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New Weighing Matrices

Christos Koukouvinos Jennifer Seberry?

Department of Mathematics Department of Computer Science National Technical University of Athens and University of Wollongong Zografou 157 73 Wollongong
Athens NSW 2522
Greece Australia

Abstract

New weighing matrices and skew weighing matrices are given for many orders 4t <= 100. These are constructed by finding new sequences with zero autocorrelation. These results enable us to determine for the first time that for 4t <= 84 a W (4t; k) exists for all k = 1, ... , 4t ? 1 and also that there exists a skew-weighing matrix (also written as an OD(4t;1; k)) for 4t <= 80, t odd, k = a2 + b2 + c2; a; b; c integers except k = 4t ? 2 must be the sum of two squares.
Key words and phrases: Autocorrelation, construction, associated polynomial,sequence. AMS Subject Classification: Primary 62K05, 62K10, Secondary 05B20

1 Introduction

An orthogonal design of order n and type (s1; s2; : : : ; su) (si > 0), denoted OD(n; s1; s2; : : : ; su), on the commuting variables x1; x2; : : : ; xu is an n?n matrix A with entries from f0; ?x1; ?x2; : : : ; ?xug such that

AAT = ( uX

i=1 six2i )In

Alternatively, the rows of A are formally orthogonal and each row has precisely si entries of the type ?xi. In [3], where this was first defined, it was mentioned that

ATA = ( uX

i=1 six2i )In

and so our alternative description of A applies equally well to the columns of A. It was also shown in [3] that u <= ae(n), where ae(n) (Radon's function) is defined by ae(n) = 8c + 2d, when n = 2ab, b odd, a = 4c + d, <= d < 4.
A weighing matrix W = W (n; k) is a square matrix with entries ; ?1 having
k non-zero entries per row and column and inner product of distinct rows zero. Hence W satisfies WWT = kIn, and W is equivalent to an orthogonal design OD(n;k). The number k is called the weight of W .

?Research funded by ARC grant A48830241.

Given the sequence A = fa1; a2; : : : ; ang of length n the non-periodic autocorrelation function NA(s) is defined as

NA(s) = n?sX
i=1 aiai+s; s = ; 1; : : : ; n ? 1; (1)

If A(z) = a1 + a2z + : : : + anzn?1 is the associated polynomial of the sequence A, then

A(z)A(z?1) = nX

i=1

nX

j=1 aiajzi?j = NA(0) + n?1X
s=1 NA(s)(zs + z?s); z <> : (2)

Given A as above of length n the periodic autocorrelation function PA(s) is defined, reducing i + s modulo n, as

PA(s) = nX
i=1 aiai+s; s = ; 1; : : : ; n ? 1: (3)

Sequences with zero periodic autocorrelation function or zero non-periodic autocorrelation function, of length t, are used to form the first rows of four circulant matrices which are then used in the Goethals-Seidel array to form matrices of order 4t of the required type (if orthogonal designs) or weight (if weighing matrices). In the case of sequences with zero non-periodic autocorrelation function, the sequences are first padded with sufficient zeros added to the end to make their length t.
The results used are:

Theorem 1 [5, Theorem 4.49] If there exist four circulant matrices A1, A2, A3, A4 of order n satisfying 4X

i=1 AiATi = fI

where f is the quadratic form Puj=1 sjx2j ; then there is an orthogonal design OD(4n; s1; s2; : : : ; su).

Corollary 1 If there are four f0; ?1g-sequences of length n and weight w with zero periodic or non-periodic autocorrelation function then these sequences can be used as the first rows of circulant matrices which can be used in the GoethalsSeidel array to form OD(4n;w) or a W(4n;w). If one of the sequences is skewtype then they can be used similarly to make an OD(4n; 1; w). We note that if there are sequences of length n with zero non-periodic autocorrelation function then there are sequences of length n + m for all m >= .

Weighing matrices have long been studied because of their use in weighing experiments as first studied by Hotelling [7] and later by Raghavarao [12] and others. For more applications of weighing matrices see Banerjee [1] and Harwit and Sloane [6]. There are a number of conjectures concerning weighing matrices:

Conjecture 1 There exists a weighing matrix W(4t; k) for k 2 f1; : : : ; 4tg.

Conjecture 2 When n ? 4(mod 8), there exists a skew-weighing matrix (also written as an OD(n; 1; k)) when k <= n? 1, k = a2 + b2 + c2; a; b; c integers except that n ? 2 must be the sum of two squares.

Conjecture 3 When n ? 0(mod 8), there exists a skew-weighing matrix (also written as an OD(n; 1; k)) for all k <= n ? 1.

The reader in referred to Geramita and Seberry [5] for all other undefined terms.In Geramita and Seberry [5] the status of the weighing matrix conjecture is given for W (4t; k); k 2 f1; : : : ; 4tg and t 2 f1; : : : ; 21g: We give new results including resolving the conjecture in the affirmative for all 4t <= 84.

2 Some new sequences with zero autocorrelation

Tables 1 to 6 give new results which allow us to settle in the affirmative Conjecture 2 for orders 68 and 76. They also eliminate most unsolved cases for Conjecture 1 in order 92 and Conjecture 2 in order 84.

Length=19 Sequences with zero non-periodic autocorrelation function 1,62 f0 ++ +? +? + a ? + ?+ ?? ?0g,
f+ ? ++ +? ?+ + ?? +?0 +++ +g,
f+ ? ++ +? ?+ + ++ ? +0 ??? ?g,
f0 ++ +? +? + + ?+ ?+ ++ g
1,66 f0 ++ +? +? ++ a ?? +?+ ?? ? g,
f0 ++ +? +? ++ + +? +? ++ + g,
f+ + +? +? ?? ?? ?++ ?+ +? g,
f+ + +? +? ?? ++ +?? +? ?+ g
1,68 f+ + ?+ +? ?? a ++ +? ?+? ?g,
f+ + +? ++ ?+ + ++ ?+ ?? +++g,
f+ + +? ++ ?+ + ?? +? ++ ? ??g,
f+ + ?+ +? ?? ??? + + ?+ +g
1,72 f? ? ++ ++ +? + a ? +? ?? ??++g,
f+ ? +? ++ ?+ + +? ++ +? ?++g,
f+ ? +? ++ ?+ + ?+ ?? ?+ + ??g,
f? ? ++ ++ +? + +? ++ ++ + ??g

Table 1: Sequences of length 19 with zero non-periodic autocorrelation function

3 Numerical consequences

We use the tables of Appendix H and extend Theorem 4.149 of Geramita and Seberry [5].

We note that sequences with periodic autocorrelation function zero exist for W(4t; 4t ? 1), W(4t; 4t) for all t 2 f1; : : : ; 31g [15]. We use the (OD; 1; 1; 64) for n >= 17 and the (OD; 1; 1; 80) for n >= 21 from [11]. Hence using the results of [8], [9], [10], [11] and those given in Tables 1, 2 and 3, we have:

Length=19 Sequences with zero periodic autocorrelation function 1,61 f0 ?+ +? +? ? a ++? +? ?+ g,
f+ ++ +++ ?? ?+ ?+ ?+ ++ g,
f+ +? +++ ?? ++ ?+ ?+ + g,
f? + ? ??+ +? ++ ++ +? ?? g
1,67 f+ ?? ? ++ ?+ a ? +? ? ++ + ?g,
f0 ++ ++ +? ++ + + ?++ ?? + ?g,
f? ++ ++? +? ++ ?? ?+ ?+ + g,
f? ++? ++ +? ++ ++ ??? ?+ g
1,69 f + ?? +++ ?+ a ?+ ?? ?+ +? g,
f? ++ +++ ?? ?+ ?+ ?? + + ?g,
f+ +? + ?+ ?? +? ?? +++ ++?g,
f? +++ ++ ++ +? +? +?+ ++?g,
1,70 f+ ++ ++? +? + a ? +? +? ?? ??g,
f+ ?+? ++ ?? +? ++ +?+ +++g,
f+ +? +++ ?? ++ +? ?+ +?+ g,
f? ++ + ?+ +? ++ +? ??? ++g
1,1,74 f+ +? +?+ ++ ? a + ?? ?+ ?+ ??g,
f? +? ?+? ?? ? b + +++ ?+ +?+g,
f? ++ ??? ?? ++ +? ?? ??+ + ?g,
f? ++ ?+? ++ ++ ++ +? +?+ + ?g,

Table 2: Sequences of length 19 with zero periodic autocorrelation function

Theorem 2 There exists an orthogonal design OD(4n; 1; k) when

(i) for n >= t, t = 3, 5, 7, 9 with k 2fx : x <= 4t ? 1; x = a2 + b2 + c2g;

(ii) for n >= 11, with k 2fx : x <= 43; x = a2 + b2 + c2; x <> 42g;

(iii) for n >= 13, with k 2fx : x <= 51; x = a2 + b2 + c2g;

(iv) for n >= 15, with k 2fx : x <= 59; x = a2 + b2 + c2g;

(v) for n >= 17, with k 2fx : x <= 67; x = a2 + b2 + c2; x <> 61; 66g;

(vi) for n >= 19, with k 2fx : x <= 75; x = a2 + b2 + c2; x <> 61g.

(vii) for n >= 21, with k 2fx : x <= 83; x = a2 + b2 + c2; x <> 61; 77; 78; 82g.

(viii) for n >= 23, with k 2fx : x <= 91; x = a2 + b2 + c2; x <> 61; 77; 78; 82; 85; 86; 89; 90; 91g.

(ix) for n >= 25, with k 2fx : x <= 99; x = a2 + b2 + c2; x <> 61; 77; 78; 82; 85; 86; 89; 90; 91; 92; 93; 94; 97; 98; 99g.

All are constructed by using four circulant matrices in the Goethals-Seidel array. Proof. We have the results for n >= 13 for all k except 46 and 49 from [11]. The OD(52;1; 46) is given in [10, Table 2]. An OD(52;1; 49) is given in [5, Theorem 8.38]. The OD(4n; 1; 46) and OD(4n; 1; 49) for n >= 15 are given in Table 1 of [8]. The existence of OD(4n; 1; 57) n >= 15 and OD(4n;1; 67) for n >= 17 is established in [9].

For n >= 17 [11] gives the result except for k = 46, 49, 61, 62. The result for 62 is given in Table 1. An OD(68; 1; 66) does not exist as its existence would imply the existence of an OD(68;1; 1; 66) by the Geramita-Verner Theorem [5, Theorem 2.20] and the existence of that design requires that 66 should be the sum of two squares which it is not. Hence the OD(68;1; 66) does not exist. The value 60 was erroneously included in [11] Theorem 8: it should not have been included as 60 is a number of the form 4a(8b + 7) and so it cannot be written as three squares as required by the conditions of Conjecture 2.
OD(4n; 1; k) for k = 70 exist for n = 19 from Table 2, k = 21 from [9], and for n >= 23 from Table 5.

OD(4n; 1; k) for k = 68, 72 and n >= 19 are given in Table 1, the result for k = 66 and 68 with n >= 19 is given in [5, Table H.2]. OD(4n; 1; 76) for n >= 21 is given in Table 4 and OD(4n; 1; k) for k = 68, 72 and 83 are given in [9]. The OD(4n; 1; k) for k = 84, 88 and n >= 23 are given in Table 4. The OD(4n;1; 75) for n = 21 is from Table 3 and for n >= 23 from Table 5.

The OD(4n;1;k) for k = 69, 74 and 75 exist for n =19 from Table 2 and for n >= 21 from [9]. The OD(76;1; 73) is given in [8] and the OD(4n; 1; 73), n >= 21, in [9].The OD(4n; 1; 96), n >= 25 is given in Table 6. 2

Length=21 Sequences with zero periodic autocorrelation function 1,57 f? ?? ?+? ? a + +? +++ +g,
f+ + ++ +? ++ +?+ ? ?++g,
f+ ? ++ + + ? ++ ?? ?+ ? g,
f+ +? ? + + + +? ? ? + + ?g
1,61 f+ ++ ?+ ? + + a ? ? + ? +? ??g,
f+ ?+ ?++ ?? + + ++ ?++ + g,
f+ ++ ?+ + ++ +? ?+ ??+ g,
f+ + ? ? ++ ??? ?+ ? + +g
1,62 f? ?? ?+ ? ? a + + ? ++ ++g,
f+ ++? +? ? + + ++ +? +? ++g,
f+ ++ + ?? +? +? + ? + ? 0+g,
f + +? ?++ ? ? +?+ ? + ++ ?g
1,67 f ? ?+ ++? +? a +? +? ??+ + g,
f+ ?+? +? + + + ++ ?? ++ ++g,
f+ + + ?+? ? ? ?+ +++ ?+ +? g,
f ? ??+ ++ ++ +? ++ ?+? + ? g
1,73 f+ ++ +++ ?+ ? a + ?+ ?? ?? ??g,
f? ++ ? ++ +? + + ++ ? ?+ ++?g,
f+ +? + +? ++ ?+ ?+ ? ++ ++ ?g,
f? +? +++ ?? ?? ++ ?? +? + ++g
1,74 f+ +? ??? +? +? a +? +? ++ ++ ? ?g,
f? ++ +++ ?? + + ?++ ++ +? + ?g,
f+ ? ++ ++ +++ ?+ ?? ?+ ??+g,
f+ +? +?+ ++ ?? ?+ +? + ?+g

Table 3 : Sequences of length 21 with zero periodic autocorrelation function

Length=21 Sequences with zero periodic autocorrelation function 1,75 f+ +? +?+ ++ + a ? ?? ?+ ?+ ??g,
f? + ++ ?? ?+ ++ ?+ ++? ++ +?g,
f? ++ +++ +? ++ ?? ?+ ?++ + ?g,
f+ +? +?+ +? ++ ++ +? ?+? + ?g
1,76 f+ +? ??+ ?+ +? a +? ?+ ?+ ++ ? ?g,
f? ? ++ ++ ++ ++ ?+ ??+ + + ?g,
f? ++ ++? +? ++ ?+ ?+ ?++ + + ?g,
f+ ? +? ?? ++ ?? ?+ ?++ + + +g
1,77 f? ++ +++ ?+ ? a + ?+ ?? ?? ?+g,
f? +++ +? +? ++ +? ++? ?+ ++?g,
f+ + + ++ + ?+ ?+ +? +? ++? ? ?g,
f? +? ++ ? ?+ +? ?+ ?++ ++ ++?g
79 f+ +? + ?? ?+ +? ++ ++ + +? ++g,
f + ++ ??? ++ +? + + +? ++? ?+g,
f? +? +++ ?+ ++ +? ?? +?? + + ?g,
f+ ?+ +?? +? ?+ ?+ ?+ ??? ++ ++g
1,1,82 f? ?+ ??+ ++ ?? a ++ ?? ?+ +? ++g,
f+ ++ +?? ?+ ?? b ++? ++ +? ?? ?g,
f? +? ++? +? +? +? +? +?+ +? + ?g,
f? +? ?++ ++ ++ ++ ++ +++ ?? + ?g

Table 3 (cont): Sequences of length 21 with zero periodic autocorrelation function

Length=21 Sequences with zero non-periodic autocorrelation function 1,68 f+ + ++ ?+ +? a +? ?+ ?? ? ?g,
f? + +? ++ + ? ? ++ ?? ?++ ?+g,
f? + +? ++ + ? ? ?? ++ +?? +?g,
f+ + ++ ?+ +? ? + + ?+ + ++g
1,72 f +? ++ ?? ++ + a ??? ++ ?? +? g,
f+ + ?+ ?+ +? + ?? ?+ ++ ++ 0+g,
f+ + ?+ ?+ +? + ++ +? ?? ?? ?g,
f0 +? ++ ?? +++ ++ +? ?+ +? + g
74 f? + ++ ++ ++ ? +? + ? ?+++g,
f? + ++ ++ ++ + ?+ ? + +? ??g,
f? + ?+ +? ?+ +? ++? ?? ?+ ?+ +g,
f? + ?+ +? ?+ +? ??+ ++ +? +? ?g
1,76 f+ + + ? ?+ ?? + a ? ++ ?+ + ? ??g,
f? + ++ +? ++ ++ ++? ?+ ?+ ?+ ?g,
f? + ++ +? ++ ++ ??+ +? +? +? +g,
f+ + + ? ?+ ?? + +? ?+ ?? + ++g

Table 4: Sequences of length 21 with zero non-periodic autocorrelation function

Length=23 Sequences with zero non-periodic autocorrelation function 1,70 f0 ?+ ++ ++? ?+ a ? ++ ?? ?? ? + g,
f? + + + + + ?+ ? ++ ?+ + + ??g,
f? + + + + + ?+ + ?? +? ? ? ++g,
f0 ?+ ++ ++? ?+ +? ?+ ++ + +? g
1,74 f? ? ++ ?++ ? a +? ?+ ?? + 0+g,
f? + ++ + + ++ ?+ ? +? ++ +? ?? ++g,
f? + ++ + + ++ ?+ + ?+ ?? ?+ ++ ??g,
f? ? ++ ?++ ? ? ++ ?+ + ? ?g
1,84 f+ ? ++ ?+ ++ ?? ? a + ++ ?? ?+ ??+?g, f+ + ?+ + ? ++ ++ ? +? +? ++ ++?g,
f+ + ?+ + ? ++ ++ + ?+ ?+ ?? ??+g,
f+ ? ++ ?+ ++ ?? ? ?? ?+ ++ ?+ +?+g
1,88 f? + +? ?+ ?+ ++ + a ? ?? ?+ ?+ +??+g, f+ ? ?+ ++ ?+ ++ + ?+ ?+ ?+ ++ + ??g,
f+ ? ?+ ++ ?+ ++ + +? +? +? ?? ?++g,
f? + +? ?+ ?+ ++ + ++ ++ ?+ ?? ++?g

Table 5: Sequences of length 23 with zero non-periodic autocorrelation function

Length=25 Sequences with zero non-periodic autocorrelation function 1,96 f+ ?+ +? ++ ++? ?? a + ++ ?? ??+ ?? + ?g, f+ ++ ?+ ?+ ++? ++ +? ??+ +? ++ +? +g,
f+ ++ ?+ ?+ ++? ++ ?+ ++? ?+ ?? ?+ ?g,
f+ ?+ +? ++ ++? ?? ?? ?++ ++ ?+ +? +g,

Table 6: Sequences of length 25 with zero non-periodic autocorrelation function

Theorem 3 There exists a W (4n; k) when

(i) for n >= t, t = 3, 5, 7, 9, 11, 13, 15, 17, 19 with k 2fx : x <= 4tg;

(ii) for n >= 21, with k = 1, : : : , 78, 80, 81, 82;

(iii) for n >= 23, with k = 1, : : : , 78, 80, 81, 82, 85, 86, 88, 89, 90, 92.

All are constructed by using four circulant matrices in the Goethals-Seidel array. Proof. We use Theorem 2, the W(4t; 4t) and W(4t; 4t ? 1) from [15] and Theorem 3 from [11].

The result for 75 in (i) comes from Theorem 1, Tables 3 and 5 and for 77, in (ii) from Theorem 1 and from Table 4. The result for 78, 80, 81, 82 and 84 in (ii) comes from [11, Table 8] as does the result for 86, 88, 90 and 92 in (iii). Table 5 gives 71, 85 and 89 of (iii). 2

Lemma 1 The necessary conditions are sufficient for the existence of OD(4n; 1; k) for n = 3; 5; : : : ; 19 and k <= 4n ? 1. All are constructed from four circulant matrices in the Goethals-Seidel array.

Proof. Use Theorem 2 for all the results for n <= 11.

The OD(52;1; 46) and OD(60;1; 57) required are given in Tables 2 and 4 of [10].The OD(68;1;k) for k = 57, 61, and 67 are given in [11, Table 6] and [10, Table 6]. An OD(68;1; 66) does not exist as was shown in the proof of Theorem 2 above. The value 60 was erroneously included in [11]: an OD(68;1; 60) does not exist.

The OD(76;1;k) for k = 57, 61, 66, 67, 69, 70, 73, 74 and 75 are given in Tables 2 and 3 of [8], and Table 2 of this paper. This gives us the result of the enunciation. 2

Lemma 2 There exists an OD(84;1;k) for k 2fx : x <= 83; x = a2 + b2 + c2g with the possible exception of 70; 78 which are undecided. All may be constructed using four circulant matrices in the Goethals-Seidel array.

Proof. The results for k = 57, 61, 67, 73, 75, 77, 82 and 83 which may be constructed using four circulants in the Goethals-Seidel array are given in Table 3 of this paper. 2

Lemma 3 There exists an OD(92; 1; k) for n >= 23, with k 2fx : x <= 91; x = a2 + b2 + c2 g with the possible exception of 61; 77; 78; 82; 85; 86; 89; 90; 91 which are undecided.

Proof. Use Theorem 2. 2

Lemma 4 There exists a W(4n;k) for k 2 fx : <= x <= 4ng with n =
1; 3; : : : ; 21. All are constructed from four circulant matrices in the GoethalsSeidel array.

Proof. Follows from [11], Lemma 1 and Tables 1 and 2. k = 71 is given in [9]. The sequences for W (84; 79) and W(84;77) are given in Tables 3 and 4. 2

Lemma 5 There exists a W(92;k) for all k except possibly 79; 87, which are undecided. All are constructed by using four circulant matrices in the GoethalsSeidel array.

Proof. The paper [11] gives all k except possibly 71, 73, 75, 77, 79, 83, 85, 87 and 89. Koukouvinos [9] gives k = 83. The sequences for 71, 73, 75, 77, 85, and 89 are given in Tables 2, 4 and 5. 2

Lemma 6 There exists a W(100;k) for all k except possibly 87; 91; 93; 95 which are undecided. All are constructed by using four circulant matrices in the Goethals-Seidel array.

Proof. The paper [11] gives all k except possibly 71, 77, 83, 87, 89, 91, 93, 95, 97. Koukouvinos [9] gives k = 83. The sequences for 71, 77 and 89 are given in Tables 4 and 5. The OD(100;1; 96) in Table 6 gives k = 97.

Applicable Unresolved Applicable Unresolved
Order Conjecture Cases Conjecture Cases
4 1 true 2 true
8 1 true 3 true
12 1 true 2 true
16 1 true 3 true
20 1 true 2 true
24 1 true 3 true
28 1 true 2 true
32 1 true 3 true
36 1 true 2 true
40 1 true 3 true
44 1 true 2 true*
48 1 true 3 true
52 1 true 2 true
56 1 true 3 true
60 1 true 2 true
64 1 true 3 true
68 1 true 2 true*
72 1 true 3 true
76 1 true 2 true
80 1 true 3 true
84 1 true 2 70,78
88 1 true 3 true
92 1 79,87 2 61,77,78,82
85,86,89,90,91
96 1 true 3 true
100 1 87,91,93,95 2 61,77,78,82,85,
86,89,90,91,92,
93,94,97,98
104 1 95 3 94,95
112 1 true 3 true
120 1 true 3 true

Table 7: Summary of the Conjectures.
True signifies the conjecture is verified.
?OD(n; 1; n ? 2) is not possible as n ? 2 is not the sum of two squares.

References

[1] K. S. Banerjee, Weighing Designs for Chemistry, Medicine, Economics, Operations Research and Statistics, Marcel Dekker, New York, 1975.

[2] P. Eades and J. Seberry Wallis, An infinite family of skew-weighing matrices, Combinatorial Mathematics IV, in Lecture Notes in Mathematics, Vol 560, Springer-Verlag, Berlin- Heidelberg-New York, 1976, 27{60.

[3] A.V.Geramita, J.M.Geramita, and J.Seberry Wallis, Orthogonal designs, Linear and Multilinear Algebra, 3 (1976), 281-306.

[4] A. V. Geramita, N. J. Pullman and J. Seberry Wallis, Family of weighing matrices, Bull. Austral. Math. Soc., 10, (1974), 119{122.

[5] A. V. Geramita and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel, 1979.

[6] M. Harwit and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, New York, 1979.

[7] H. Hotelling, Some improvements in weighing and other experimental techniques, Ann. Math. Stat., 15, (1944), 297{306.

[8] C. Koukouvinos, Construction of some new weighing matrices, Utilitas Math., 44, (1993), 51{55.

[9] C. Koukouvinos, Some new weighing matrices, Utilitas Math., 46, (1994), 81{89.

[10] C. Koukouvinos and J. Seberry, Some new weighing matrices using sequences with zero autocorrelation function, Austral. J. Combin., 8, (1993), 143{152.

[11] C. Koukouvinos and J. Seberry, On weighing matrices, Utilitas Math., 43, (1993), 101{127.

[12] D. Raghavarao, Constructions and Combinatorial Problems in Design of Experiments, J. Wiley and Sons, New York, 1971.

[13] Jennifer Seberry, An infinite family of skew{weighing matrices, Ars Combinatoria, 10, (1980), 323{329.

[14] Jennifer Seberry, The skew-weighing matrix conjecture, University of Indore Research J. Science, 7, (1982), 1{7.

[15] Jennifer Seberry and Mieko Yamada, Hadamard matrices, sequences and block designs, in Contemporary Design Theory - a Collection of Surveys, eds J. Dinitz and D.R. Stinson, John Wiley and Sons, New York, (1992), 431{560.

[16] Jennifer Seberry Wallis, Hadamard matrices, Part IV, Combinatorics: Room Squares, sum free sets and Hadamard Matrices, Lecture Notes in Mathematics, Vol 292, eds. W. D. Wallis, Anne Penfold Street and Jennifer Seberry Wallis, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

[17] J. (Seberry) Wallis, Orthogonal (0, 1, ?1)-matrices, Proceedings of the First Australian Conference on Combinatorial Mathematics, (ed Jennifer Wallis and W. D. Wallis), TUNRA Ltd, Newcastle, Australia, (1972), 61-84.