Sankhy¯ a : The Indian Journal of Statistics 2002, Volume 64, Series B, Pt.1, pp 26-36
NEW INFINITE CLASSES OF WEIGHING MATRICES By S. GEORGIOU AND C. KOUKOUVINOS National Technical University of Athens, Greece SUMMARY. In this paper we use a new algorithm to find weighing matrices W (2n, 9) constructed using two circulant matrices. The basic idea of this algorithm is to investigate all possible ways the weight 9 can be split into two weights of the corresponding sequences. Solutions for each split, if they exist, are given for weighing matrices W (2n, 9), for all n up to 100. Many of these sequences are new and given here for the first time. Using these sequences we can obtain many new infinite classes of weighing matrices W (2n, k).
1.
Introduction and Preliminaries
A weighing matrix W = W (n, k) is a square matrix of order n with entries 0, ±1 having k non-zero entries per row and column and inner product of distinct rows zero. Hence, W satisfies W W T = kIn . The number k is called the weight of W . Note that W T can be looked upon as the design matrix of a chemical balance weighing design. Under standard linear model assumptions, following Kiefer (1975), it is not hard to check that such a design is universally optimal in the class of all n-observation chemical balance weighing designs, with n objects, such that at most k objects are used in each weighing. Hence, the study of weighing matrices is important from the perspective of optimal statistical design theory. In the present paper, we propose a method to obtain new results on weighing matrices. For the construction of the new infinite classes of weighing matrices given in this paper we need the following definitions and notations. Paper received February 2001; revised December 2001. AMS (1991) subject classification. Primary 62K05, 62K10; secondary 05B20. Key words and phrases. Weighing matrices, sequences, autocorrelation, construction, algorithm, linear model.
27
new infinite classes of weighing matrices
Given a set of ` sequences, A = {Aj : Aj = {aj1 , aj2 , ..., ajn }, j = 1, . . . , `}, of length n, the periodic autocorrelation function (abbreviated as PAF) PA (s) is defined, reducing i + s modulo n, as PA (s) =
n ` X X
aji aj,i+s ,
s = 0, 1, ..., n − 1.
(1)
j=1 i=1
Notation. We use the following notation throughout this paper. 1. We use a to denote −a. 2. A circulant weighing matrix of order n and weight k will be denoted as CW (n, k). 3. We use R = (rij ) to (denote the n × n back diagonal matrix whose 1, when i + j = n + 1 elements satisfy rij = i, j = 1, 2, . . . , n 0, otherwise. 4. Let A = {a1 , a2 , . . . , an } where ai ∈ {0, ±1}. Then the positive describing set of A is the set P OSA = {i : ai = 1} and the negative describing set of A is the set N EGA = {i : ai = −1}. We will use either A or A = {a1 , a2 , . . . , an } in order |P OSA | ≥ |N EGA | where |S| is the number of elements in set S. 5. The support of A is the set SPA = {i : ai 6= 0}. Hence, SPA = P OSA ∪ N EGA . A sequence X can be fully described by its support if we set SPX = P OSX ∪ N EGX . That is SPX = {p1 , p2 , . . . , pi , n1 , n2 , . . . , nj } where p` ∈ P OSX , ∀` = 1, 2, . . . , i and n` ∈ N EGX , ∀` = 1, 2, . . . , j and i + j = |SPX |. Theorem 1 (Geramita and Seberry (1979), Theorem 4.46) If there exist two circulant matrices A1 , A2 of order n with elements from the set {0, 1, −1}, satisfying 2 X
Ai ATi = kI,
i=1
then there exists a W (2n, k). The construction of the corresponding weighing matrices can be obtained by the following arrays Ã
D=
A1 −AT2
A2 AT1
!
Ã
or
D=
A1 A2 R −A2 R A1
!
.
(2)
28
s. georgiou and c. koukouvinos
Lemma 1 (Koukouvinos and Seberry (1999)) If there exists a CW (n, k) there exists a CW (pn, k) for all p ≥ 1. Lemma 2 (Koukouvinos and Seberry (1999)) If there exist two circulant matrices which give a W (2n; k), then there exist two circulant matrices which give a W (2pn; k) and a W (2pn; 2k) for all integers p ≥ 1. Lemma 3 (Strassler (1997)) A CW (n, 9) exist if and only if n is multiple of 13 or 24 (i.e. 13 | n or 24 | n). Definition 1 We say that two pairs of (0, ±1) sequences of length n (A,B) and (C,D) are equivalent iff one can be obtained from the other if we apply some of the following transformations. 1. Multiply one or both sequences of the pair by −1. 2. Reverse one or both sequences of the pair. 3. Take circulant permutation of one or both sequences of the pair. 4. Multiply the support of both sequences by `, (`, n) = 1. We call the corresponding weighing matrices constructed from two circulant matrices whose first rows are the sequences (A, B) and (C, D) equivalent. 2.
The Algorithm
Earlier algorithms to search for W (2n, k) constructed from two circulant matrices have been functions of n rather than k. This has meant that for n ≥ 25 an exhaustive search could not be carried out in reasonable time. The algorithm we use here is based on the weight, k, of a weighing matrix. When the weight is small enough the performance of this algorithm is excellent: even for n ≈ 100 we can carry out an exhaustive search using 56 hours of CPU time on a 400 MHz PC with 64 MB RAM and 4 GB hard disk. We will apply the algorithm in the case of weighing matrices of order 2n and weight 9. We will search for two sequences A, B which satisfy |SPA | + |SPB | = 9 and PA (s) + PB (s) = 0, ∀s = 1, 2, . . . , n − 1. These sequences can be used as the first rows of the corresponding circulant matrices in the constructions given by (2) to obtain the weighing matrices W (2n, k).
new infinite classes of weighing matrices
29
It is well known that if there exist a W (2n, k) constructed from two circulant matrices of order n, then k = a2 + b2 , where a and b are the row (and column) sums of A and B respectively. Lemma 4 We split 9 into two parts |SPA | + |SPB | = 9. If (|SPA |, |SPB |) 6= (9, 0)}, then we get three other possible cases that could satisfy both of the above equations. These are (|SPA |, |SPB |) ∈ {(5, 4), (6, 3), (7, 2)}. Proof. The case (|SPA |, |SPB |) = (9, 0), which is the case where a circulant weighing matrix exists, was investigated in Strassler (1997). This case implies that one sequence of weight 9 exists. Thus, CW (n, 9), CW (2n, 9) exist (see lemma 3) and a CW (2n, 9), constructed from two circulant matrices, exists (take one to be the matrix with all elements zero and the other to be the CW (n, 9) as it is given in Strassler (1997) or in Table 1 below). For more details on this case see Strassler (1997). If (|SPA |, |SPB |) = (8, 1) and PA (s) + PB (s) = 0, ∀ s = 1, 2, . . . , n − 1, then there exists a W (2n, 9), constructed from two circulant matrices, such that 9 = a2 + 12 which is not possible since 8 is not a perfect square. So the case (8, 1) is not permitted. 2 Using the fact that SPS = P OSS ∪ N EGS , S ∈ {A, B} and that 9 = 32 +02 = (|P OSA |−|N EGA |)2 +(|P OSB |−|N EGB |)2 these 3 cases become: (i) (|SPA |, |SPB |) = (5, 4). Here |P OSA | = 4, |N EGA | = 1, |P OSB | = |N EGB | = 2. In the first sequence we will have four positive elements and one negative. Those can occur in only one way (ordering positive and negative elements), that is the negative element first and the four positive elements following. n1 , p1 , p2 , p3 , p4 , where p1 , p2 , p3 , p4 ∈ P OSA and n1 ∈ N EGA All other subcases p1 , p2 , p3 , n1 , p4 ; p1 , p2 , n1 , p3 , p4 ; p1 , n1 , p2 , p3 , p4 ; p1 , p2 , p3 , p4 , n1 are permutations of this one. In the second sequence we will have two positive and two negative elements. These can occur in only two ways (ordering positive and negative elements), that is, the two positive elements first and the two negative elements afterwards or one positive, one negative, one positive and one negative: p1 , p2 , n1 , n2 and p1 , n1 , p2 , n2
30
s. georgiou and c. koukouvinos where p1 , p2 ∈ P OSB and n1 , n2 ∈ N EGB . All other subcases p1 , n1 , n2 , p2 ;
n1 , p1 , p2 , n2 ;
n1 , p1 , n2 , p2
are permutations of these two. In total, for case (i), we have to check 1 × 2 = 2 subcases. (ii) (|SPA |, |SPB |) = (6, 3). Then |P OSA | = |N EGA | = 3, |P OSB | = 3, |N EGB | = 0. In the first sequence we will have three positive elements and three negative. Those can occur only in three ways (ordering positive and negative elements) as follows : p1 , p2 , p3 , n1 , n2 , n3 ;
p1 , p2 , n1 , n2 , p3 , n3 ;
p1 , n1 , p2 , n2 , p3 , n3
where pi ∈ P OSA , ni ∈ N EGA , i = 1, 2, 3. All other subcases are permutations of these three. In the second sequence we will have three positive elements and no negatives. Those can occur only in one way. That is p1 , p2 , p3 where p1 , p2 , p3 ∈ P OSB There are no other subcases. In total, for case (ii), we have to check 3 × 1 = 3 subcases. (iii) (|SPA |, |SPB |) = (7, 2). Then |P OSA | = 5, |N EGA | = 2, |P OSB | = |N EGB | = 1. In the first sequence we will have five positive elements and two negative. Those can occur only in three ways (ordering positive and negative elements) : p1 , p2 , p3 , p4 , p5 , n1 , n2 ; p1 , p2 , p3 , p4 , n1 , p5 , n2 ; p1 , p2 , p3 , n1 , p4 , p5 , n2 where p1 , p2 , p3 , p4 , p5 ∈ P OSA and n1 , n2 ∈ N EGA . All other subcases are permutations of these three. In the second sequence we will have one positive element and one negative. These can occur only in one way (ordering positive and negative elements), that is, the positive element first and the negative element following. p1 , n 1 ,
p1 ∈ P OSB and n1 ∈ N EGB
The other subcase n1 , p1 is a permutation of this one. In total, for case (iii), we have to check 3 × 1 = 3 subcases.
31
new infinite classes of weighing matrices
We will need to examine these three cases in order to perform an exhaustive search for all lengths n < 100. The idea of the new algorithm is that we only have to find the sets P OSA , P OSB , N EGA , N EGB . Finding these sets we can archive the results more efficiently than for any previously known algorithm.. We will give the steps of our algorithm for the first subcase of case (i). For all the other cases and subcases the steps can be written similarly. The steps of this algorithm are: Step 1. Fix the first element of sequence A to be −1. Step 2. Find all possible positions of +1. That gives us Ã
n−1 4
!
=
n4 − 10n3 + 35n2 − 50n + 24 24
cases.
Save those sequences and their associated PA (s), s = 1, 2, . . . , sorted in lexicographic order.
£n¤ 2
Step 3. Fix the first element of sequence B to be 1. Ã
!
n−3 Step 4. Find all possible positions for the other +1. That is, = 1 n − 3 possible positions. We do not use the last two positions of B because the two −1 elements should occur after the second +1. Step 5. For all cases found in step 4 find all positions of the two −1. Thus, we have in total n−1 X i=2
Ã
i 2
!
=
n3 − 6n2 + 11n − 6 6
cases.
Save those sequences and their associated PB (s), s = 1, 2, . . . , sorted into lexicographic order.
£n¤ 2
Step 6. Check all pairs of sequences A and B found above using the fastest search known, the binary search. If PA (s)+PB (s) = 0, ∀s=1, 2, . . . , [ n2 ] then save the sequences as a solution. We have about the same number of sequences (O(n4 )) to check in the second subcase of (i). Thus, this algorithm needs to check only O(n4 ) sequences in order to find two sequences of type (0, ±1) in case (i). The algorithm in all subcases of all other cases work just the same.
s. georgiou and c. koukouvinos
32
In total, for an exhaustive search, we have to check O(n6 ) sequences for all subcases of all cases. For example, for n = 81 we must check in about 816 = 324 sequences of the total 32·81 = 3162 sequences. If conjecture 1 is true we only have to check 814 = 316 sequences to verify that a solution do not exist. 3.
The Constructions
We set: S1 = S(5,4)
=
{5, 7, 8, 10. . .17, 19. . .26, 28. . .30, 32. . .36, 38. . .46, 48. . .52, 55. . .58, 60, 63. . .66, 68. . .70, 72, 75. . .78, 80, 82, 84. . .88, 90. . .92, 95, 96, 98, 99}.
S(6,3)
=
{7, 10. . .15, 17, 19. . . 26, 28. . .30, 33. . .36, 38. . .40, 42, 44 . . .46, 48. . .52, 55. . .58, 60, 63, 65, 66, 68. . .70, 72, 75. . .78, 80, 84, 85, 87, 88, 90. . .92, 95, 96, 98, 99}.
S(7,2)
=
{10, 13. . .15, 19, 20, 23, 25, 26, 28, 30, 38. . .40, 42, 45, 46, 50, 52, 56, 57, 60, 65, 69, 70, 75, 76, 78, 80, 84, 90. . .92, 95, 98}.
S(9,0)
= =
{13, 24, 26, 39, 48, 52, 65, 72, 78, 91, 96} = {13k, k = 1, 2, 3, 4, 5, 6, 7} ∪ {24k, k = 1, 2, 3, 4}.
Theorem 2 There exist pairs of sequences with zero PAF which give a weighing matrix W(2n,9) for all lengths n ∈ S1 . Moreover, there exist pairs of sequences with weight distribution (5, 4), (6, 3), (7, 2), (9, 0) with zero PAF for all lengths n ∈ S(5,4) , S(6,3) , S(7,2) , S(9,0) respectively. Thus, there exist many inequivalent W (2n, 9) constructed by these sequences. Proof. Use the sequences given in Table 1.
2
Remark 1 In theorem 2 the sequences of lengths n = 29, 41, 43 give new weighing matrices W (2kn, 9), k = 1, 2, . . . , constructed from two circulant matrices. For example, new W (2·29, 9), W (2·58, 9), W (2·87, 9), W (2·41, 9), W (2 · 82, 9), W (2 · 43, 9), W (2 · 86, 9) are constructed from two circulant matrices. Moreover, some of the above sequences were known, but not for all distributions of weight 9 in the two sequences as we have given in this paper. Hence, we have found many new sequences corresponding to different distributions for all lengths n, n < 100 and thus many new, up to equivalence, weighing matrices.
new infinite classes of weighing matrices
33
Remark 2 We observe that S1 = S(5,4) ⊇ S(6,3) ⊇ S(7,2) Conjecture 1 The existence of two sequences of weight 9 and distribution (6, 3) with zero PAF implies the existence of two sequences of weight 9 and distribution (5, 4). The existence of two sequences of weight 9 and distribution (7, 2) with zero PAF implies the existence of two sequences of weight 9 and distribution (6, 3). We set: S2 = {6, 9, 18, 27, 31, 37, 47, 53, 54, 59, 61, 62, 67, 71, 73, 74, 79, 81, 83, 89, 93, 94, 97}. Theorem 3 For every length n ∈ S2 there is no pair of sequences of length n and weight 9 with zero PAF and hence no weighing matrices W (2n, 9) constructed using two circulant matrices. Proof. The result is obtained by an exhaustive search.
2
Remark 3 In theorem 3 above all results for lengths n ∈ S2 and n > 18 are new. Conjecture 2 A weighing matrix W (2n, 9) cannot be constructed from two circulant matrices for any n > 43, n prime. Conjecture 3 A weighing matrix W (2n, 9) cannot be constructed from two circulant matrices for any n ∈ S, S = {3a , p prime > 43, 3b ·p : p prime}. These conjectures have been verified by an exhaustive search for orders 2n, n < 100 (see Theorem 3).
4.
The Results
In Table 1 we present the results of an exhaustive search for weighing matrices W (2n, 9), for n ≤ 99. In this table 2n denotes the order of the weighing matrices which are constructed using two circulant matrices of order n, and N denotes the number of inequivalent weighing matrices as described in definition 1. Moreover, x ¯ denotes −x. To save space, we give only one representative for each case. The complete list of solutions for each case is available on request. If in the N column there is an S, then the reader is referred to Strassler (1997) for this case.
34
s. georgiou and c. koukouvinos
Table 1. The results of an exhaustive search for the existence of W (2n, 9) constructed via two circulant matrices. n SPA SPB N n SPA SPB N 5 1,2,3,4,5 1,2,3,5 1 7 1,2,3,4,7 1,2,4,5 2 7 1,2,3,4,5,6 1,2,4 6 8 1,2,3,7,8 1,2,3,6 6 10 1,2,3,510 1,2,4,7 7 10 1,2,4,5,6,10 1,3,7 18 10 1,2,3,4,6,8,10 1, 5 4 11 1,2,3,9,11 1,3 ,4,7 22 1,2,6 60 12 1,3,5,6,12 1,3,5,9 3 11 1,2,3,4,8,10 12 1,2,4,6,7,10 1,3,5 6 13 1,2,3,5,13 1,2,6,8 42 13 1,2,3,4,7,13 1,3,6 228 13 1,3,4,6,10,12,13 1,2 10 13 1,2,3,4,5,6,8,10,11 ∅ S 14 1,2,3,6,12 1,3,5,7 26 14 1,3,5,7,9,11 1,3,7 6 14 1,3,4,5,7,8,14 1,3 6 1,2,3,4,7 1,2,6,11 14 15 1,2,6,7,12,14 1,4,10 120 15 15 1,2,7,9,10,12,14 1,4 4 16 1,2,4,5,8 1,3,6,12 18 17 1,2,3,7,9 1,3,5,13 30 17 1,2,4,5,12,16 1,4,9 114 19 1,2,3,4,7 1,2,7,11 54 19 1,2,5,6,10,17 1,6,11 258 1,7,8,13,16,17,18 1,2 16 20 1,2,3,4,20 1,3,7,15 33 19 1,3,9 78 20 1,3,7,11,13,15,19 1,5 4 20 1,2,11,12,15,19 21 1,2,8,9,15 1,4,7,16 14 21 1,2,6,11,19,21 1,4,10 78 22 1,2,3,4,22 1,3,7,13 62 22 1,3,7,5,15,19 1,3,11 60 23 1,2,3,6,8 1,4,8,14 34 23 1,2,5,6,17,22 1,3,9 156 1,12,13,15,19,20,21 1,2 19 24 1,3,7,11,13 1,5,9,17 11 23 24 1,3,11,7,13,19 1,5,9 6 24 1,2,5,8,9,13,14,20,21 ∅ S 25 1,2,3,17,19 1,5,9,16 27 25 1,2,6,7,17,23 1,4,12 204 25 1,6,8,12,14,21,22 1,2 18 26 1,3,5,7,11 1,3,11,15 54 1,5,11 228 26 1,3,5,9,11,15,23 1,3 12 26 1,3,5,7,13,25 26 4,6,8,10,12,16,18,20,26 ∅ S 28 1,2,8,15,16 1,3,5,9 56 28 1,5,13,9,17,21 1,5,13 6 28 1,3,4,7,8,14,19 1,3 30 29 1,2,3,10,22 1,3,8,13 22 29 1,2,5,22,27,29 1,9,18 210 30 1,3,5,7,13 1,3,11,21 33 30 1,3,11,13,23,27 1,7,19 138 1,11,13,17,21,25,27 1,7 8 32 1,3,7,9,15 1,5,11,23 18 30 33 1,4,7,10,28 1,7,10,19 42 33 1,4,10,7,22,28 1,4,16 60 34 1,3,5,13,17 1,5,9,25 42 34 1,3,7,9,23,31 1,7,17 114 35 1,6,11,16,21 1,6,16,21 3 35 1,6,16,11,21,26 1,6,16 6 36 1,4,10,16,19 1,7,13,25 3 36 1,4,16,10,19,28 1,7,13 6 38 1,3,5,7,13 1,3,13,21 56 38 1,3,9,11,19,33 1,11,21 258 38 1,9,11,13,19,25,29 1,19 18 39 1,4,7,10,16 1,4,16,22 52 39 1,4,7,10,19,37 1,7,16 228 39 1,10,16,22,28,31,37 1,16 12 39 3,6,9,12,18,21,24,33,39 ∅ S 40 1,3,5,7,39 1,5,13,29 43 1,5,17 78 40 1,9,13,17,25,29,33 1,9 4 40 1,3,21,23,29,37 41 1,2,3,5,39 1,6,13,30 32 42 1,3,15,17,29 1,7,13,31 54 42 1,3,11,21,37,41 1,7,19 78 42 1,7,10,13,19,22,40 1,7 6 43 1,2,3,5,41 1,6,13,25 34 44 1,3,5,7,43 1,5,13,25 62 44 1,5,13,9,29,37 1,5,21 60 45 1,4,7,10,19 1,4,16,31 23 45 1,4,16,19,34,40 1,10,28 120 45 1,16,19,25,31,37,40 1,10 4 46 1,3,5,11,15 1,7,15,27 34 46 1,3,9,11,33,43 1,5,17 156 46 1,23,25,29,37,39,41 1,3 22 48 1,4,10,13,22 1,7,16,34 21 48 1,3,9,15,17,25,27,39,41 ∅ S 48 1,5,21,13,25,37 1,9,17 6
new infinite classes of weighing matrices
35
Table 1. (cont.) n 49 50 50 51 52 52 55 56 57 57 58 60 63 64 65 65 66 68 69 70 70 72 75 75 76 77 78 78 80 80 84 84 85 87 88 90 90 91 91 92 95 95 96 98 98 99
SPA 1,8,15,22,29 1,3,5,33,37 1,7,11,19,21,25,49 1,4,10,13,34,46 1,5,9,13,25,49 4,8,12,16,24,28,32,44,52 1,6,16,11,36,46 1,9,25,17,33,41 1,4,7,10,19 1,7,22,31,40,52,55 1,3,9,43,53,57 1,4,31,34,43,55 1,4,22,25,43 1,5,13,17,29 1,6,11,16,31,61 5,10,15,20,30,45,50,55,65 1,7,19,13,43,55 1,5,13,17,45,61 1,4,13,16,49,64 1,6,11,26,56 1,11,16,46,51,61,66 1,4,13,22,25,37,40,58,61 1,4,7,49,55 1,4,16,25,34,37,61 1,5,17,21,37,65 1,8,15,22,64 1,7,13,19,31 1,7,13,31,37,43,67 1,5,9,13,77 1,9,25,41,49,57,73 1,4,22,43,46 1,4,37,49,55,58,76 1,6,16,21,56,76 1,4,7,28,64 1,5,9,13,85 1,7,13,19,37 1,7,37,49,55,67,79 1,8,15,22,43,85 1,8,15,29,57,64,71 1,5,17,21,65,85 1,6,11,16,31 1,6,11,51,66,76,86 1,5,17,29,33,49,53,77,81 1,8,15,36,78 1,29,36,43,71,78,92 1,10,28,19,64,82
SPB 1,8,22,29 1,9,17,31 1,23 1,10,25 1,9,21 ∅ 1,6,26 1,9,25 1,4,19,31 1,10 1,17,35 1,7,25 1,10,19,46 1,9,21,45 1,11,26 ∅ 1,7,31 1,13,33 1,7,25 1,11,21,31 1,11 ∅ 1,13,25,46 1,28 1,21,41 1,15,22,43 1,7,31,43 7,19 1,9,25,57 1,17 1,7,13,25 1,7 1,16,41 1,7,22,37 1,9,25,49 1,7,31,61 1,19 1,15,36 1,15 1,9,33 1,6,31,51 1,21 ∅ 1,15,29,43 1,29 1,10,46
N 6 39 24 114 228 S 60 6 56 17 210 204 16 18 228 S 60 114 156 51 10 S 53 24 258 46 56 12 55 4 77 30 114 28 64 33 8 234 12 156 61 17 S 38 6 60
n SPA SPB N 49 1,8,22,15,29,36 1,8,22 6 50 1,3,11,13,33,45 1,7,23 222 51 1,4,7,19,25 1,7,13,37 40 52 1,5,9,13,21 1,5,21,29 48 52 1,5,9,21,25,29,45 1,9 12 55 1,6,11,16,46 1,11,16,31 45 56 1,3,15,29,31 1,5,9,17 70 56 1,5,7,13,15,27,37 1,5 30 57 1,4,13,16,28,49 1,16,31 258 58 1,3,5,19,43 1,5,15,25 20 60 1,4,7,10,58 1,7,19,43 66 60 1,7,13,19,31,43,55 1,25 8 63 1,4,16,31,55,61 1,19,28 78 65 1,6,11,16,26 1,6,26,36 55 65 1,6,11,21,26,36,56 1,6 12 66 1,4,7,10,64 1,7,19,37 70 68 1,5,9,25,33 1,9,17,49 38 69 1,4,7,16,22 1,10,22,40 36 69 1,4,37,46,49,52,64 1,4 21 70 1,8,22,29,36,64 1,15,43 24 72 1,7,19,31,37 1,13,25,49 7 72 1,7,31,19,37,55 1,13,25 6 75 1,4,16,19,49,67 1,10,34 324 76 1,5,9,13,25 1,5,25,41 60 76 1,5,21,41,65,69,73 1,9 18 77 1,8,22,15,50,64 1,8,36 66 78 1,7,13,19,37,73 1,13,31 228 78 7,13,19,31,37,43,49,55,67 ∅ S 80 1,5,41,45,57,73 1,9,33 78 82 1,3,5,9,77 1,11,25,59 32 84 1,5,21,41,73,81 1,13,37 84 85 1,6,11,31,41 1,11,21,61 37 86 1,3,5,9,81 1,11,25,49 36 87 1,4,13,64,79,85 1,25,52 210 88 1,9,25,17,57,73 1,9,41 60 90 1,7,31,37,67,79 1,19,55 138 91 1,8,15,22,36 1,8,36,50 54 91 8,15,22,29,36,50,71,78,85 ∅ S 92 1,5,9,21,29 1,13,29,53 38 92 1,5,9,29,37,45,77 1,9 22 95 1,6,21,26,46,81 1,26,51 258 96 1,7,19,25,43 1,13,31,67 27 96 1,9,41,25,49,73 1,17,33 6 98 1,15,43,29,57,71 1,15,43 6 99 1,10,19,28,82 1,19,28,55 44
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s. georgiou and c. koukouvinos
In the following example we illustrate how the weighing matrix W (10, 9) is constructed from the sequences of length 5 as these are given in Table 1. Example 1 In Table 1 we can see that for n = 5 we have SPA = {1, 2, 3, 4, ¯ 5} and SPB = {1, 2, ¯3, ¯5}. From the notation of SPX (see notation 4) we have that SPX = P OSX ∪ N EGX , X ∈ {A, B}. Thus, P OSA = {1, 2, 3, 4}, N EGA = {5}, P OSB = {1, 2} and N EGB = {3, 5}. Now by notation of P OSX and N EGX , X ∈ {A, B} (see notation 5) we obtain that A = (1, 1, 1, 1, −1) and B = (1, 1, −1, 0, −1). Let A1 = circ(A) and A2 = circ(B) be the 5×5 circulant matrices with first row A and B respectively. To obtain the desirable weighing matrix W (10, 9) we use one of the arrays given in (2). 2 Acknowledgment. We thank the referees for their comments and suggestions that substantially improved the presentation of this paper. References Geramita, A.V. and Seberry, J. (1979). Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel. Kiefer, J. (1975). Construction and optimality of generalized Youden designs, in Statistical Design and Linear Models, J.N. Srivastava, ed., North-Holland, Amsterdam, 333-353. Koukouvinos, C. and Seberry, J. (1999). New weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function — a review, J. Statist. Plann. Inference, 81, 153-182. Strassler, Y. (1997). The Classification of Circulant Weighing Matrices of Weight 9, Ph.D. Thesis, Bar-Ilan University, Ramat-Gan. S. Georgiou and C. Koukouvinos Department of Mathematics National Technical University of Athens Zografou 15773, Athens, Greece E-mail:
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