On 0-1 matrices and small excluded submatrices

We say that a 0-1 matrix A avoids another 0-1 matrix (pattern) P if no matrix P^′ obtained from P by increasing some of the entries is a submatrix of A. Following the lead of (SIAM J. Discrete Math. 4 (1991) 17-27; J. Combin. Theory Ser. A 55 (1990) 316-320; Discrete Math. 103 (1992) 233-251) and other papers we investigate n by n 0-1 matrices avoiding a pattern P and the maximal number ex(n,P) of 1 entries they can have. Finishing the work of [8] we find the order of magnitude of ex(n,P) for all patterns P with four 1 entries. We also investigate certain collections of excluded patterns. These sets often yield interesting extremal functions different from the functions obtained from any one of the patterns considered.     

An infinite family of Hadamard matrices of Williamson type

In this paper a new proof is given of the following theorem of Turyn: Let q = 2n ? 1 be a prime power ≡1 (mod 4); then there exists an Hadamard matrix of order 4n that is of the Williamson type.     

Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10

All Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2-(63,31,15) designs having line spreads by Rahilly’s construction [A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291-303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(n,p^m) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71-87].     

A new construction for Williamson-type matrices

It is shown that ifq is a prime power then there are Williamson-type matrices of order

1. 1/2q ²(q + 1) whenq ≡ 1 (mod 4).
2. 1/4q ²(q + 1) whenq ≡ 3 (mod 4) and there are Williamson-type matrices of order 1/4(q + 1).

This gives Williamson-type matrices for the new orders 363, 1183, 1805, 2601, 3174, 5103. The construction can be combined with known results on orthogonal designs to give an Hadamard matrix of the new order 33396 = 4 ? 8349.     

Circulant partial Hadamard matrices

A question arising in stream cypher cryptanalysis is reframed and generalized in the setting of Hadamard matrices as follows: For given n, what is the maximum value of k   for which there exists a k×n$k\times n$(±1)$(\pm 1)$-matrix A   such that AA^T=n_Ik$A{A}^{\mathsf{T}}=n{I}_k$, with each row after the first obtained by a cyclic shift of its predecessor by one position? For obvious reasons we call such matrices circulant partial Hadamard matrices. Further, what is the maximum value of k subject to the condition that the row sums are equal to r?     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

New proof of a theorem of Szekeres

A Hadamard matrix H of order 16t² is constructed for all t for which there is a Hadamard matrix of order 4t, in such a way that each row of H contains exactly 8t² + 2t ones. As a consequence a new method of constructing the symmetric block designs with parameters (16t², 8t² + 2t, 4t² + 2t) for all t for which there is a Hadamard matrix of order 4t is given.     

Hadamard matrices of order 28m, 36m and 44m

We show that if four suitable matrices of order m exist then there are Hadamard matrices of order 28m, 36m, and 44m. In particular we show that Hadamard matrices of orders 14(q + 1), 18(q + 1), and 22(q + 1) exist when q is a prime power and q ≡ 1 (mod 4).Also we show that if n is the order of a conference matrix there is an Hadamard matrix of order 4mn.As a consequence there are Hadamard matrices of the following orders less than 4000: 476, 532, 836, 1036, 1012, 1100, 1148, 1276, 1364, 1372, 1476, 1672, 1836, 2024, 2052, 2156, 2212, 2380, 2484, 2508, 2548, 2716, 3036, 3476, 3892.All these orders seem to be new.     

Lower Bounds For Multidimensional Zero Sums

Let f(n, d) denote the least integer such that any choice of f(n, d) elements in contains a subset of size n whose sum is zero. Harborth proved that (n-1)2 ^d +1 f(n,d) (n-1)n ^d +1. The upper bound was improved by Alon and Dubiner to c _d n. It is known that f(n-1) = 2n-1 and Reiher proved that f(n-2) = 4n-3. Only for n = 3 it was known that f(n,d) > (n-1)2 ^d +1, so that it seemed possible that for a fixed dimension, but a sufficiently large prime p, the lower bound might determine the true value of f(p,d). In this note we show that this is not the case. In fact, for all odd n 3 and d 3 we show that .     

More orthogonal double covers of complete graphs by Hamiltonian paths

An orthogonal double cover (ODC) of the complete graph K_n by a graph G is a collection G of n spanning subgraphs of K_n, all isomorphic to G, such that any two members of G share exactly one edge and every edge of K_n is contained in exactly two members of G. In the 1980s Hering posed the problem to decide the existence of an ODC for the case that G is an almost-Hamiltonian cycle, i.e. a cycle of length n-1. It is known that the existence of an ODC of K_n by a Hamiltonian path implies the existence of ODCs of K_4n and of K_16n, respectively, by almost-Hamiltonian cycles. Horton and Nonay introduced two-colorable ODCs and showed: If there are an ODC of K_n by a Hamiltonian path for some n?3 and a two-colorable ODC of K_q by a Hamiltonian path for some prime power q?5, then there is an ODC of K_qn by a Hamiltonian path. In [U. Leck, A class of 2-colorable orthogonal double covers of complete graphs by hamiltonian paths, Graphs Combin. 18 (2002) 155-167], two-colorable ODCs of K_n and K_2n, respectively, by Hamiltonian paths were constructed for all odd square numbers n?9. Here we continue this work and construct cyclic two-colorable ODCs of K_n and K_2n, respectively, by Hamiltonian paths for all n of the form n=4k²+1 or n=(k²+1)/2 for some integer k.     

Hadamard matrices of order 4(2p + 1)

It is shown in this paper that if p is a prime and q = 2p ? 1 is a prime power, then there exists an Hadamard matrix of order 4(2p + 1).     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.     

A direct method to solve block banded block Toeplitz systems with non-banded Toeplitz blocks

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman-Morrison-Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diagonal matrix with Toeplitz blocks, and the resulting Toeplitz systems are solved by means of a fast Toeplitz solver.The computational complexity in the case one uses fast Toeplitz solvers is equal to ξ(m,n,k)=O(mn³)+O(k³n³) flops, there are m block rows and m block columns in the matrix, n is the order of blocks, 2k+1 is the bandwidth. The validity of the approach is illustrated by numerical experiments.     

Integral circulant graphs of prime power order with maximal energy

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count n and a set D of divisors of n in such a way that they have vertex set Z_n and edge set {{a,b}:a,b∈Z_n,gcd(a-b,n)∈D}. Using tools from convex optimization, we analyze the maximal energy among all integral circulant graphs of prime power order p^s and varying divisor sets D. Our main result states that this maximal energy approximately lies between s(p-1)p^s-1 and twice this value. We construct suitable divisor sets for which the energy lies in this interval. We also characterize hyperenergetic integral circulant graphs of prime power order and exhibit an interesting topological property of their divisor sets.     

A fast Fourier-collocation method for second boundary integral equations

In this paper we develop a fast collocation method for second boundary integral equations by the trigonometric polynomials. We propose a convenient way to compress the dense matrix representation of a compact integral operator with a smooth kernel under the Fourier basis and the corresponding collocation functionals. The compression leads to a sparse matrix with only O(nlog²n) number of nonzero entries, where 2n+1 denotes the order of the matrix. Thus we develop a fast Fourier-collocation method. We prove that the fast Fourier-collocation method gives the optimal convergence order up to a logarithmic factor. Moreover, we design a fast scheme for solving the corresponding truncated linear system. We establish that this algorithm preserves the quasi-optimal convergence of the approximate solution with requiring a number of O(nlog³n) multiplications.     

Bounds on the number of Hadamard designs of even order

A new lower bound on the number of non‐isomorphic Hadamard symmetric designs of even order is proved. The new bound improves the bound on the number of Hadamard designs of order 2n given in [12] by a factor of 8n ? 1 for every odd n > 1, and for every even n such that 4n ? 1 > 7 is a prime. For orders 8, 10, and 12, the number of non‐isomorphic Hadamard designs is shown to be at least 22,478,260, 1.31 × 10¹⁵, and 10²⁷, respectively. For orders 2n = 14, 16, 18 and 20, a lower bound of (4n ? 1)! is proved. It is conjectured that (4n ? 1)! is a lower bound for all orders 2n ≥ 14. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 363‐378, 2001     

On a series of Hadamard matrices of order 2 t and the maximal excess of Hadamard matrices of order 22t

In this paper we give a new series of Hadamard matrices of order 2 ^t . When the order is 16, Hadamard matrices obtained here belong to class II, class V or to class IV of Hall's classification [3]. By combining our matrices with the matrices belonging to class I, class II or class III obtained before, we can say that we have direct construction, namely without resorting to block designs, for all classes of Hadamard matrices of order 16.Furthermore we show that the maximal excess of Hadamard matrices of order 2^2t is 2^3t , which was proved by J. Hammer, R. Levingston and J. Seberry [4]. We believe that our matrices are inequivalent to the matrices used by the above authors. More generally, if there is an Hadamard matrix of order 4n ² with the maximal excess 8n ³, then there exist more than one inequivalent Hadamard matrices of order 2^2t n ² with the maximal excess 2^3t n ³ for anyt 2.     

Orthogonal designs II

Orthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show that orthogonal designs of type (1,k) and ordern exist for everyk < n whenn = 2 ^t+2?3 andn = 2 ^t+2?5 (wheret is a positive integer). We also find orthogonal designs that exist in every order 2n and others that exist in every order 4n. Coupled with some results of earlier work, this means that theweighing matrix conjecture ‘For every ordern ≡ 0 (mod 4) there is, for eachk ?n, a square {0, 1, ? 1} matrixW = W(n, k) satisfyingWW ^t =kIn’ is resolved in the affirmative for all ordersn = 2^t+1?3,n = 2^t+1?5 (t a positive integer). The fact that the matrices we find are skew-symmetric for allk < n whenn ≡ 0 (mod 8) and because of other considerations we pose three other conjectures about weighing matrices having additional structure and resolve these conjectures affirmatively in a few cases. In an appendix we give a table of the known results for orders ? 64.     

Skew Hadamard designs and their codes

Skew Hadamard designs (4n – 1, 2n – 1, n – 1) are associated to order 4n skew Hadamard matrices in the natural way. We study the codes spanned by their incidence matrices A and by I + A and show that they are self-dual after extension (resp. extension and augmentation) over fields of characteristic dividing n. Quadratic Residues codes are obtained in the case of the Paley matrix. Results on the p-rank of skew Hadamard designs are rederived in that way. Codes from skew Hadamard designs are classified. An optimal self-dual code over GF(5) is rediscovered in length 20. Six new inequivalent [56, 28, 16] self-dual codes over GF(7) are obtained from skew Hadamard matrices of order 56, improving the only known quadratic double circulant code of length 56 over GF(7).