TWO-CIRCULANT HADAMARD MATRICES, WEIGHING MATRICES, AND RYSER'S CONJECTURE

(1963). The Search for Hadamard Matrices. The American Mathematical Monthly: Vol. 70, No. 1, pp. 12-17.

An Hadamard matrix H is a square matrix of order n all of whose entries are ± 1 such thatThere are matrices of order 1 and 2and for all other Hadamard matrices the order n is a multiple of 4, n = 4m. It is a reasonable conjecture that Hadamard matrices exist for every order which is a multiple of 4 and the lowest order in doubt is 268. With every Hadamard matrix H4m a symmetric design D exists with

The multilevel (n-ary) Hadamard matrices, which are related to orthogonal matrices or orthogonal designs, are investigated in this paper. It is assumed that all matrix elements are integers, which make the binary Hadamard matrices as special cases of multilevel (n-ary) Hadamard matrices. It is shown that the multilevel Hadamard matrices of odd order do exist if only two different matrix elements are contained; but, except an unknown case, multilevel (n-ary) Hadamard matrices of odd order do not exist when all matrix elements are successive integers.

The A DFT (Discrete Fourier Transform) has seen studied and applied to signal processing and communication theory. The relation between the Fourier matrix and the Hadamard transform was developed in [Ahmed & Rao, 1975; Whelchel & Guinn, 1968] for signal representation and classification and the Fast Fourier-Hadamand Transform(FFHT) was proposed. This idea was further investigated in [Lee & Lee, 1998] as an extension of the conventional Hadamard matrix. Lee et al [Lee & Lee, 1998] has proposed the Reverse Jacket Transform(RJT) based on the decomposition of the Hadamard matrix into the Hadamard matrix(unitary matrix) itself and a sparse matrix. Interestingly, the Reverse Jacket(RJ) matrix has a strong geometric structure that reveals a circulant expansion and contraction properties from a basic 2x2 sparse matrix. The discrete Fourier transform (DFT) is an orthogonal matrix with highly practical value for representing signals and images [Ahmed & Rao, 1975; Lee, 1992; Lee, 2000]. Recently, the Jacket matrices which generalize the weighted Hadamard matrix were introduced in [Lee, 2000], [Lee & Kim, 1984, Lee, 1989, Lee & Yi, 2001; Fan & Yang, 1998]. The Jacket matrix1 is an abbreviated name of a reverse Jacket geometric structure. It includes the conventional Hadamard matrix [Lee, 1992; Lee, 2000; Lee et al., 2001; Hou et. al., 2003], but has the weights,ω , that are j or 2 , where k is an integer, and 1 j = − , located in the central part of Hadamard matrix. The weighted elements' positions of the forward matrix can be replaced by the non-weighted elements of its inverse matrix and the signs of them do not change between the forward and inverse matrices, and they are only as element inverse and transpose. This reveals an interesting complementary matrix relation. Definition 1: If a matrix m J ⎡ ⎤ ⎣ ⎦ of size m m × has nonzero elements

This paper discusses the factorization of a bent function type complex Hadamard matrix of order N=p/sup n/ with a prime p. The bent function type complex Hadamard matrix is the generalization of the bent function type binary Hadamard matrix. It is shown that the bent function type complex Hadamard matrix is closely related to the complex Hadamard matrix generalized the Sylvester type binary Hadamard matrix, and can be expressed by using 2n-2 matrices of order N with pN non-zero elements at most. The application of the bent function type complex Hadamard matrix to synchronous spread spectrum communication is also discussed.

Purpose.To investigate more fully, than what was done in the past, certain families of symmetric Hadamard matrices of small orders by using the so called propus construction.Methods.Orbit method for the search of three cyclic blocks to construct Hadamard matrices of propus type. This method speeds up the classical search of required sequences by distributing them into different bins using a hash-function.Results. Our main result is that we have constructed, for the first time, symmetric Hadamard matrices of order 268, 412, 436 and 604. The necessary difference families are constructed by restricting the search to the families which admit a nontrivial multiplier. A wide collection of new symmetric Hadamard matrices was obtained and tabulated, according to the feasible sets of parameters.Practical relevance.Hadamard matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in the mathematical network “Internet” together with executable on-line algorithms.

Quasi-orthogonal Hadamard matrices and Mersenne matrices with two element values are considered, which are used in digital data processing, orthogonal image transformations, and also as a basis for error-correcting codes. Attention is paid to the structures of cyclic matrices with symmetries and antisymmetries. The relationship between symmetry and antisymmetry of structures of cyclic matrices on orders equal to prime numbers, product of close prime numbers, composite numbers, powers of a prime number is shown. Separately, the orders equal to the power of the prime number 2, both the orders of the Hadamard matrices and the basis of the composite orders of the Mersenne matrices of block structures with two element values, are singled out separately.

Calculating symmetrical Hadamard matrices of Balonin-Seberry construction for coding and masking

Maximal inequalities for the signed vector summands are established. Probabilistic estimations for the sets of appropriate signs are given. By use of the “transference technique” appropriate maximal inequalities are derived for the permutations. One application for orthogonal and Hadamard matrices is suggested.

Symmetric balanced incomplete block designs (SBIBD) and Hadamard matrices

It was established in (5) that the existence of a Hadamard matrix of order 4t is equivalent to the existence of a symmetrical balanced incomplete block design with parameters v = 4t — 1, k = 2t — 1, and λ = t — 1. A block design is completely characterized by its so-called incidence matrix. The existence of a block design with parameters v, k, and λ such that the corresponding incidence matrix is cyclic was shown in (3) to be equivalent to the existence of a cyclic difference set with parameters v, k, and λ. For certain values of the parameters, Hadamard matrices, block designs, and difference sets do coexist.

Using Hadamard matrices and mutually orthogonal Latin squares, we construct two new quasi-symmetric designs, with parameters 2 ? (66,30,29) and 2 ? (78,36,30). These are the first examples of quasi-symmetric designs with these parameters. The parameters belong to the families 2 ? (2u 2 ? u,u 2 ? u,u 2 ? u ? 1) and 2 ? (2u 2 + u,u 2,u 2 ? u), which are related to Hadamard parameters. The designs correspond to new codes meeting the Grey---Rankin bound.

Given two Hadamard matrices of the same order, it can be quite difficult to decide whether or not they are equivalent. There are some criteria to determine Hadamard inequivalence. Among them, one of the most commonly used is the 4-profile criterion. In this paper, a reformulation of this criterion in the cocyclic framework is given. The improvements obtained in the computation of the 4-profile of a cocyclic Hadamard matrix are indicated.

Hadamard matrix is a square matrix where any components are -1 or +1, and where any pairs of rows are mutually orthogonal. On the other hand, Hadamard-type matrix on finite fields is a similar one, but has multi-valued components on finite fields. To be more specific, we consider n × n matrices H that have their elements on the given finite fields GF(p), and satisfy HHT = nI under modulo p, where I is an identity matrix. Any additions and multiplications should be executed under modulo p. In the authors' previous studies, some properties of Hadamard-type matrices on finite fields have been proven. For example, it has been shown that the order of a Hadamard-type matrix of odd order on GF(p) is limited to a quadratic residue of a given prime p. On the other hand, it is not clear how many and how various Hadamard-type matrices on GF(p) exist in general. In this paper, we count all possible Hadamard-type matrices on finite fields when the order n of a matrix is small. We also categorize Hadamard-type matrices into six different types when n = 3 and p = 11. In addition, we prove that for any prime p, a Hadamard-type matrix over GF(p) in each of the six types always exists if and only if the order ‘3’ is a quadratic residue of a given p.

A digital image is subjected to the most manipulation. This is driven by the easy manipulating process through image editing software which is growing rapidly. These problems can be solved through the watermarking model as an active authentication system for the image. One of the most popular methods is Singular Value Decomposition (SVD) which has good imperceptibility and detection capabilities. Nevertheless, SVD has high complexity and can only utilize one singular matrix S, and ignore two orthogonal matrices. This paper proposes the use of the Walsh matrix with dyadic ordering to generate a new S matrix without the orthogonal matrices. The experimental results showed that the proposed method was able to reduce computational time by 22% and 13% compared to the SVD-based method and similar methods based on the Hadamard matrix respectively. This research can be used as a reference to speed up the computing time of the watermarking methods without compromising the level of imperceptibility and authentication.