Skew Hadamard Matrices Research Articles

On the classification of skew Hadamard matrices of order $$\varvec{36}$$ and related structures

On the classification of skew Hadamard matrices of order $$\varvec{36}$$ and related structures

Conference matrices from Legendre C-pairs

Introduction: There are just a few known methods for the construction of symmetric C-matrices, due to the lack of a universal structure for them. This obstruction is fundamental, in addition, the structure of C-matrices with a double border is incompletely described in literature, which makes its study especially relevant. The purpose: To describe the two-border two-circulant construction in detail with the proposal of the concept of C-pairs Legendre. Results: The paper deals with C-matrices of order n=2v+2 with two borders and extends the so called generalized Legendre pairs, v odd, to a wider class of Legendre C-pairs with even and odd v, defined on a finite abelian group G of order v. Such a pair consists of two functions a, b: G→Z, whose values are +1 or −1 except that a(e)=0, where e is the identity element of G and Z is the ring of integers. To characterize the Legendre C-pairs we use the subsets X={xÎG: a(x)=–1} and Y={xÎG: b(x)=–1} of G. We show that a(x−1)=(−1)v a(x) for all x. For odd v we show that X and Y form a difference family, which is not true for even v. These difference families are precisely the so called Szekeres difference sets, used originally for the construction of skew-Hadamard matrices. We introduce the subclass of the special Legendre C-pairs and prove that they exist whenever 2v+1 is a prime power. In the last two sections of the paper we list examples of special cyclic Legendre C-pairs for lengths v<70. Practical relevance: C-matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of conference matrices and a library of constructed matrices are used in the mathematical network “mathscinet.ru” together with executable on-line algorithms.

The SAT+CAS method for combinatorial search with applications to best matrices

In this paper, we provide an overview of the SAT+CAS method that combines satisfiability checkers (SAT solvers) and computer algebra systems (CAS) to resolve combinatorial conjectures, and present new results vis-\`a-vis best matrices. The SAT+CAS method is a variant of the Davis$\unicode{8211}$Putnam$\unicode{8211}$Logemann$\unicode{8211}$Loveland $\operatorname{DPLL}(T)$ architecture, where the $T$ solver is replaced by a CAS. We describe how the SAT+CAS method has been previously used to resolve many open problems from graph theory, combinatorial design theory, and number theory, showing that the method has broad applications across a variety of fields. Additionally, we apply the method to construct the largest best matrices yet known and present new skew Hadamard matrices constructed from best matrices. We show the best matrix conjecture (that best matrices exist in all orders of the form $r^2+r+1$) which was previously known to hold for $r\leq6$ also holds for $r=7$. We also confirmed the results of the exhaustive searches that have been previously completed for $r\leq6$.

Difference Families, Skew Hadamard Matrices, and Critical Groups of Doubly Regular Tournaments

In this paper we investigate the structure of the critical groups of doubly-regular tournaments (DRTs) associated with skew Hadamard difference families (SDFs) with one, two, or four blocks. Brown and Ried found that the existence of a skew Hadamard matrix of order $n+1$ is equivalent to the existence of a DRT on $n$ vertices. A well known construction of a skew Hadamard matrix order $n$ is by constructing skew Hadamard difference sets in abelian groups of order $n-1$. The Paley skew Hadamard matrix is an example of one such construction. Szekeres and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with two blocks. Wallis and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with four blocks. In this paper we consider the critical groups of DRTs associated with skew Hadamard matrices constructed from skew Hadamard difference families with one, two or four blocks. We compute the critical groups of DRTs associated with skew Hadamard difference families with two or four blocks. We also compute the critical group of the Paley tournament and show that this tournament is inequivalent to the other DRTs we considered. Consequently we prove that the associated skew Hadamard matrices are not equivalent.

Construction of special entangled basis based on generalized weighing matrices

A special entangled basis (SEBk) is a set with orthonormal entangled pure states in whose nonzero Schmidt coefficients are all equal to . When k is equal to the minimum of d and , we get a maximally entangled basis. In this paper, we present how to construct a special entangled basis via some special matrices which are known as weighing matrices. Specifically, using a skew Hadamard matrix of order k + 1, we derive a weighing matrix that is useful for constructing SEBk in whenever . These results are further progress of those studied by Guo et al (2015 J. Phys. A: Math. Theor. 48 245301). We also disprove two conjectures proposed by them.

Constructions of skew Hadamard difference families

In this paper, we generalize classical constructions of skew Hadamard difference families with two or four blocks in the additive groups of finite fields given by Szekeres (1969, 1971), Whiteman (1971) and Wallis–Whiteman (1972). In particular, we show that there exists a skew Hadamard difference family with 2u−1 blocks in the additive group of the finite field of order qe for any prime power q≡2u+1(mod2u+1) with u⩾2 and any positive integer e. In the aforementioned papers of Szekeres, Whiteman, and Wallis–Whiteman, the constructions of skew Hadamard difference families with 2u−1 (u=2 or 3) blocks in (Fqe,+) work only for restricted e;namely e≡1,2, or 3(mod4) when u=2, and e≡1(mod2) when u=3, respectively. Our more general construction, in particular, removes the restrictions on e. As a consequence, we obtain new infinite series of skew Hadamard difference families with two or four blocks, and hence skew Hadamard matrices.

Robust Hadamard Matrices, Unistochastic Rays in Birkhoff Polytope and Equi-Entangled Bases in Composite Spaces

We study a special class of (real or complex) robust Hadamard matrices, distinguished by the property that their projection onto a 2-dimensional subspace forms a Hadamard matrix. It is shown that such a matrix of order n exists, if there exists a skew Hadamard matrix or a symmetric conference matrix of this size. This is the case for any even nle 20, and for these dimensions we demonstrate that a bistochastic matrix B located at any ray of the Birkhoff polytope, (which joins the center of this body with any permutation matrix), is unistochastic. An explicit form of the corresponding unitary matrix U, such that B_{ij}=|U_{ij}|^2, is determined by a robust Hadamard matrix. These unitary matrices allow us to construct a family of orthogonal bases in the composed Hilbert space of order n times n. Each basis consists of vectors with the same degree of entanglement and the constructed family interpolates between the product basis and the maximally entangled basis. In the case n=4 we study geometry of the set {mathcal U}_4 of unistochastic matrices, conjecture that this set is star-shaped and estimate its relative volume in the Birkhoff polytope {mathcal B}_4.

On the classification of self-dual [20,10,9] codes over GF(7)

It is shown that the extended quadratic residue code of length 20 over GF(7) is a unique self-dual [20,10,9] code C such that the lattice obtained from C by Construction A is isomorphic to the 20-dimensional unimodular lattice D20+, up to equivalence. This is done by converting the classification of such self-dual codes to that of skew-Hadamard matrices of order 20.

On Skew E–W Matrices

AbstractAn E–W matrix M is a ( − 1, 1)‐matrix of order , where t is a positive integer, satisfying that the absolute value of its determinant attains Ehlich–Wojtas' bound. M is said to be of skew type (or simply skew) if is skew‐symmetric where I is the identity matrix. In this paper, we draw a parallel between skew E–W matrices and skew Hadamard matrices concerning a question about the maximal determinant. As a consequence, a problem posted on Cameron's website [7] has been partially solved. Finally, codes constructed from skew E–W matrices are presented. A necessary and sufficient condition for these codes to be self‐dual is given, and examples are provided for lengths up to 52.

A class of cyclic (v; k1, k2, k3; λ) difference families with v ≡ 3 (mod 4) a prime

Abstract We construct several new cyclic (v; k1, k2, k3; λ) difference families, with v ≡ 3 (mod 4) a prime and λ = k1 + k2 + k3 − (3v − 1)/4. Such families can be used in conjunction with the well-known Paley-Todd difference sets to construct skew-Hadamard matrices of order 4v. Our main result is that we have constructed for the first time the examples of skew Hadamard matrices of orders 4 · 239 = 956 and 4 · 331 = 1324.

Note on best possible bounds for determinants of matrices close to the identity matrix

We give upper and lower bounds on the determinant of a small perturbation of the identity matrix. The lower bounds are best possible, and in most cases they are stronger than well-known bounds due to Ostrowski and other authors. The upper bounds are best possible if a skew-Hadamard matrix of the same order exists.

Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach

Abstract An n by n skew-symmetric type (-1; 1)-matrix K =[ki;j ] has 1’s on the main diagonal and ±1’s elsewhere with ki;j =-kj;i . The largest possible determinant of such a matrix K is an interesting problem. The literature is extensive for n ≡ 0 mod 4 (skew-Hadamard matrices), but for n ≡ 2 mod 4 there are few results known for this question. In this paper we approach this problem constructing cocyclic matrices over the dihedral group of 2t elements, for t odd, which are equivalent to (-1; 1)-matrices of skew type. Some explicit calculations have been done up to t =11. To our knowledge, the upper bounds on the maximal determinant in orders 18 and 22 have been improved.

A characterization of skew Hadamard matrices and doubly regular tournaments

We give a new characterization of skew Hadamard matrices of order n in terms of spectral data for tournaments of order n-2.

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).

On skew-Hadamard matrices

Skew-Hadamard matrices are of special interest due to their use, among others, in constructing orthogonal designs. In this paper, we give a survey on the existence and equivalence of skew-Hadamard matrices. In addition, we present some new skew-Hadamard matrices of order 52 and improve the known lower bound on the number of the skew-Hadamard matrices of this order.

Self-dual codes over some prime fields constructed from skew-Hadamard matrices

Self-dual codes are an important class of linear codes, both theoretically and for practical reasons. In this paper, we give a new construction method for self-dual codes using skew-Hadamard matrices. By applying this method we construct several “good” self-dual codes over some prime fields.

New skew-Hadamard matrices of order [formula omitted] and new D-optimal designs of order [formula omitted

We give two new circlulant D-optimal designs of order 118, which are of Ehlich's type. We note this is the first time that a D-optimal has been constructed for order 118 = 2 · 59 . As a consequence new skew-Hadamard matrices of order 236 = 4 · 59 are also constructed for the first time.

Construction of new skew Hadamard matrices and their use in screening experiments

Screening experiments aim to identify the relevant variables within some process potentially depending on a large number of variables. Recently a new class of experimental designs called edge designs has been introduced. These designs allow a model-independent estimate of the set of relevant variables, thus providing more robustness than traditional designs. Some new skew Hadamard matrices of order 36 are constructed and used to produce new edge designs. An illustrative simulated example using an edge design in 70 runs is also presented.

Necessary and sufficient conditions for three and four variable orthogonal designs in order 36

We use a new algorithm to find new sets of sequences with entries from {0,± a,± b,± c,± d}, on the commuting variables a, b, c, d, with zero autocorrelation function. Then, we use these sequences to construct a series of new three and four variable orthogonal designs in order 36. We show that the necessary conditions plus ( s 1, s 2, s 3, s 4) not equal to 1 2 8 16 1 2 8 25 1 4 4 25 1 8 8 16 1 9 13 13 2 2 9 16 2 2 13 13 2 3 4 24 2 6 7 21 2 8 9 9 3 6 8 16 3 8 10 15 4 8 8 9 8 8 9 9 are sufficient for the existence of an OD(36; s 1, s 2, s 3, s 4) constructed using four circulant matrices in the Goethals–Seidel array. Of the 154 theoretically possible cases 133 are known. We also show that the necessary conditions plus (s 1,s 2,s 3)≠(2,8,25), (6,7,21), (8,9,17) or (9,13,13) are sufficient for the existence of an OD(36; s 1, s 2, s 3) constructed using four circulant matrices in the Goethals–Seidel array. Of the 433 theoretically possible cases 429 are known. Further, we show that the necessary conditions are sufficient for the existence of an OD(36; s 1, s 2,36− s 1− s 2) in each of the 54 theoretically possible cases. Further, of the 27 theoretically possible OD(36;s 1,s 2,s 3,36−s 1−s 2−s 3), 23 are known to exist, and four, (1,2,8,25), (1,9,13,13), (2,6,7,21) and (3,8,10,15), cannot be constructed using four circulant matrices. By suitably replacing the variables by ±1 these lead to more than 200 potentially inequivalent Hadamard matrices of order 36. By considering the 12 OD(36;1,s 1,35−s 1) and suitably replacing the variables by ±1 we obtain 48 potentially inequivalent skew-Hadamard matrices of order 36. A summary with all known results in order 36 is presented in the tables.

The Many Names of (7, 3, 1)

In the world of discrete mathematics, we encounter a bewildering variety of topics with no apparent connection between them. But appearances are deceptive. For example, combinatorics tells us about difference sets, block designs, and triple systems. Geometry leads us to finite projective planes and Latin squares. Graph theory introduces us to round–robin tournaments and map colorings, linear algebra gives us (0, 1)–matrices, and quadratic residues are among the many pearls of number theory. We meet the torus, that topological curiosity, while visiting the local doughnut shop or tubing down a river. Finally, in these fields we encounter such names as Euler, Fano, Fischer, Hadamard, Heawood, Kirkman, Singer and Steiner. This is a story about a single object that connects all of these. Commonly known as (7, 3, 1), it is all at once a difference set, a block design, a Steiner triple system, a finite projective plane, a complete set of orthogonal Latin squares, a doubly regular round-robin tournament, a skew-Hadamard matrix, and a graph consisting of seven mutually adjacent hexagons drawn on the torus. We are going to investigate these connections. Along the way, we’ll learn about all of these topics and just how they are tied together in one object—namely, (7, 3, 1). We’ll learn about what all of those people have to do with it. We’ll get to know this object quite well! So let’s find out about the many names of (7, 3, 1).