Hadamard ideals and Hadamard matrices with two circulant cores

Transcription

1 Hadamard ideals and Hadamard matrices with two circulant cores Ilias S. Kotsireas a,1,, Christos Koukouvinos b and Jennifer Seberry c a Wilfrid Laurier University, Department of Physics and Computer Science, 75 University Avenue West, Waterloo, Ontario N2L 3C5, Canada b Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece c Centre for Computer Security Research, School of Information Technology and Computer Science, University of Wollongong, Wollongong, NSW 2522, Australia Abstract We apply Computational Algebra methods to the construction of Hadamard matrices with two circulant cores, given by Fletcher, Gysin and Seberry. We introduce the concept of Hadamard ideal for this construction to systematize the application of Computational Algebra methods. Our approach yields an exhaustive search construction of Hadamard matrices with two circulant cores for this construction for the ten first admissible values 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. For each of these ten parameter values, the number of non-equivalent such Hadamard matrices is proportional to the square of the parameter. Key words: Hadamard Matrices, Computational Algebra, Hadamard ideal, Hadamard equivalence, algorithm MSC: 05B20, 13P10. Corresponding author. Address: Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada address: ikotsire@wlu.ca (Ilias S. Kotsireas). 1 Supported in part by a grant from the Research Office of Wilfrid Laurier University and a grant from the Natural Sciences and Engineering Research Council of Canada. Preprint submitted to Elsevier Science 5 November 2003

2 1 Introduction In [3] the authors introduce Legendre sequences and generalised Legendre pairs (GL pairs). They show how to construct an Hadamard matrix of order 2l + 2 from a GL pair of length l. This Hadamard matrix is constructed via two circulant matrices. The authors review the known constructions for GL pairs and use the discrete Fourier transform (DFT) and power spectral density (PSD) to complete an exhaustive search for GL pairs for lengths l 45 and partial results for other l. In this paper we introduce Hadamard ideals for two circulant cores as a means of applying computational algebra technique in the aforementioned Hadamard matrix construction. 2 GL-pairs Georgiou, Koukouvinos and Seberry [?] point out that GL-pairs, which can be used to construct Hadamard matrices of order 2l + 2 with two circulant cores, exist for (i) l is a prime (see for example [23]). (ii) 2l + 1 is a prime power (these arise from Szekeres difference sets, see for example [23] or [25]). (iii) l = 2 k 1, k 2 (two Galois sequences are a GL-pair, see for example [27]). (iv) l = p(p + 2) where p and p + 2 are both primes (two such sequences are a GL-pair,see for example, [28,29]). (v) l = 49, 57 (these have been found by a non-exhaustive computer search that uses generalized cyclotomy and master-switch techniques, see [25,26]). (vi) l = 3, 5,..., 45 (these have been found and classified by exhaustive computer searches, see [23]). (vii) l = 47, 49, 51, 53 and 55 (these have been found and classified by partial computer searches, see [23]). (viii) l = 143 (also verified the results for l = 3, 5, 7, 11, 13, 15, 17, 19, 23, 25, 31, 35, 37, 41, 43, 53, 59, 61, 63 see [24]). GL-pairs do not exist for even lengths. It is indicated in [23] that the following lengths l 200 are unresolved: 77, 85, 87, 91, 93, 115, 117, 121, 123, 129, 133, 145, 147, 159, 161, 169, 171, 175, 177, 185, 187 and 195. We note here that a GL-pair for length l = 143 is constructed easily since 143 = is a product of twin primes. 2

3 3 Hadamard matrices with two circulant cores An Hadamard matrix of order n is an n n matrix with elements ±1 such that HH T = H T H = ni n, where I n is the n n identity matrix and T stands for transposition. For more details see the books of J. Seberry cited in the bibliography. We detail a construction of Hadamard matrices with two circulant cores from [3]. Let A and B be two circulant matrices of order l. Then the matrix formed as indicated below is a Hadamard matrix of order 2l A B H 2l+2 = B T A T + (1) The two circulant matrices A and B satisfy the matrix equation AA T + BB T = (2l + 2)I l 2J l where I l is the identity matrix or order l and J l is a matrix of order l whose elements are all equal to 1. The matrices A and B of order l are circulant, i.e, a ij = a 1,j i+1(mod l) and b ij = b 1,j i+1(mod l). Since 2l + 2 must be equal to a multiple of 4 we have that l must be an odd integer for this construction to yield a Hadamard matrix. The following matrix is an example of a Hadamard matrix of order 8 with two circulant cores of 3

4 order l = 3 each: Hadamard matrices with circulant cores An Hadamard matrix of order n is an n n matrix with elements ±1 such that HH T = H T H = ni n, where I n is the n n identity matrix and T stands for transposition. For more details see the books of Jennifer Seberry cited in the bibliography. An Hadamard matrix of order 2p + 2 which can be written in one of the two equivalent forms C 1 C C2 T C1 T 1 or 1 1. C 1 C C2 T C1 T where C k = (c ij ), k= 1, 2 is a circulant matrix of order p i.e. c ij = c 1,j i+1(mod p), is said to have two circulant cores. The following matrices are examples for order 12. 4

5 where stands for 1 to conform with the customary notation for Hadamard matrices. The two forms are equivalent as described in section 4.1. In this paper we use the second form as described in section 4.3. To Come about existence: We group these results as a theorem Theorem 1 (Two Circulant Cores Hadamard Construction Theorem) An Hadamard matrix of order 2p + 2 with with two circulant cores can be constructed if (1) p 1(mod 4) is a prime [12]; (2) more to come 4.1 Equivalent Hadamard matrices Two Hadamard matrices H 1 and H 2 are called equivalent (or Hadamard equivalent, or H-equivalent) if one can be obtained from the other by a sequence of row negations, row permutations, column negations and columns permutations. More specifically, two Hadamard matrices are equivalent if one can be obtained by the other by a sequence of the following transformations: Multiply rows and/or columns by -1. Interchange rows and/or columns. For a detailed presentation of Hadamard matrices and their constructions see [6], [21], [15] and for inequivalent Hadamard matrices see [5] and [4]. Remark 1 For a given set X of Hadamard matrices of arbitrary but fixed dimension n, the relation of H-equivalence (noted H here) is an equivalence relation. Indeed, H-equivalence is reflexive (H H H, H X) symmetric (H 1 H H 2 implies H 2 H H 1, H 1, H 2 X) and transitive (H 1 H H 2 and 5

6 H 2 H H 3 imply H 1 H H 3, H 1, H 2, H 3 X). Therefore, one can study the equivalence classes and define representatives for each class. To define H more formally, suppose P and Q are two monomial matrices of order n (monomial means elements 0, +1, 1 and only none non zero entry in each row and column) where P P T = QQ T = I n. Then two Hadamard matrices of order n are said to be equivalent if A = P BQ. 4.2 Construction of Hadamard matrices with two circulant cores We detail the construction of Hadamard matrices with circulant core with an eye to producing a set of nonlinear polynomial equations and study the structure of the associated ideal which we will call a Hadamard Ideal. Let n = 4k, for k = 1, 2,... and consider the vector of n 2 unknowns (a 1,..., a 2k 1, b 1,..., 2k 1 ). This vector generates the two cyclic n n 2 matrix (supplemented from underneath by a row of 1s and another row of half 1s and half -1s) a 1 a 2... a 2k 2 a 2k 1 b 1 b 2... b 2k 2 b 2k 1 a 2k 1 a 1... a 2k 3 a 2k 2 b 2k 1 b 1... b 2k 3 b 2k a 2 a 3... a 2k 1 a 1 b 2 b 3... b 2k 1 b 1 b 1 b 2k 1 b 2k 2... b 2 a 1 a 2k 1 a 2k 2... a 2 B n 2 = b 2 b 1 b 2k 1... b 3 a 2 a 1 a 2k 1... a b 2k 1 b 2k 2... b 2 b 1 a 2k 1 a 2k 2... a 2 a Now we must have B t n 2 B n 2 = ni n 2 with the additional constraints: {a 1,..., a 2k 1, b 1,..., b 2k 1 } { 1, +1} n 1 (amounts to a set of n 2 quadratic constraints); a 1 + a a 2k 1 + b 1 + b b 2k 1 = 2 (linear constraint). Once we have constructed the matrix B n 2, an n n Hadamard matrix is 6

7 obtained by supplementing it with one column of 1s and another column of half 1s and half 1s from the left 1 1 a 1 a 2... a 2k 2 a 2k 1 b 1 b 2... b 2k 2 b 2k a 2k 1 a 1... a 2k 3 a 2k 2 b 2k 1 b 1... b 2k 3 b 2k a 2 a 3... a 2k 1 a 1 b 2 b 3... b 2k 1 b 1 1 b H n = 1 b 2k 1 b 2k 2... b 2 a 1 a 2k 1 a 2k 2... a 2. 1 b 2 b 1 b 2k 1... b 3 a 2 a 1 a 2k 1... a b 2k 1 b 2k 2... b 2 b 1 a 2k 1 a 2k 2... a 2 a Hadamard ideals We detail the construction of Hadamard matrices with circulant core with an eye to producing a set of nonlinear polynomial equations and study the structure of the associated ideal which we will call a Hadamard Ideal. Consider two vectors of l unknowns each (a 1,..., a l ) and (b 1,..., b l ). These two vectors generate two circulant l l matrices A l and B l : a 1 a 2... a l b 1 b 2... b l a A l = l a 1... a l 1 b, B l = l b 1... b l a 2 a 3... a 1 b 2 b 3... b 1 Once we have constructed the two circulant matrices A l and B l, the Fletcher- Gysin-Seberry construction of Hadamard matrices with two circulant cores (see [3]) stipulates that an Hadamard matrix of order 2l + 2 is obtained by supplementing these matrices and their transposes as in (1). The additional constraints {a 1,..., a l,b1,...,b l } { 1, +1} n 1 arise from the fact that the elements of a Hadamard matrix are required to be ±1. A succinct algebraic description of these quadratic constraints given above is provided by 7

8 the following set of 2l algebraic equations: a = 0,..., a 2 l 1 = 0, b = 0,..., b 2 l 1 = 0. Another way to express this, is to say that we want to target some elements of the variety which are located inside the subvariety defined by { 1, +1}... { 1, +1}. }{{} 2l terms The matrix equation H 2l+2 = (2l+2)I 2l+2 gives rise to the following categories of equations: a set of (l 1)/2 equations s 1 = 0,..., s (l 1)/2 = 0 where each s i is a quadratic polynomial with 2l terms plus the constant term 2. The 2l quadratic terms are divided into two separate classes with l terms each. One class contains monomials in a i alone and the other class contains monomials in b i alone. the equation of the form a a 2 l + b b 2 l = 2l which is satisfied trivially, since a 2 1 =... = a 2 l = b 2 1 =... = b 2 l = 1; the two equations l l a i = b i and i=1 i=1 which imply the simpler equations: l l a i b i = 2 i=1 i=1 l a i = 1 i=1 and l b i = 1. i=1 To systematize the study of the system of polynomial equations that arise in the Fletcher-Gysin-Seberry construction of Hadamard matrices with two circulant cores, we introduce the notion of Hadamard Ideal. This allows us to apply numerous tools of computational algebra to the study of Hadamard matrices with two circulant cores. This connection between an important combinatorial problem and ideals in multivariate polynomial rings is exploited in this paper from both the theoretical and the computational points of view. Hadamard Ideals are also defined for other constructions of Hadamard matrices based on the concept of circulant core [10]. The ideals that arise in all of these constructions share numerous similar characteristics and this justifies using the term Hadamard Ideal to describe all of them. When it is not clear which construction we are referring to, the name of the construction may be mentioned explicitly, to remove any potential ambiguities. 8

9 Definition 1 For any odd natural number l = 3, 5, 7,... set (l 1)/2, l = 3, 5, 7, 9 m =. 5, l 11 Then the l-th Hadamard ideal H l (associated with the two circulant cores construction by Fletcher, Gysin and Seberry) is defined by: H l = s 1,... s m, a a l 1, b b l 1, a 2 1 1,..., a 2 l 1, b 2 1 1,..., b 2 l 1 where s 1,... s m are quadratic equations defined by: l ( ) s 1 = 2 + ai a (i+1) mod l + b i b (i+1) mod l i=1... l ( ) s m = 2 + ai a (i+m) mod l + b i b (i+m) mod l i=1 Remark 2 H l is generated by m l polynomials. Notation 1 We will also use the notation H l = h 1,... h m, h m+1, h m+2,..., h m+2+2l in direct one-to-one correspondence with the definition of H k given above. Remark 3 Since mod l = 2 and l + 1 mod l = 1, equation s 1 starts with the term a 1 a 2 and finishes with the term a l a 1 (and the analogous b-terms). Property 1 H l is a zero-dimensional ideal. (This is evident, because all points in H l are also points of { 1, +1} 2l which is in turn, a finite set). Remark 4 The sum s s m is equal to an expression involving e a 2 + e b 2 where e a 2 is the second elementary symmetric function in the unknowns a 1,..., a l and e b 2 is the second elementary symmetric function in the unknowns b 1,..., b l, but only for l = 3, 5, 7, 9, 11. For l > 13 the sum s s m contains less terms than the sum e a 2 + e b 2. This can be seen by a simple term count (disregarding the constant term 2) as 9

10 follows: l = 3, m = 1, 6, (= 3 2) terms l = 5, m = 2, 30, (= 6 5) terms l = 7, m = 3, 42, (= 7 6) terms l = 9, m = 4, 72, (= 9 8) terms l = 11, m = 5, 110, (= 11 10) terms l = 13, m = 5, 130, ( 156 = 13 12) terms In ( ) general, the second elementary symmetric function e 2 in l variables contains l 2 = l(l 1) terms and therefore the sum e a e b 2 will contain 2 ( ) l 2 = l(l 1) terms. Lemma 1 For any l = 3, 5, 7,..., the (5l + 3)/2 generators of the Hadamard ideal L l are not algebraically independent. More specifically we have the syzygy... something similar with H k holds here Definition 2 Two solutions of the system corresponding to the Hadamard ideal L l will be termed equivalent, if the corresponding Hadamard matrices are equivalent. 5 Structure of the variety V (H l ) We summarize in the following table the computational results obtained using the Hadamard ideals H 3,..., H 21 : (the symbol V (H l ) stands for the number 10

11 of solution of the system corresponding to the Hadamard ideal H l ). l V (H l ) 3 9 = = = = , 904 = , 098 = , 200 = , 214 = 3, , 240, 848 = 28, , 269, 462 = 261, (2) The next theorem clarifies a basic fact about the number of solutions of the system corresponding to the Hadamard ideal H l : Theorem 2 For any odd l 3, the total number of solutions of the system corresponding to the Hadamard ideal H l is proportional to l 2. In symbols V (H l ) = h l l 2, where h l is an integer proportionality constant. Proof Consider a specific solution a 1,..., a l, b 1,..., b l as a finite sequence of length 2l. Consider the l cyclic permutations of the a 1,..., a l and the l cyclic permutations of the b 1,..., b l. Combining each of the l permutations of the a i with each of the l permutations of the b i, gives a solution of the system. Therefore, each solution gives rise to l 2 solutions, which implies that the total number of solutions of the system is proportional to l 2. Examining table (2) we see that the first ten integer proportionality constants predicted by theorem (2) are given by: l h l , , , 382 In light of theorem (2) and since Hadamard ideals provide a means of performing exhaustive searches for the associated Hadamard matrices, the above 11

12 computational results can be stated concisely as: Theorem 3 For the first ten values l = 3,..., 21, the resolution of the system corresponding to the Hadamard ideal H l indicates that the total number of Hadamard matrices with 2 circulant cores constructed by the method of Fletcher, Gysin and Seberry is proportional to l 2. The proportionality constants are given by the sequence 1, 2, 4, 12, 24, 42, 432, 3326, 28368, Conclusion In this paper we introduce the concept of Hadamard ideals to the study of Hadamard matrices with two circulant cores for the construction of Fletcher, Gysin and Seberry. Exhaustive searches are performed for the first ten values l = 3,..., 21. A Computational results For l = 3, 5, 7 the results of solving the systems attached to the Hadamard ideals H 3, H 5, H 7 are given below: A.1 l = 3 H 3 = 2+a 1 a 3 +a 2 a 1 +a 3 a 2 +b 1 b 3 +b 2 b 1 +b 3 b 2, a 1 +a 2 +a 3 1, b 1 +b 2 +b 3 1, a 2 1 1,... b Solutions: 1, {a1 = -1, a2 = 1, a3 = 1, b1 = -1, b2 = 1, b3 = 1} 2, {a1 = -1, a2 = 1, a3 = 1, b1 = 1, b2 = 1, b3 = -1} 3, {a1 = -1, a2 = 1, a3 = 1, b1 = 1, b2 = -1, b3 = 1} 4, {a1 = 1, a2 = 1, a3 = -1, b1 = -1, b2 = 1, b3 = 1} 5, {a1 = 1, a2 = 1, a3 = -1, b1 = 1, b2 = 1, b3 = -1} 6, {a1 = 1, a2 = 1, a3 = -1, b1 = 1, b2 = -1, b3 = 1} 12

13 7, {a1 = 1, a2 = -1, a3 = 1, b1 = -1, b2 = 1, b3 = 1} 8, {a1 = 1, a2 = -1, a3 = 1, b1 = 1, b2 = 1,, b3 = -1} 9, {a1 = 1, a2 = -1, a3 = 1, b1 = 1, b2 = -1, b3 = 1} A.2 l = 5 H 5 = 2+a1 a5 +a2 a1 +a3 a2 +a4 a3 +a5 a4 +b1 b5 +b2 b1 +b3 b2 +b4 b3 +b5 b4 2+a1 a4 +a2 a5 +a3 a1 +a4 a2 +a5 a3 +b1 b4 +b2 b5 +b3 b1 +b4 b2 +b5 b3 a1 + a2 + a3 + a4 + a5 1, b1 + b5 + b4 + b3 + b2 1, a 2 1 1,... b

14 1, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 2, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 3, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 4, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 5, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 6, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 7, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 8, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 9, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 10, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 11, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 12, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 13, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 14, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 15, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 16, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 17, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 18, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 19, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 20, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 21, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 22, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 23, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 24, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 25, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 14

15 26, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 27, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 28, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 29, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 30, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 31, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 32, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 33, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 34, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 35, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 36, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 37, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 38, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 39, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 40, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 41, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 42, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 43, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 44, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 45, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 46, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 47, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 48, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 49, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] 50, [a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, b5 = 1] A.3 l = 7 H 7 = 2 + a1 a7 + a2 a1 + a3 a2 + a4 a3 + a5 a4 + a6 a5 + a7 a6 + b1 b7 + b2 b1 + b3 b2 + b4 b3 + b5 b4 + b6 b5 + b7 b6 15

16 2 + a1 a5 + a2 a6 + a3 a7 + a4 a1 + a5 a2 + a6 a3 + a7 a4 + b1 b5 + b2 b6 + b3 b7 + b4 b1 + b5 b2 + b6 b3 + b7 b4 2 + a1 a6 + a2 a7 + a3 a1 + a4 a2 + a5 a3 + a6 a4 + a7 a5 + b1 b6 + b2 b7 + b3 b1 + b4 b2 + b5 b3 + b6 b4 + b7 b5 a1 + a2 + a3 + a4 + a5 + a6 + a7 1, b1 + b2 + b3 + b4 + b5 + b6 + b7 1, a 2 1 1,... b2 7 1 There are 196 solutions, see References [1] L. D. Baumert, Cyclic Difference Sets, Lecture Notes in Mathematics, Vol. 182, Springer-Verlag, Berlin-Heidelberg-New York, [2] D. Cox, J. Little and D. O Shea,Ideals, Varieties, and Algorithms : an Introduction to Computational Algebraic Geometry and Commutative Algebra UTM, Springer-Verlag, New York, [3] R. J. Fletcher, M. Gysin and J. Seberry, Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices, Australas. J. Combin., 23 (2001), [4] S. Georgiou and C. Koukouvinos, On equivalence of Hadamard matrices and projection properties, Ars Combinatorica, (to appear) [5] S. Georgiou, C. Koukouvinos and J. Seberry, Hadamard matrices, orthogonal designs and construction algorithms, Chapter 7, in Designs 2002: Further Computational and Constructive Design Theory, ed. W.D. Wallis, Kluwer Academic Publishers, Norwell, Massachusetts, 2003, [6] A.V. Geramita and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel, [7] M. Hall Jr, A survey of difference sets, Proc. Amer. Math. Soc., 7 (1956), [8] M. Hall Jr, Combinatorial Theory, 2nd Ed., Wiley, 1998 [9] J.-H. Kim and H.-Y. Song, Existence of cyclic Hadamard difference dets and its relation to binary sequences with ideal autocorrelation, J. of Comm. and Networks, 1 (1999), [10] I.S. Kotsireas, C. Koukouvinos and J. Seberry, Hadamard ideals and Hadamard matrices with circulant core. In preparation, (2003). [11] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, [12] R.E.A.C. Paley, On orthogonal matrices, J. Math. Phys., 12 (1933), [13] R.L. Plackett and J.P. Burman, The design of optimum multifactorial experiments, Biometrika, 33 (1946),

17 [14] J. S. Rose, A Course on Group Theory, Cambridge University Press, Cambridge; New York, [15] J. Seberry and M. Yamada, Hadamard matrices, sequences, and block designs, in Contemporary Design Theory: A Collection of Surveys, eds. J. H. Dinitz and D. R. Stinson, John Wiley, New York, pp , [16] J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938) [17] N. J. A. Sloane, AT&T on-line encyclopedia of integer sequences, ñjas/sequences/ [18] R.G. Stanton and D.A. Sprott, A family of difference sets, Can. J. Math., 10 (1958), [19] B. Sturmfels, Algorithms in Invariant Theory, Texts and monographs in symbolic computation, Springer-Verlag, Wien; New York, [20] B. Sturmfels, Solving Systems of Polynomial Equations, American Mathematical Society, CBMS Regional Conference Series in Mathematics, 97, [21] W.D. Wallis, A.P. Street and Jennifer Seberry Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard matrices, Lecture Notes in Mathematics, Springer-Verlag, Vol. 292, [22] A.L. Whiteman, A family of difference sets, Illinois J. Math., 6 (1962), [23] R.J. Fletcher, M. Gysin, and J. Seberry, Application of the discrete Fourier transform to the search for generalized Legendre pairs and Hadamard matrices, Australas. J. Combin., 23 (2001), [24] S. Georgiou and C. Koukouvinos, On generalized Legendre pairs and multipliers of the corresponding supplementary difference sets, Utilitas Math., 61 (2002), [25] A.V. Geramita, and J. Seberry, Orthogonal designs: Quadratic forms and Hadamard matrices, Marcel Dekker, New York-Basel, [26] M. Gysin and J. Seberry, An experimental search and new combinatorial designs via a generalization of cyclotomy, J. Combin. Math. Combin. Comput., 27 (1998), [27] M.R. Schroeder, Number Theory in Science and Communication, Springer Verlag, New York, [28] R.G. Stanton and D.A. Sprott, A family of difference sets, Canad. J. Math., 10 (1958), [29] A.L. Whiteman, A family of difference sets, Illinois Journal of Mathematics, 6 (1962),