Williamson Matrices Research Articles

On Goethals and Seidel Array

Objectives: In this article, we aim to find a series of Hadamard matrices by suitable selection of the special class of matrices given in the Goethals and Seidel array and study the pattern obtained. Methods: In the presented work, the search technique of Hadamard matrices has been done by selecting special class of (0,1) negacyclic matrices instead of the back diagonal identity matrix given in Geothals and Seidel arrays and the possible existence of negacyclic matrices for the corresponding four matrices have been explored in each case. Findings: Corresponding to the special class of (0,1) negacyclic matrices, no sets of four negacyclic matrices have been obtained in the Goethal Seidel array, for even orders. For odd orders, except in the case when all four matrices are equal and the case when there is a pair of equal matrices, many outputs have been obtained for the remaining cases, the search domain being upto 11,9 and 7 for the case of two different, three different and four different matrices respectively, in the Goethal Seidel array. Novelty: The selection of special class of negacyclic matrices instead of the back diagonal identity matrix and finding the corresponding four negacyclic matrices in Goethals and Seidel arrays for constructing Hadamard matrices provides a new approach to the construction of Hadamard matrices. Keywords: Hadamard matrix, Circulant matrix, Williamson matrices, Orthogonal array, Goethals and Seidel array

On Some Examples of Williamson Matrices

On Some Examples of Williamson Matrices

Applying computer algebra systems with SAT solvers to the Williamson conjecture

We employ tools from the fields of symbolic computation and satisfiability checking—namely, computer algebra systems and SAT solvers—to study the Williamson conjecture from combinatorial design theory and increase the bounds to which Williamson matrices have been enumerated. In particular, we completely enumerate all Williamson matrices of even order up to and including 70 which gives us deeper insight into the behaviour and distribution of Williamson matrices. We find that, in contrast to the case when the order is odd, Williamson matrices of even order are quite plentiful and exist in every even order up to and including 70. As a consequence of this and a new construction for 8-Williamson matrices we construct 8-Williamson matrices in all odd orders up to and including 35. We additionally enumerate all Williamson matrices whose orders are divisible by 3 and less than 70, finding one previously unknown set of Williamson matrices of order 63.

The SAT+CAS paradigm and the Williamson conjecture

We employ tools from the fields of symbolic computation and satisfiability checking---namely, computer algebra systems and SAT solvers---to study the Williamson conjecture from combinatorial design theory and increase the bounds to which Williamson matrices have been enumerated. In particular, we completely enumerate all Williamson matrices of orders divisible by 2 or 3 up to and including 70. We find one previously unknown set of Williamson matrices of order 63 and construct Williamson matrices in every even order up to and including 70. This extended abstract outlines a preprint currently under submission [4].

A SAT+CAS Method for Enumerating Williamson Matrices of Even Order

We present for the first time an exhaustive enumeration of Williamson matrices of even order n < 65. The search method relies on the novel SAT+CAS paradigm of coupling SAT solvers with computer algebra systems so as to take advantage of the advances made in both the field of satisfiability checking and the field of symbolic computation. Additionally, we use a programmatic SAT solver which allows conflict clauses to be learned programmatically, through a piece of code specifically tailored to the domain area. Prior to our work, Williamson matrices had only been enumerated for odd orders n < 60, so our work increases the bounds that Williamson matrices have been enumerated up to and provides the first enumeration of Williamson matrices of even order. Our results show that Williamson matrices of even order tend to be much more abundant than those of odd orders. In particular, Williamson matrices exist for every even order n < 65 but do not exist in orders 35, 47, 53, and 59.

English

English

Combining SAT Solvers with Computer Algebra Systems to Verify Combinatorial Conjectures

We present a method and an associated system, called MathCheck, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflict-driven clause-learning SAT solver. SAT+CAS systems, a la MathCheck, can be used as an assistant by mathematicians to either find counterexamples or finitely verify open universal conjectures on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS. Such a SAT+CAS system combines the efficient search routines of modern SAT solvers, with the expressive power of CAS, thus complementing both. The key insight behind the power of the SAT+CAS combination is that the CAS system can help cut down the search-space of the SAT solver, by providing learned clauses that encode theory-specific lemmas, as it searches for a counterexample to the input conjecture (just like the T in DPLL (T)). In addition, the combination enables a more efficient encoding of problems than a pure Boolean representation. In this paper, we leverage the capabilities of several different CAS, namely the SAGE, Maple, and Magma systems. As case studies, we study three long-standing open mathematical conjectures, two from graph theory regarding properties of hypercubes, and one from combinatorics about Hadamard matrices. The first conjecture states that any matching of any d-dimensional hypercube can be extended to a Hamiltonian cycle; the second states that given an edge-antipodal coloring of a hypercube there always exists a monochromatic path between two antipodal vertices; the third states that Hadamard matrices exist for all orders divisible by 4. Previous results on the graph theory conjectures have shown the conjectures true up to certain low-dimensional hypercubes, and attempts to extend them have failed until now. Using our SAT+CAS system, MathCheck, we extend these two conjectures to higher-dimensional hypercubes. Regarding Hadamard matrices, we demonstrate the advantages of SAT+CAS by constructing Williamson matrices up to order 42 (equivalently, Hadamard up to order $$4\times 42=168$$4×42=168), improving the bounds up to which Williamson matrices of even order have been constructed. Prior state-of-the-art construction was only feasibly performed for odd numbers, where possible.

Turyn type Williamson matrices up to order 99

In 2008 Holzmann et al. introduced a new algorithm to search for symmetric circulant Williamson matrices. We adapted the algorithm for Turyn type Williamson matrices. For each order up to 99 where Turyn type Williamson matrices exist our computer search found, up to equivalence, exactly one such matrix.

Williamson matrices up to order 59

A recent result of Schmidt has brought Williamson matrices back into the spotlight. In this article, a new algorithm is introduced to search for hard to find Williamson matrices. We find all nonequivalent Williamson matrices of odd order n up to n = 59. It turns out that there are none for n = 35, 47, 53, 59 and it seems that the Turyn class may be the only infinite class of these matrices.

A New Method for Constructing Williamson Matrices

For every prime power q ? 1 (mod 4) we prove the existence of (q; x, y)-partitions of GF(q) with q=x2+4y2 for some x, y, which are very useful for constructing SDS, DS and Hadamard matrices. We discuss the transformations of (q; x,y)-partitions and, by using the partitions, construct generalized cyclotomic classes which have properties similar to those of classical cyclotomic classes. Thus we provide a new construction for Williamson matrices of order q2.

Williamson Matrices and a Conjecture of Ito‘s

We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t,2,4t,2t)-difference sets in the dicyclic groups Q_{8t}=\la a,b|a^{4t}=b^4=1, a^{2t}=b^2, b^{-1}ab=a^{-1}\ra for all t of the form t=2^a\cdot 10^b \cdot 26^c \cdot m with a,b,c\ge 0, m\equiv 1 (\mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over \Z_m. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t,2,4t,2t)-difference sets in Q_{8t} for every positive integer t. We also give simpler alternative constructions for relative (4t,2,4t,2t) -difference sets in Q_{8t} for all t such that 2t-1 or 4t-1 is a prime power. Relative difference sets in Q_{8t} with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito‘s conjecture for all t\le 46.

Williamson matrices of order 4 n for n=33,35,39

Nonequivalent Williamson matrices (always with symmetric circulant blocks) of indicated orders are enumerated. Since we have found a new Williamson matrix of order 4·33, the result of Koukouvinos and Kounias (1988) is incomplete. The claim made in the title of their paper (1990) is erroneous as we have found a Williamson matrix of order 4·39

Some infinite classes of special Williamson matrices and difference sets

In this paper we show that for any integer n ⩾ 1, primes p 1, …, p n , p i ≡ 3 (mod 4), p i > 3, i = 1, …, n, and all nonnegative integers r, r 1, …, r n there exist Hadamard matrices of special Williamson kind and difference sets of order 4 × 3 2 r × ( p 1 r 1 ··· p n r n ) 4.

Williamson matrices of orders 4·29 and 4·31

It is established that there is only one, up to equivalence, Williamson matrix of order 4·29 and there are only two non-equivalent such matrices of order 4·31.

An infinite class of supplementary difference sets and Williamson matrices

In this paper we prove that there exist 4-{ v; k, k, k, k; λ} supplementary difference sets (SDSs) with v = q 2, q ≡ 1 (mod 4) a prime power, k = q(q − 1) 2 , λ = 4 k − v, and Williamson matrices of order 4 tv for t ϵ S = {2 k + 1:0 ⩽ k ⩽ 16} ∪ {37, 59, 61, 67 } ∪ {2 i · 10 j · 26 k +1: i, j, k ⩾0}.

On composition of four-symbol 𝛿-codes and Hadamard matrices

It is shown that key instruments for composition of four-symbol δ \delta -codes are the Lagrange identity for polynomials, a certain type of quasisymmetric sequences (i.e., a set of normal or near normal sequences) and base sequences. The following is proved: If a set of base sequences for length t t and a set of normal (or near normal) sequences for length n n exist then four-symbol δ \delta -codes of length ( 2 n + 1 ) t ( or n t ) \left ( {2n + 1} \right )t\left ( {{\text {or }}nt} \right ) can be composed by application of the Lagrange identity. Consequently a new infinite family of Hadamard matrices of order 4 u w 4uw can be constructed, where w w is the order of Williamson matrices and u = ( 2 n + 1 ) t ( or n t ) u = \left ( {2n + 1} \right )t\left ( {{\text {or }}nt} \right ) . Other related topics are also discussed.

A Hadamard matrix of order 268

The existence of Hadamard matrices of order 268 is established. More generally, suppose that there exist Williamson matrices of orderr. It is shown that this implies the existence of a Hadamard matrix of order 268r. The existence of Baumert-Hall arrays of order 335, and 603 is established as well.

A special class of Williamson matrices and difference sets

A construction is given of a very special class of Hadamard matrices. This yields Hadamard matrices of the Williamson kind and difference sets of order 4 · 3 2 m .

Lagrange identity for polynomials and 𝛿-codes of lengths 7𝑡 and 13𝑡

It is known that application of the Lagrange identity for polynomials (see [1]) is the key to composing four-symbol δ \delta -codes of length ( 2 s + 1 ) t (2s + 1)t for s = 2 a 10 b 26 c s = {2^a}{10^b}{26^c} and odd t ⩽ 59 t \leqslant 59 or t = 2 d 10 e 26 f + 1 t = {2^d}{10^e}{26^f} + 1 , where a a , b b , c c , d d , e e and f f are nonnegative integers. Applications of the Lagrange identity also lead to constructions of four-symbol δ \delta -codes of length u u for u = 7 t u = 7t or 13 t 13t . Consequently, new families of Hadamard matrices of orders 4 u w 4uw and 20 u w 20uw can be constructed, where w w is the order of Williamson matrices. Related topics on zero correlation codes are also discussed.

Hadamard Matrices and δ-Codes of Length 3n

It is found that four-symbol 6-codes of length t = 3n can be composed for odd n 0. Consequently new families of Hadamard matrices of orders 4tw and 20tw can be constructed, where w is the order of Williamson matrices. Introduction. An Hadamard matrix Hn = (hij) of order n is an n X n matrix with entries 1 or -1 such that HnHT = nIn, where In is the n X n identity matrix and T indicates the transposed matrix. In Hn, row vectors vi = (hil, hi2, ... , hin) are orthogonal, i.e. vi vj = En 1 hikhjk = 0, i $4 j. Hn exists only if n = 1, 2, or 4k. A sequence of vectors V = (vk)n (v1, v2,..., vn)4 where vk is one of m orthonormal vectors il, i2, ...i, or their negatives, is said to be an m-symbol 6-code of length n, if (I) v(j) = 0 for j $4 0, where v(j) Vk . Vk+j is the nonperiodic auto-correlation function of V. Another characterization of V = (vk)n being an m-symbol 6-code is that its associated polynomial V(z) _ Ekv=1 VkZZ,m 1 Pj(z)ij, where Pj(z) = EZn pjkzk-1, 1 0) will be called Turyn base sequences for length 2m + p (abbreviated as TBS(2m + p)) if they satisfy (3) u(j) + w(j) + x(j) + y(j) = 0 for j 7$ 0. Condition (3) is also equivalent to 1U12 + IW12 + IX12 + lYl2 = 2(2m + p) for any z E K. If TBS(2m + p): U, W; X and Y exist, then TS(2m + p) can be formed (cf. [1]) as follows: I (U' + W, 0), I (U W, 0), I (O', X + Y), and I (O', X Y), where o = 0m (the sequence of zeros of length m) and O' = 0m+p. THEOREM. Let U = (Uk)m+p, W = (Wk)m+p; X = (Xk)m and Y (Yk)m be TBS(n) for n = 2m + p. Then the following are TS(3n): 2 Q = (U+W,X+Y;O/,O;(U W)*,O), 2 R -(U W, X =Y; O' O; (U + W) O), (4) 2 1 S = -(0', 0; U + W, -(X + Y); 0', (X -Y)*) 2 1 T = (?' 0; U WI (X Y); O', (X + Y)*) 2 or Q = ((U W)*,O;U+W,X+Y;O',O), 2 R = -(-(U+ W)*,;U -W,X -Y;O',O), (5) 2 1 S =2(0', (X Y)*;O',O ; U+W w (X+ Y)) 2 1 T = (?'' (X + Y)*; O', O; U W (X Y)), 2 where A* (aN, aN-1, .. ., a,) is the reverse of A = (a,, a2, .. ., aN). LEMMA. Let a, b, c and d be polynomials with real coefficients in z E K. And lete =a+b+c, f =a -b+d,g =a -c -d, andh =b -c+d. Then IeI2 + If 12 + 1g12 + Ihi2 = 3(aI2 + IbI2 + Ic12 + IdI2) for any z E K. The Lemma can be proved easily by straightforward computations and by observing that IpI2 = pp', where p' = p(z-1) for any z E K. PROOF OF THEOREM. Let a = U, b = -zn-mX, C = z2n-my* , and d = -z2nW* in the Lemma. Then as sequences, e = (U, -X; O', -Y*), f = (U,X;O',0; W*), g = (U,0;O',Y*;W*) and h = (0, X;O',Y*; W*). Consequently g* = (W, Y; O', 0; U*) and h* = (-W, Y; O', -X*; 0'). In case (4), we have Q = (f +g*)/2, R = (f g*)/2, S = zn(e-h*)/2, and T = zn(e+h*)/2. By noting that IzI = 1 and Ip*12 p12 since Ip*(z)I = Ip(z-')I, we obtain IQI2 + 2This neat form of case (4), which contains less *'s than my original one, was suggested by R. J. Turyn. This content downloaded from 207.46.13.57 on Mon, 08 Aug 2016 04:56:54 UTC All use subject to http://about.jstor.org/terms