Regular Hadamard Matrices Research Articles

Self-Dual Hadamard Bent Sequences

A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application ( Sol\'e et al, 2021). We study the self dual class in length at most $196.$ We use three competing methods of generation: Exhaustion, Linear Algebra and Groebner bases. Regular Hadamard matrices and Bush-type Hadamard matrices provide many examples. We conjecture that if $v$ is an even perfect square, a self-dual bent sequence of length $v$ always exist. We introduce the strong automorphism group of Hadamard matrices, which acts on their associated self-dual bent sequences. We give an efficient algorithm to compute that group.

On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs

It is proved that a code L(q) which is monomially equivalent to the Pless symmetry code C(q) of length $$2q+2$$ contains the (0,1)-incidence matrix of a Hadamard 3- $$(2q+2,q+1,(q-1)/2)$$ design D(q) associated with a Paley–Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley–Hadamard matrix of type I. If $$q=5, 11, 17, 23$$ , then the full permutation automorphism group of L(q) coincides with the full automorphism group of D(q), and a similar result holds for the ternary extended quadratic residue codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by codewords of the Pless symmetry code C(17) are enumerated and classified up to equivalence. There are two equivalence classes of such matrices: the Paley–Hadamard matrix H of type I with a full automorphism group of order 19584, and a second regular Hadamard matrix $$H'$$ such that the symmetric 2-(36, 15, 6) design D associated with $$H'$$ has trivial full automorphism group, and the incidence matrix of D spans a ternary code equivalent to C(17).

Matrix vitrages and regular Hadamard matrices

Introduction: The Kronecker product of Hadamard matrices when a matrix of order n replaces each element in another matrix of order m, inheriting the sign of the replaced element, is a basis for obtaining orthogonal matrices of order nm. The matrix insertion operation when not only signs but also structural elements (ornamental patterns of matrix portraits) are inherited provides a more general result called a "vitrage". Vitrages based on typical quasi-orthogonal Mersenne (M), Seidel (S) or Euler (E) matrices, in addition to inheriting the sign and pattern, inherit the value of elements other than unity (in amplitude) in a different way, causing the need to revise and systematize the accumulated experience. Purpose: To describe new algorithms for generalized product of matrices, highlighting the constructions that produce regular high-order Hadamard matrices. Results: We have proposed an algorithm for obtaining matrix vitrages by inserting Mersenne matrices into Seidel matrices, which makes it possible to expand the additive chains of matrices of the form M-E-M-E-… and S-E-M-E-…, obtained by doubling the orders and adding an edge. The operation of forming a matrix vitrage allows you to obtain matrices of high orders, keeping the ornamental pattern as an important invariant of the structure. We have shown that the formation of a matrix vitrage inherits the logic of the Scarpi product, but is cannot be reduced to it, since a nonzero distance in order between the multiplicands M and S simplifies the final regular matrix ornamental pattern due to the absence of cyclic displacements. The alternation of M and S matrices allows you to extend the multiplicative chains up to the known gaps in the S matrices. This sheds a new light on the theory of a regular Hadamard matrix as a product of Mersenne and Seidel matrices. Practical relevance: Orthogonal sequences with floating levels and efficient algorithms for finding regular Hadamard matrices with certain useful properties are of direct practical importance for the problems of noise-proof coding, compression and masking of video data.

The Construction of Regular Hadamard Matrices by Cyclotomic Classes

For every prime power $$q \equiv 7 \; mod \; 16$$, there are (q; a, b, c, d)-partitions of GF(q), with odd integers a, b, c, and d, where $$a \equiv \pm 1 \; mod \; 8$$ such that $$q=a^2+2(b^2+c^2+d^2)$$ and $$d^2=b^2+2ac+2bd$$. Many results for the existence of $$4-\{q^2; \; \frac{q(q-1)}{2}; \; q(q-2) \}$$ SDS which are simple homogeneous polynomials of parameters a, b, c and d of degree at most 2 have been found. Hence, for each value of q, the construction of SDS becomes equivalent to building a $$(q; \; a, \; b, \; c, \; d)$$-partition. Once this is done, the verification of the construction only involves verifying simple conditions on a, b, c and d which can be done manually.

A Note on Siamese Twin Designs Intersecting in a BIBD and a PBD

Suppose there exists a Hadamard 2- $$(m,\frac{m-1}{2},\frac{m-3}{4})$$ design with skew incidence matrix, and a conference graph with v vertices, where $$v = 2m-1$$ . Under this assumption we prove that there exists a Siamese twin Menon design with parameters $$(4m^{2},2m^{2}-m,m^{2}-m)$$ intersecting in a balanced incomplete block design $$\mathrm {BIBD}(2m^{2} - m, m^{2} - m, m^{2} - m - 1)$$ and a pairwise balanced design $$\mathrm {PBD}(2m^{2} - m, \{m^{2}, m^{2} - m\}, m^{2} - m - 1)$$ . These Menon designs lead to regular amicable Hadamard matrices of orders not previously constructed. Further we construct complex orthogonal designs of order $$4m^2$$ and Butson Hadamard matrices $$\mathrm {BH}(4m^{2},2k)$$ for all k. Some results regarding automorphisms of the constructed Menon designs are proven.

Orbit matrices of Hadamard matrices and related codes

In this paper we introduce the notion of orbit matrices of Hadamard matrices with respect to their permutation automorphism groups and show that under certain conditions these orbit matrices yield self-orthogonal codes. As a case study, we construct codes from orbit matrices of some Paley type I and Paley type II Hadamard matrices. In addition, we construct four new symmetric (100,45,20) designs which correspond to regular Hadamard matrices, and construct codes from their orbit matrices. The codes constructed include optimal, near-optimal self-orthogonal and self-dual codes, over finite fields and over Z4.

Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs

Suppose there exists a Hadamard 2-(m,m−12,m−34) design having skew incidence matrix. If there exists a conference graph on 2m−1 vertices, then there exists a regular Hadamard matrix of order 4m2. A conference graph on 2m+3 vertices yields a regular Hadamard matrix of order 4(m+1)2.

On some double circulant codes of Crnković and disproof of his two conjectures

Crnković (2014) introduced a self-orthogonal [2q,q−1] code and a self-dual [2q+2,q+1] code over the finite field Fp arising from orbit matrices for Menon designs, for every prime power q, where q≡1(mod4) and p a prime dividing q+12. He showed that if q is a prime and q=12m+5, where m is a non-negative integer, then the self-dual [2q+2,q+1] code over F3 is equivalent to a Pless symmetry code. However for other values of q, he remarked that these codes, up to his knowledge, do not belong to some previously known series of codes. In this paper, we describe an equivalence between his self-dual codes and the known codes introduced by Gaborit in 2002. On the other hand, Crnković (2014) also conjectured that if p=q+12 is a prime, the self-orthogonal code and the self-dual code have minimum distance p+3. We disprove this conjecture by giving two counter-examples in the case of the self-orthogonal code and the self-dual code, respectively when q=25 and p=13.

Application of weighing matrices to simultaneous driving technique for capacitive touch sensors

This paper generalizes the capacitive touch sensing problem in a form of weighing matrices which cover Walsh-Hadamard, 4k-order Hadamard, and other types of weighing matrices. The existing regular Hadamard matrix, conference matrix, and other matrices that have a small sum of column weights, increase the signal sensitivity as well as signal-to-noise ratio for the capacitive signals. Experiments show that the proposed design approach and the various types of weighing matrices can meet the different design requirements. The results of this research show that a sophisticated weighing matrix improves sensitivity three times higher than the time-multiplexed method.

REGULAR HADAMARD MATRIX OF ORDER 196 AND SIMILAR MATRICES

Purpose: This note discusses two level quasi-orthogonal matrices which were first highlighted by J. J. Sylvester; Hadamard matrices, symmetric conference matrices, and weighing matrices are the best known of these matrices with entries from the unit disk. The goal of this note is to develop a theory of such matrices based on preliminary research results. Methods: Our new regular Hadamard matrix constructed for order 196, suggests a source of ideas to construct regular Hadamard matrices of orders n = 1 + p × q = 1 + p × (1 + 2m), where p, q are twin odd integer (q – p = 2); m = (q – 1)/2, prime, order of inner blocks. Results: We present a new method aimed to give regular Hadamard matrix of order 196 and similar matrices. Such kinds of regular Hadamard matrix of order 36 were done by Jennifer Seberry (1969), that inspired to find matrices of orders 4k2, k integer, 36, 100, 196, ..., 1444 and many others. We apply this result to the family of regular matrices obtaining a new infinite family of Cretan matrices with orders 4t + 1, t an integer, 37, 101, 197, ..., 1445, etc. Practical relevance: Web addresses are given for other illustrations and other matrices with similar properties. Algorithms to construct regular matrices have been implemented in developing software of the research program-complex.

Classes of self-orthogonal or self-dual codes from orbit matrices of Menon designs

For every prime power q, where q≡1(mod4), and p a prime dividing q+12, we construct a self-orthogonal [2q,q−1] code and a self-dual [2q+2,q+1] code over the field of order p. The construction involves Paley graphs and the constructed [2q,q−1] and [2q+2,q+1] codes admit an automorphism group Σ(q) of the Paley graph of order q. If q is a prime and q=12m+5, where m is a non-negative integer, then the self-dual [2q+2,q+1]3 code is equivalent to a Pless symmetry code. In that sense we can view this class of codes as a generalization of Pless symmetry codes. For q=9 and p=5 we get a self-dual [20,10,8]5 code whose words of minimum weight form a 3-(20, 8, 28) design.

2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order, and their related Hadamard matrices and codes

We present the full classification of Hadamard 2-(31,15,7), Hadamard 2-(35, 17,8) and Menon 2-(36,15,6) designs with automorphisms of odd prime order. We also give partial classifications of such designs with automorphisms of order 2. These classifications lead to related Hadamard matrices and self-dual codes. We found 76166 Hadamard matrices of order 32 and 38332 Hadamard matrices of order 36, arising from the classified designs. Remarkably, all constructed Hadamard matrices of order 36 are Hadamard equivalent to a regular Hadamard matrix. From our constructed designs, we obtained 37352 doubly-even [72,36,12] codes, which are the best known self-dual codes of this length until now.

Infinite families of non‐embeddable quasi‐residual Menon designs

AbstractA Menon design of order h2 is a symmetric (4h2,2h2‐h,h2‐h)‐design. Quasi‐residual and quasi‐derived designs of a Menon design have parameters 2‐(2h2 + h,h2,h2‐h) and 2‐(2h2‐h,h2‐h,h2‐h‐1), respectively. In this article, regular Hadamard matrices are used to construct non‐embeddable quasi‐residual and quasi‐derived Menon designs. As applications, we construct the first two new infinite families of non‐embeddable quasi‐residual and quasi‐derived Menon designs. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 53–62, 2009

On some Menon designs

We describe a construction of symmetric designs with parameters (324,153,72), (576,276,132), and (900,435,210), admitting an automorphism group isomorphic to Frob17·8×Z9, Frob23·11×Z12, and Frob29·14× Z15 respectively. The derived designs of the constructed designs, with respect to the fixed block, are cyclic.

A series of Menon designs and 1-rotational designs

Let p and 2 p + 3 be prime powers and p ≡ 3 ( mod 4 ) . We describe a construction of a symmetric design D with parameters ( 4 ( p + 1 ) 2 , 2 p 2 + 3 p + 1 , p 2 + p ) . If p and 2 p + 3 are primes, then a derived design of D is 1-rotational.

On Two Classes of Menon Designs

Let p and 2p−1 be prime powers and p≡3(mod4). Then there exists a symmetric design with parameters (4p2,2p2−p,p2−p). Thus there exists a regular Hadamard matrix of order 4p2. In a similar way we construct a symmetric (4(p+1)2,2p2+3p+1,p2+p) design D when p and 2p+3 are prime powers, such that p≡3(mod4).

On a new class of productive regular Hadamard matrices

We show that for each integer n for which there is a Hadamard matrix of order 4 n and 8 n 2 - 1 is a prime number, there is a productive regular Hadamard matrix of order 16 n 2 ( 8 n 2 - 1 ) 2 . As a corollary, by applying a recent result of Ionin, we get many parametrically new classes of symmetric designs whenever either of 4 n ( 8 n 2 - 1 ) - 1 or 4 n ( 8 n 2 - 1 ) + 1 is a prime power.

A Series of Regular Hadamard Matrices

Let p and 2p?1 be prime powers and p ? 3 (mod 4). Then there exists a symmetric design with parameters (4p2, 2p2 ? p, p2 ? p). Thus there exists a regular Hadamard matrix of order 4p2.

New Hadamard matrices of order [formula omitted] obtained from Jacobi sums of order 16

Let p ≡ 7 mod 16 be a prime. Then there are integers a , b , c , d with a ≡ 15 mod 16 , b ≡ 0 mod 4 , p 2 = a 2 + 2 ( b 2 + c 2 + d 2 ) , and 2 ab = c 2 - 2 cd - d 2 . We show that there is a regular Hadamard matrix of order 4 p 2 provided that p = a ± 2 b or p = a + δ 1 2 b + 4 δ 2 c + 4 δ 1 δ 2 d with δ i = ± 1 .

Regular Hadamard Matrices Generating Infinite Families of Symmetric Designs

If H is a regular Hadamard matrix with row sum 2h, m is a positive integer, and q e (2h − 1)2, then (4h2(qm + 1 − 1)/(q −1),(2h2 − h)qm,(h2-h)qm) are feasible parameters of a symmetric designs. If q is a prime power, then a balanced generalized weighing matrix BGW((qm +1 − 1)/(q−1),qm,qm−qm−1) can be applied to construct such a design if H satisfies certain structural conditions. We describe such conditions and show that if H satisfies these conditions and B is a regular Hadamard matrix of Bush type, then B×H satisfies these structural conditions. This allows us to construct parametrically new infinite families of symmetric designs.