7. Let us consider these separately. Case (a). If μ' = 6, then (3.22), (3.15), (3.17) and (3.21) give (a) 9ki + llk ^ 160, (b) 0 < k < 9, (c) k^lk,^ 28μ 2 -11£. (7.1)
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J.N.SRIVASTAVA AND D.V.CHOPRA
CH. 35
Now, using (3.17), (3.18), we get (k, k^ = (7, 5). From (3.18), (3.22), we have μ" = 3 and w = 2. Now (3.21) gives u = 0, and hence * 2 = * 6 = 0; and (3.15), (3.16) give x = 35 and v = 14. Therefore, the SDE reduce to (a) 35* χ + 5* 3 + 35 = 70μ0, (b) 35*! +30JC 3 + 560+ 5*5 = 280μΐ9 (c) 30*3 + 1260 + 30x5 = 1680, (7.2) (d) 5*3 + 560 + 30*5 + 35* 7 = 280μ3, (e) 35 + 5* 5 + 35* 7 = 70μ4. From (7.2a,c,d,e) we find that 7|* 3 , 2|(* 1 +* 3 + l), 7|* 5 , 2|(* 1 +* 5 ), 2|(* 3 + * 7 ) and 2|(* 5 +* 7 + l). Also, v = 14, w = 2, together with the above conditions, give us (*3, * 5 ) = (0, 14), (7, 7) or (14, 0), and (xl9 * 7 ) = (0, 2), (1, 1) or (2, 0). Now (xl9 * 7 ) = (0, 2) and (*3, * 5 ) = (0, 14) contradict (7.2b), and (xl9 * 7 ) = (2, 0) and (*3, * 5 ) = (14, 0) contradict (6.4d). Finally, (*l5 * 7 ) = (1,1) and (*3, * 5 ) = (7, 7) is rejected by (7.2). Thus there does not exist a B-array having * 4 = 35, u = 0, v = 14 and w = 2. Case (b). When μ' = 7, we have 9k± + 17A: ^ 144. It can be easily checked that the only possible (k, kx) in this case are (9, 1), (8, 2), (7, 3), (7, 5), (6, 6), (0, 16). Of these, the pair (9, 1) is rejected because of (7.1c), and for the remaining possible values of (k, kx) we find that N ^ 56. This completes the proof. Theorem 7.2. There does exist a B-array Γ(8, N, 4; μ') having N = 56, μ2 = 4, and index set (4, 6, 4, 1, 0). The array is obtained by writing the 56 ^-vectors of weight 3 each. This array is unique, apart from an interchange of 0 and 1. Furthermore, this is the only array with N = 56. Proof. Using the result in Section 4, check that N = 56 implies (k, k^ = (0, 16). Also, then, μ' = 7, μ" = 4, * = u = w = 0, v = 56. Using the SDE, 5|(μ3 —1). Hence |t/| = 56. This gives the result. Theorem 7.3. For a B-array T of strength 4, with m ^ 8 and μ2 = 5, we have N ^ 66. (However, it is not known whether a B-array of strength 4 with m = 8, μ2 = 5 and N = 66 exists.) Proof. We have TV = 6μ2 + 4μ' + μ" = 30 + 4μ' + μ", which implies that μ' and μ" have to be minimum for N to be the smallest possible. Moreover, from (3.22), μ' is such that 16μ' ^ 64μ2 —[17^ + 9^!], where [*] denotes the greatest integer less than or equal to *, and therefore we take that value of μ' which satisfies
* - Η.~~ϊ6~-\|·
(7 3)
·
It is obvious from (7.3) that 17k+9k1 has to be as large as possible if μ' is to be minimum. From (3.15), we obtain 0 < k ^ 11. Now, using (3.17) and (3.23), we find that k φ 11, and that the possible values of (k, kx) (with k1 a maximum) are (0, 20), (1, 17), (2, 16), (3, 15), (4, 12), (5, 11), (6, 10), (7, 9),
CH. 35
427
BALANCED ARRAYS AND ORTHOGONAL ARRAYS
(8,6), (9,5) and (10,4). Furthermore, we observe that Ws(\7k+9k1)] assumes the largest value 12 when (k, k^) = (6, 10), (7, 9), (9, 5) and (10,4). Of these four competing pairs, the minimum value of N is given by that pair for which μ" is the least. It can be checked from (3.18) that (k, kx) = (6,10) gives the least value of μ", which equals 4. Hence the minimum number of assemblies required is 66. Theorem 7.4. Let T(m xN) be a B-array of strength 4, with μ2 = 6, and m > 8. Then Tmust have N ^ 70. Proof. From (2.9), we have μ' ^ 8. If μ' > 8, then N ^ 4μ' + 6μ2 ^ 72. When μ' = 8, the only values of μχ and μ 3 satisfying (2.9) are μί = μ 3 = 4. For these cases, we have N ^ 68, and the non-existence of arrays with N = 68 or 69 will complete the proof of the theorem. Possible μ' with N = 68 or 69 are (0,4,6,4,0), (1,4,6,4,0) and (0,4,6,4,1). For μ' = (0, 4, 6, 4, 0), from the SDE (3.1), (3.3), we have x5 = 224, which contradicts N < 70. Similar contradictions are obtained for the other two values of μ'. 8. Orthogonal arrays Before closing, we make a few remarks on orthogonal arrays of strength 4 and index μ. The following result throws light on how some of the preceding development can be brought to bear on orthogonal arrays. Theorem 8.1. An orthogonal array of strength 4, m rows and index μ exists if and only if there exists a trim balanced array with the same parameters, and index set (μ0, μ, μ, μ, μ4) with μ0 < μ, and μ 4 =ζ μ. J*
It was in view of this result that in Section 3, we used the symbol" — " to indicate the expressions one obtains when μ± = μ2 = μ3. Theorems 2.1 and 2.2 can be used to get existence conditions for orthogonal arrays with 5 and 6 rows, respectively. Thus, for example, for an orthogonal array of index μ and m = 5, we have ψ11 = 0, ψ12 = μ, so that there are 1 + μ non-isomorphic arrays. For six-rowed arrays, we have the following interesting result. Theorem 8.2. A necessary and sufficient condition that there exists a B-array with m — 6, t = 4, and index set μ', is that there exist integers d*(^ 0) and i (0 < i < 5) such that {d^ = dt = rf* + 1 , di+1 = d6 = d*) is a solution of (2.2), where for i = 0 the equality (dt = dt = d* +1) is to be ignored. Proof. The result is established by observing that the coefficient matrix (say A) of the vector d' = (dl9..., d6) on the left side of (2.2) is such that each element in any column of A is larger than or equal to the corresponding element in the preceding column of A. Notice that the above theorem greatly simplifies the inequalities (2.2), since the left hand side of (2.2) now uses only one variable d* instead of six
428
J.N.SRIVASTAVA AND D.V.CHOPRA
CH. 35
variables dl9..., d6. For orthogonal arrays with (a general) index μ, there is further simplification since now the right hand side of (2.2) involves just μ (instead of μθ9. .., μ4). Thus it becomes a simple matter to check the existence of orthogonal arrays with m = 6. Indeed, using the above, we would easily find that orthogonal arrays with m = 6, t = 4 exist for all μ, except for μ = 1 and 3 (Seiden and Zemach [1966]). The considerable simplification occurring for m = 6 holds out promise for the study of existence of orthogonal arrays with m ^ 7. References A. Barlotti, 1965, Some topics infinitegeometrical structures, University of North Carolina, Institute of Statistics, Mimeo Series No. 439 (Chapel Hill, N. Car.). R. C. Bose, 1947, Mathematical theory of the symmetrical factorial design, Sankhya 8, 249-256. R. C. Bose, 1961, On some connections between the design of experiments and information theory, Bull. Intern. Statist. Inst. 38, part IV. R. C. Bose and K. A. Bush, 1952, Orthogonal arrays of strength two and three, Ann. Math. Statist. 23, 508-524. R. C. Bose and J. N. Srivastava, 1964, Multidimensional partially balanced designs and their analysis, with applications to partially balanced factorial fractions, Sankhya Ser. A 26, 145-168. I. M. Chakravarti, 1956, Fractional replication in asymmetrical factorial designs and parti ally balanced arrays, Sankhya 17,143-164. D. V. Chopra and J. N. Srivastava, 1971, Optimal balanced 27 fractional factorial designs of resolution V, N < 41 (submitted for publication). P. Dembowski, 1968, Finite Geometries (Springer, New York). T. A. Dowling, 1971, Codes, packings and the critical problem, Proc. Conf. on Combinatorial Geometry and Its Applications (A. Barlotti, ed.; Perugia, Italy). J. A. Rafter, 1971, Contributions to the theory and construction of partially balanced arrays, Ph.D. dissertation (under Prof. E. Seiden), Michigan State University, East Lansing, Mich. C. R. Rao, 1947, Factorial arrangements derivable from combinatorial arrangements of arrays, Suppl. J. Roy. Statist. Soc. 9,123-139. E. Seiden and R. Zemach, 1966, On orthogonal arrays, Ann. Math. Statist. 37, 1355-1370. J. N. Srivastava, 1967, Investigations on the basic theory of 2™3η fractional factorial designs of resolution V and related orthogonal arrays (Abstract), Ann. Math. Statist. 38, 637. J. N. Srivastava, 1971, Some general existence conditions for balanced arrays of strength t and 2 symbols, /. Combin. Theory, to appear. J. N. Srivastava and D. A. Anderson, 1971, Factorial subassembly association schemes with application to the construction of multi-dimensional partially balanced designs, Ann. Math. Statist. 42, 1167-1181. J. N. Srivastava and D. V. Chopra, 1971a, On the characteristic roots of the information matrix of balanced fractional 2m factorial designs of resolution V, with applications, Ann. Math. Statist. 42, 722-734. J. N. Srivastava and D. V. Chopra, 1971b, Balanced optimal 2m fractional factorial designs of resolution V, m < 6, Technometrics 13, 257-269.