D-Optimal Designs and Group Divisible Designs Hiroki Tamura Graduate School of Information Sciences, Tohoku University Aramaki aza Aoba 09, Aoba-ku Sendai-city Miyagi-pref. 980-8579, Japan, E-mail: [email protected] Received May 13, 2005; revised November 10, 2005

Published online 15 February 2006 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.20103

Abstract: We obtain new conditions on the existence of a square matrix whose Gram matrix has a block structure with certain properties, including D-optimal designs of order n ≡ 3 (mod 4), and investigate relations to group divisible designs. We also find a matrix with large determinant for n = 39. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 451–462, 2006 Keywords: D-optimal design; divisible design; divisible difference set; Bruck-Ryser theorem

1.

INTRODUCTION

Let Xn be the set of all n × n matrices with entries ±1. A D-optimal design of order n is an element of Xn having maximal determinant md(n). Hadamard’s inequality gives the upper bound md(n) ≤ nn/2 . Equality holds only if n = 1, 2 or n is divisible by 4, and it is conjectured that the condition is also sufficient. For other values of n, better upper bounds are given by Ehlich [4,5]. If n ≡ 1 (mod 4), md(n) ≤ (2n − 1)1/2 (n − 1)(n−1)/2 . Equality holds only if 2n − 1 is a square, and if and only if there exists X ∈ Xn with XXT = (n − 1)In + Jn , where Jn is the matrix whose all entries are 1. If n ≡ 2 (mod 4), md(n) ≤ (2n − 2)(n − 1)(n−2)/2 . Equality holds only if n − 1 is the sum of two squares, and if and only if there exists X ∈ Xn with XXT = I2 ⊗ [(n − 2)In/2 + 2Jn/2 ]. If n ≡ 3 (mod 4), which is the most difficult case, md(n) ≤ (det Bn (l))1/2 where l = 6 if 15 ≤ n ≤ 59 and l = 7 if n ≥ 63, Bn (l) is the n × n matrix with block structure whose diagonal blocks are of the form (n − 3)Im + 3Jm , m = [n/ l] or m = [n/ l] + 1 and all other © 2006 Wiley Periodicals, Inc.

451

452

TAMURA

entries are −1. It was shown by Cohn [3] that equality holds only if n ≥ 63, n is divisible by 7, and 4n/7 − 3 is a square. In Section 2, we give necessary conditions for the existence of a square matrix X of order n = kl satisfying XXT = Il ⊗ (aIk + bJk ) + cJn which generalizes the theorem of Bose and Connor, and as a special case, we give more restrictions on the parameter n such that a matrix X ∈ Xn with XXT = Bn (7) exists, and hence show that n ≥ 511. We also find a matrix X ∈ X39 with large determinant satisfying XXT = B39 (13), from a nonabelian divisible difference set. As a divisible difference set is a special case of symmetric (and hence square) group divisible design (GDD), it is a natural question to ask whether a matrix X ∈ Xn with XXT = Il ⊗ (aIk + bJk ) + cJn always comes from a square GDD. In Section 3, we give a sufficient condition for this to hold, including a generalization of the result of Ionin and Shrikhande [8]. In Section 4, we give some additional conditions under which the corresponding group divisible designs is symmetric. Theorem 13 gives such a condition which depends only on parameters k, l, a, b, c.

2.

NON-EXISTENCE THEOREM

Let p be an odd prime and s = pi s , t = pj t  with p  s t  . Then the Hilbert symbol (s, t)p [6, Chap. 5.6] is defined by  (s, t)p =

−1 p

ij   j   i s t , p p

where ( px ) = 1 if x is a quadratic residue modulo p and −1 otherwise, is the Legendre symbol. The Hilbert symbol has the following properties. Let a, a1 , a2 , b, x be nonzero integers. Then (a, b)p = (b, a)p (a1 a2 , b)p = (a1 , b)p (a2 , b)p (ax2 , b)p = (a, b)p (a, −a)p = (a, 1)p = 1 (a, b)p = 1, (if a, b are prime to p)   a (a, p)p = , (if a is prime to p) p (a1 , b)p = (a2 , b)p . (ifa1 ≡ a2 ≡ 0

There is another useful property [1, Lemma III], (a, b)p = (a + b, −ab)p .

(mod p))

453

D-OPTIMAL AND GROUP DIVISIBLE DESIGNS

The equation ax2 + by2 = z2 has a nontrivial integral solution if and only if at least one of a and b is positive and (a, b)p = +1 for all odd prime p. In particular, (a, b)p = +1 if a + b is a perfect square. Two symmetric matrices A, B are called rationally equivalent (∼) if there is a nonsingular rational matrix S such that A = S T BS. By Witt’s theorem, diag(a1 , . . . , an , b1 , . . . , bm ) ∼ diag(a1 , . . . , an , c1 , . . . , cm ) holds if and only if diag(b1 , . . . , bm ) ∼ diag(c1 , . . . , cm ). Theorem 1 [Hasse-Minkowski]. Two nonsingular rational diagonal matrices A = diag(a1 , . . . , am ), B = diag(b1 , . . . , bm ) are rationally equivalent if and only if det A = x2 det B for some rational  x, #{i|ai > 0} = #{i|bi > 0} and cp (A) = cp (B) for all odd prime p, where cp (A) = 1≤i 0, (2) a + bk + cn is a square, k−1 l−1 k−1 (3) (a, (−1) 2 k)p = (a + bk, (−1) 2 l)p , or equivalently (a, (−1) 2 b)p = (a + bk, (−1)

l+1 2

abc)p for all odd primes p.

Proof. For (1) and (2), a, a + bk, a + bk + cn are all eigenvalues of XXT with multiplicity (k − 1)l, l − 1, 1 and det XXT = (det X)2 = a(k−1)l (a + bk)l−1 (a + bk + cn)

(1)

must be a square. For (3), the proof of the former equation is almost the same as that of a theorem of Bose and Connor [1, Theorem 9(c)]. We give the remaining proof. Since (a, bk)p = (a + bk, −abk)p and (a + bk, cn)p = 1, (a, (−1)

k−1 2

k)p (a + bk, (−1)

l−1 2

l)p

= (a, (−1)

k−1 2

k)p (a + bk, (−1)

= (a, (−1)

k−1 2

k · bk)p (a + bk, (−1)

= (a, (−1)

k−1 2

b)p (a + bk, (−1)

l−1 2

l+1 2

l)p (a, bk)p (a + bk, −abk)p (a + bk, cn)p l−1 2

l · (−abk) · cn)p

abc)p . 

454

TAMURA

Remark 1. When k = 1, this theorem is nothing more than the Bruck-Ryser-Chowla theorem. If we put a = n − 3, b = 4, c = −1 then we have: Cororally 3. Let n = kl ≡ 3 XXT = Bn (l) implies

(mod 4). The existence of a square rational matrix X with

(1) 4k − 3 is a square, k−1 (2) (n − 3)x2 + (−1) 2 y2 = (n + 4k − 3)z2 has a nontrivial integral solution. Proof. (1) is immediate from Theorem 2(1). k−1 l+1 (2) By Theorem 2(2), (n − 3, (−1) 2 )p = (n + 4k − 3, (−1) 2 · (−1)(n − 3))p . This implies

((−1)

k+1 2

(n − 3), (−1)

k−1 2

(n + 4k − 3))p = 1.

(2)

This means that there is a nontrivial integral solution to (−1)

k−1 2

((n + 4k − 3)z2 − (n − 3)x2 ) = y2 . 

For the D-optimal case l = 7, the values of n ≥ 63 satisfying the above conditions are 511, 1687, 5299, 7399, . . .. We will show in Example 1 and Remark 4 in Section 4, that if any of these D-optimal designs exists, it always comes from a group divisible design. Although a case l = 7 may not be D-optimal, it still gives a large determinant, if it exists and l is close to 7. Unfortunately, we have the following. Cororally 4. Let n = kl ≡ 3 (mod 4). For l = 5 or 9, there does not exist a square rational matrix X with XXT = Bn (l). Proof. We show that the Equation (2) does not hold for p = 3. As 5(5k − 3) − (9k − 3) = 4(4k − 3), 9(9k − 3) − 5(13k − 3) = 4(4k − 3) and 4(9k − 3) − 15 = 9(4k − 3) are all squares by Cororally 3 (1), we have (5(5k − 3), −(9k − 3))p = (9k − 3, −5(13k − 3))p = (9k − 3, −15)p = 1. So (5k − 3, −(9k − 3))3 = (5(5k − 3), −(9k − 3))3 (5, −1)3 (5, 9k − 3)3 = (5, 9k − 3)3 , (9k − 3, −(13k − 3))3 = (9k − 3, −5(13k − 3))3 (9k − 3, 5)3 = (5, 9k − 3)3 , (5, 9k − 3)3 = (−15, 9k − 3)3 (−3, 9k − 3)3

D-OPTIMAL AND GROUP DIVISIBLE DESIGNS

455

= (−3, 9k − 3)3 = (−3, 3k − 1)3 = (3, −1)3 = −1.  Possible values of small n = kl (k, l > 1) for the existence of a matrix X satisfying XXT = Bn (l) are (k, l) = (3, 13), (21, 3), (7, 13), (7, 17), (3, 49), (7, 21), . . . , (73, 7), . . . The smallest case (k, l) = (3, 13) is realized by a divisible difference set with parameter (13, 3, 18, 9, 8) and (det B39 (13))1/2 = 250 · 333 by (1), is 90.3% of Ehlich’s upper bound for n = 39 and exceed the record given in [10]. It is obtained as “partial complement” of a nonabelian (13, 3, 9, 0, 2)-RDS. (Recently, Rokicki discovered a circulant matrix with better record [11].) Definition 1. Let G be a group of order ml, N a subgroup of G of order m. A set D of k elements in G is called a divisible difference set (DDS) with parameters l, m, k, λ1 , λ2 , or briefly an (l, m, k, λ1 , λ2 )-DDS, if the differences di dj−1 (di , dj ∈ D, di = dj ) contain each element of N\{1} exactly λ1 times and each element of G\N exactly λ2 times. A divisible difference set is called a relative difference set (RDS) if λ1 = 0. Let G = a, b : a3 = b13 = 1, ba = ab3 be a nonabelian group of order 39. Then a (13, 3, 9, 0, 2)-RDS is given by D = {b, ab9 , a2 b3 , b5 , ab6 , a2 b2 , b4 , b10 , b12 }. Note that D satisfies DD−1 = D−1 D. The “support” R of D (the image of the map ai bj → bj ) is also a difference set giving the complement of a projective plane of order 3. Taking the complement of D on its support, a R \ D becomes a (13, 3, 18, 9, 8)-DDS. It is not known whether a matrix X ∈ X39 with XXT = B39 (13) always comes from a divisible difference set. If XT X = B39 (13) also holds, then X comes from a symmetric group divisible design, as we shall see in Section 4, Theorem 12.

3.

GROUP DIVISIBLE DESIGNS

In this section, we consider group divisible designs instead of divisible difference sets more generally. Definition 2. A group divisible design GDD(v, k, m, n, λ1 , λ2 ) is a triple (P, S, A) where

 P is a set of points, where |P| = v = mn,  S is a class of non-empty subsets of P (called groups), of size m, which partitions P, and |S| = n,

 A is a class of subsets of P (called blocks), of size k ≥ 2,  each pair of distinct points x, y where x and y are from the same group is contained in precisely λ1 blocks,

456

TAMURA

 each pair of distinct points x, y where x and y are not from the same group is contained in precisely λ2 blocks. A group divisible design is called square if v = |A|, and called symmetric if its dual (exchange of blocks and points) is also a GDD with the same parameter. Proposition 5. Let (P, S, A) be a GDD(v, k, m, n, λ1 , λ2 ). Then each point x ∈ P is contained in precisely r blocks, and b = |A|, r satisfy bk = rv, r(k − 1) = λ1 (m − 1) + λ2 (v − m). The existence of a GDD(v, k, m, n, λ1 , λ2 ) is equivalent to the existence of a v × |A| incidence matrix B satisfying J1,v B = kJ1,|A| and BBT = In ⊗ ((r − λ1 )Im + (λ1 − λ2 )Jm ) + λ2 Jv . Remark 2. Let D be a divisible difference set over a group G, then we can give a corresponding group divisible design as follows:

 points and blocks are both labeled by G,  a point x is in a block y if and only if xy−1 ∈ D. A GDD(39, 9, 3, 13, 0, 2) is given directly in [7] by using the projective plane of order 3, and it is shown to be represented as a nonabelian divisible difference set [9, Cororally 4.11]. n−2t+b+c n−2t+c If a square GDD(n, n−t , 4 ) with n = kl, t 2 = a + bk + cn exists, 2 , k, l, 4 we obtain a square matrix X satisfying XXT = Il ⊗ (aIk + bJk ) + cJn by replacing 0 with −1 in its incidence matrix. Conversely, does a matrix X ∈ Xn with XXT = Il ⊗ (aIk + bJk ) + cJn always come from a group divisible design? This holds for l = 1, n, and we do not know any counterexamples for other values of l. But we can show that it holds under certain conditions. Lemma 6. Let n = kl = a + b + c, t 2 = a + bk + cn, and let X be a matrix in Xn with equal row sum t satisfying XXT = Il ⊗ (aIk + bJk ) + cJn . Then A = 21 (Jn − X) is an n−2t+b+c n−2t+c incidence matrix of a square GDD(n, n−t , 4 ). 2 , k, l, 4 Proof. Let en be the all 1 vector of length n and u := XT en . Then nt 2 = n1 (uT en )2 ≤ uT u = eTn XXT en = n(a + bk + cn). Since equality holds, u = ten and Jn A = n−t 2 Jn , 1 1 T T T AA = 4 (nJn − XJn − Jn X + XX ) = 4 {Il ⊗ (aIk + bJk ) + (n − 2t + c)Jn }.  Theorem 7. Let n = kl = a + b + c and c = 0. Assume there exists a nonsingular matrix X ∈ Xn satisfying XXT = Il ⊗ (aIk + bJk ) + cJn and X has a block structure such that 

 X11 · · · X1l  . ..   X= .   .. Xl1 · · · Xll

457

D-OPTIMAL AND GROUP DIVISIBLE DESIGNS

where each Xij is a k × k matrix with equal column sum xij . Then there exists a square n−2t+b+c n−2t+c GDD(n, n−t , 4 ) with a + bk + cn = t 2 . 2 , k, l, 4 Remark 3. When k = 1, a k × k matrix always has equal column sum, so the above theorem means that a nonsingular matrix X ∈ Xn satisfying XXT = (n − c)In + cJn , c = 0 corresponds to some symmetric BIB design. This is equivalent to the result of Ionin and Shrikhande [8]. To prove the above theorem, we use the following lemmas. Lemma 8. Let X, Y be nonsingular matrices satisfying XXT = Y and let v be a column vector of X. Then vT Y −1 v = 1. Proof.

Since XT Y −1 X = I, the result follows.

Lemma 9.



If n = kl and a(a + bk)(a + bk + cn) = 0, then

{Il ⊗ (aIk + bJk ) + cJn }

−1

 = Il ⊗

 c 1 b Ik − Jk − Jn . a a(a + bk) (a + bk)(a + bk + cn)

Lemma 10. Let n = kl = a + b + c and let X be a nonsingular matrix in Xn satisfying XXT = Il ⊗ (aIk + bJk ) + cJn . If 

 v1  .   v=  ..  vl is a column vector of X where each vi has size k and sum vˆ i , then 2 l l

b 2 ac b+c = vˆ i + vˆ i . a + bk (a + bk)(a + bk + cn) i=1

i=1

Proof. From Lemma 8 and Lemma 9, 1 = vT {Il ⊗ (aIk + bJk ) + cJn }−1 v

2 l l



1 T b c 2 = v v− vˆ i − vˆ i , a a(a + bk) (a + bk)(a + bk + cn) i=1

i=1



l l



c b n vˆ 2i − vˆ i = − a a(a + bk) (a + bk)(a + bk + cn) i=1

2 .

i=1

 Lemma 11. If a nonsingular matrix X with equal column sum satisfies XXT = aI + bJ, then XT X = aI + bJ.

458

TAMURA

Proof. We have I = XT (aI + bJ)−1 X = XT ( a1 I + cJ)X for some c. Since X has equal column sum, I = a1 XT X + dJ for some d. As XXT = aI + bJ and XT X = aI − adJ should have the same determinant, b = −ad.  Proof of Theorem 7. Let v be a column vector of X as in Lemma 10. From (Il ⊗ Jk )XXT (Il ⊗ Jk ) = (Il ⊗ Jk ){Il ⊗ (aIk + bJk ) + cJn }(Il ⊗ Jk ), we obtain an equation Xˆ Xˆ T = (a + bk)Il + ckJl ,

(3)

where Xˆ = (xij ), i, j = 1, . . . , l. Since 

 vˆ 1  .   vˆ :=   ..  vˆ l ˆ is a column vector of X, 1 = vˆ T {(a + bk)Il + ckJl }−1 vˆ   1 ck = vˆ T Il − Jl vˆ a + bk (a + bk)(a + bk + cn) 2 l l

1 2 ck = vˆ i − vˆ i . a + bk (a + bk)(a + bk + cn) i=1

(4)

i=1

From Lemma 10 and (4), l 2

c c= vˆ i . a + bk + cn i=1

Since c = 0,

l

2 vˆ i

= a + bk + cn.

(5)

i=1

This implies that the column sum of Xˆ (and that of X) is constant up to sign. Thus Xˆ (and X) can be normalized into a matrix with equal column sum by a suitable change of signs of columns. If X is normalized to have equal column sum t, then (3), (5), and Lemma 11 imply In = XT {Il ⊗ (aIk + bJk ) + cJn }−1 X

459

D-OPTIMAL AND GROUP DIVISIBLE DESIGNS

2 l

1 T b c T = X X− Xˆ Xˆ ⊗ Jk − vˆ i Jn a a(a + bk) (a + bk)(a + bk + cn) i=1

1 b c ((a + bk)Il + ckJl ) ⊗ Jk − Jn = XT X − a a(a + bk) a + bk =

1 T b c X X − Il ⊗ J k − J n , a a a

and hence XT X = Il ⊗ (aIk + bJk ) + cJn . Since XT has equal row sum t, A = 21 (Jn − XT ) is an incidence matrix of a square GDD by Lemma 6. 

4.

RELATION TO SYMMETRIC GROUP DIVISIBLE DESIGNS

Under several conditions described in this section, we show that the group divisible design obtained from a matrix X ∈ Xn is symmetric. Theorem 12. Let n = kl = a + b + c and c(bk + cn) = 0. If a matrix X ∈ Xn satisfies XXT = XT X = Il ⊗ (aIk + bJk ) + cJn then 21 (Jn − X) is an incidence matrix of a symmetric GDD. Proof. tion

We calculate XXT (XXT − aIn )(XXT − (a + bk)In ) in two ways. From the equaXXT (XXT − aIn )(XXT − (a + bk)In ) = {Il ⊗ (aIk + bJk ) + cJn }{bIl ⊗ Jk + cJn }{Il ⊗ (−bkIk + bJk ) + cJn } = c(bk + cn)(a + bk + cn)Jn .

On the other hand,

XXT (XXT − aIn )(XXT − (a + bk)In ) = X(XT X − aIn )(XT X − (a + bk)In )XT = c(bk + cn)XJn XT = c(bk + cn)(Xen )(Xen )T . Since c(bk + cn) = 0, Xen = ten with t 2 = a + bk + cn, so we can apply Lemma 6 to n−2t+b+c n−2t+c conclude that 21 (Jn − X) is an incidence matrix of a GDD(n, n−t , 4 ). 2 , k, l, 4 1 Obviously this argument holds if we exchange X and XT , so 2 (Jn − XT ) is also an incidence matrix of a GDD with the same parameter. 

460

TAMURA

The idea of the following theorem comes from Theorem 6.1 of [2] which gives a condition for a square GDD to be symmetric. Theorem 13. Let n = kl = a + b + c be odd and t 2 = a + bk + cn such that (a, b) = 4, (ac, t 2 ) is square-free and b = 0, where (, ) is the symbol for the greatest common divisor. If a nonsingular matrix X ∈ Xn satisfies XXT = Il ⊗ (aIk + bJk ) + cJn , then there exists n−2t+b+c n−2t+c a symmetric GDD(n, n−t , 4 ). 2 , k, l, 4 Proof. Let v(1) , . . . , v(n) be the column vectors of X. From Lemma 10, ac(eTn v(i) )2 is divisible by t 2 . Since (ac, t 2 ) is square-free, eTn v(i) is divisible by t. From the evaluation of eTn XXT en , we get ni=1 (eTn v(i) )2 = nt 2 . Since eTn v(i) is odd and divisible by t, |eTn v(i) | = t, thus we can assume that X has equal column sum t. Let vˆ(i) be defined as in the proof of TheT T orem 7, then vˆ(i) vˆ(i) = a + (b + c)k by Lemma 10 and (4). If i = j, v(i) (XXT )−1 v(j) = 0 so from Lemma 10, T T ˆ + ac, (a + bk)v(i) v(j) = bvˆ(i) v(j)

and with a parameter sij , we can write T ˆ = a + (b + c)k − s (a + bk), vˆ(i) v(j) ij T

v(i) v(j) = b + c − sij b. Since T ˆ ≤ (vˆ(i) T vˆ(i) ) 21 (v(j) ˆ ) 21 ˆ T v(j) vˆ(i) v(j)

(6)

= a + (b + c)k, sij is nonnegative because a + bk > 0. Note that all the entries of vˆ(i) are odd, u(i) := 1 ˆ(i) 2 (v − el ) is an integral vector whose sum does not depend on i and T ˆ = (2u(i) + e )T (2u(j) + e ) vˆ(i) v(j) l l T

T

= 4u(i) u(j) + 2u(i) el + 2eTl u(j) + eTl el ≡l

(mod 4).

T T T ˆ modulo 4 do not depend on We also have v(i) v(j) ≡ n (mod 4). Thus v(i) v(j) , vˆ(i) v(j) T T i, j. Since vˆ(i) vˆ(i) = a + (b + c)k and v(i) v(i) = a + b + c ≡ b + c (mod 4),

sij (a + bk) ≡ sij b ≡ 0

sij a ≡ sij b ≡ 0

(mod 4),

(mod 4).

461

D-OPTIMAL AND GROUP DIVISIBLE DESIGNS

 So sij is an integer because (a, b) = 4. From (6), sii = 0 if and only if vˆ(i) = v(iˆ ) , and if sii = 0, sij = si j for all j. So if we let v(tˆ1 ) , . . . , v(tˆl ) be all the distinct vectors and ˆ = v(tˆ i ) }, then by a suitable permutation of columns of X, let ki = #{j ∈ {1, . . . , n} | v(j) T X X has a block structure with block sizes (k1 , . . . , kl ), k1 + · · · + kl = n, such that the diagonal blocks are aIki + (b + c)Jki and the off-diagonal blocks are (b + c − sti tj )Jki ,kj . Since rank(XT X − aIn ) ≤ l , XT X has an eigenvalue a with multiplicity not less than n − l and equal to that of XXT . The eigenvalues of XXT are a, a + bk, a + bk + cn with multiplicities n − l, l − 1, 1, respectively, and they are all distinct since b = 0 and cn is odd. So we obtain l ≤ l . Let

Y = (el , v(tˆ1 ) , · · · , v(tˆl ) ). Then 

 l t···t t    T  , Y Y = .   .. a + (b + c)k − (a + bk)sti tj  t 

where sti ti = 0 for convenience. Let uT = (− llt , 1, . . . , 1) then 0 ≤ uT Y T Yu 

l

l 2 t 2 2 =− + (a + (b + c)k)l − (a + bk) sti tj l i,j=1



l

  2 1 = (a + bk) l (1 − ) − sti tj . l i,j=1

Since a + bk > 0, 

l 1 1 1

1− ≥ 2 sti tj = 2  l l i,j=1 l



l

sti tj ≥ 1 −

i,j=1, i=j

1 , l

l ≥ l . Thus l = l and sti tj = 1 for all i = j. From n

T

T

(v(ti ) v(j) )2 = v(ti ) XXT v(ti )

j=1 T

T

= av(ti ) v(ti ) + bv(tˆ i ) v(tˆ i ) + c(eTn v(ti ) )2

462

TAMURA

= a(a + 2b + 2c) + b(b + 2c)k + c2 n, and

n

T



j=1

(v(ti ) v(j) )2 = (a + b + c)2 +

(b + c − sti j b)2

=

(a + b + c)2 + (b + c)2 (n − 1) − 2b(b + c)

=

(a + b + c) + (b + c) (n − 1) − 2b(b + c)(n − ki ) + b (n − ki )

=

a(a + 2b + 2c) + b(b + 2c)ki + c2 n,

j=ti



sti j + b2

j=ti 2

2

we obtain b(b + 2c)(ki − k) = 0 for all i. Since b = 0 and b + 2c ≡ 2 for all i. Thus XT X = XXT and we can apply Theorem 12.



sti j 2

j=ti 2

(mod 4), ki = k 

Example 1. Putting k = 73, l = 7, a = 508, b = 4, c = −1, Theorem 13 implies that a matrix X ∈ X511 with XXT = B511 (7) always comes from a symmetric group divisible design. Remark 4. In fact, a matrix X ∈ X7k with XXT = B7k (7) (n = 7k ≥ 63) always comes from a symmetric group divisible design. If such X exists, X is D-optimal and so is XT . Thus XT X is equal to B7k (7) by a suitable permutation or sign changes of columns of X [5], and we can apply Theorem 12.

REFERENCES [1] R. C. Bose and W. S. Connor, Combinatorial properties of group divisible incomplete block designs, Ann Math Stat 23 (1952), 367–383. [2] W. S. Connor, Some relations among the blocks of symmetrical group divisible designs, Ann Math Stat 23 (1952), 602–609. [3] J. H. E. Cohn, Almost D-optimal designs, Utilitas Math 57 (2000), 121–128. [4] H. Ehlich, Determinantenabsch¨atzungen f¨ur bin¨are Matrizen, Math Z 83 (1964), 123–132. [5] H. Ehlich, Determinantenabsch¨atzungen f¨ur bin¨are Matrizen mit N ≡ 3 (mod 4), Math Z 84 (1964), 438–447. [6] H. Hasse, Number Theory, English translation edited and prepared for publication by Horst Gunter Zimmer, Grundlehren der mathematischen Wissenschaften, Bd. 229, Springer-Verlag, New York-Berlin, 1980. [7] D. Hughes, Biplanes and semi-biplanes, “Combinatorial Mathematics”, Lecture Notes in Math 686, Springer-Verlag, Berlin, (1978), 55–58. [8] Y. J. Ionin and M. S. Shrikhande, Equidistant families of sets, Linear Algebra Appl 226–228 (1995), 223–235. [9] D. Jungnickel, On automorphism groups of divisible designs, Can J Math 34(2) (1982), 257–297. [10] W. P. Orrick, B. Solomon, R. Dowdeswell, and W. D. Smith, New lower bounds for the maximal determinant problem, arXiv preprint math.CO/0304410. [11] The Hadamard maximal determinant problem WEB page, http://www.indiana.edu/˜maxdet/

D-optimal designs and group divisible designs

(13, 3, 18, 9, 8) and (det B39(13))1/2 = 250 · 333 by (1), is 90.3% of Ehlich's upper bound for n = 39 and exceed the record given in [10]. It is obtained as “partial complement” of a nonabelian (13, 3, 9, 0, 2)-RDS. (Recently, Rokicki discovered a circulant matrix with better record [11].) Definition 1. Let G be a group of order ml, ...

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