Inequivalent Hadamard Matrices Research Articles

Inequivalent Hadamard Matrices of Order 40 using Block Designs

This article presents the classification of Hadamard matrices of order 40 with automorphisms of order 19. We determine the 2-(19,9,8) and 2-(20,10,9) designs with automorphisms of order 19. These designs are embedded into 2-(39,19,9) designs which are then extended to 3-(40,20,9) designs. We get 33 non-isomorphic 2-(39,19,9) designs out of which 18 designs gives non-isomorphic 3-(40,20,9) designs and hence 18 inequivalent Hadamard matrices of order 40 with automorphisms of order 19.

Hadamard matrices related to a certain series of ternary self-dual codes

In 2013, Nebe and Villar gave a series of ternary self-dual codes of length $$2(p+1)$$ for a prime p congruent to 5 modulo 8. As a consequence, the third ternary extremal self-dual code of length 60 was found. We show that these ternary self-dual codes contain codewords which form a Hadamard matrix of order $$2(p+1)$$ when p is congruent to 5 modulo 24. In addition, we show that the ternary self-dual codes found by Nebe and Villar are generated by the rows of the Hadamard matrices. We also demonstrate that the third ternary extremal self-dual code of length 60 contains at least two inequivalent Hadamard matrices.

A META-SOFTWARE SYSTEM FOR ORTHOGONAL DESIGNS AND HADAMARD MATRICES

In this paper, we construct inequivalent Hadamard matrices based on several new and old full orthogonal designs, using circulant and symmetric block matrices. Not all orthogonal designs produce inequivalent Hadamard matrices, because the corresponding systems of equations do not possess solutions. The systems of equations arising when we search for inequivalent Hadamard matrices from full orthogonal designs using circulant and symmetric block matrices, can be concisely described using the periodic autocorrelation function of the generators of the block matrices. We use Maple, Magma, C and Unix tools to find many new inequivalent Hadamard matrices.

Improving the lower bounds on inequivalent Hadamard matrices through orthogonal designs and meta-programming techniques

In this paper, we construct inequivalent Hadamard matrices based on several new and old full orthogonal designs, using circulant and symmetric block matrices. Not all orthogonal designs produce inequivalent Hadamard matrices, because the corresponding systems of equations do not possess solutions. In addition, we give some new constructions for orthogonal designs derived from sequences with zero autocorrelation. The orthogonal designs used to construct the inequivalent Hadamard matrices are produced from theoretical and algorithmic constructions.

New two-variable full orthogonal designs and related experiments with linear regression models

Factorial designs are very important when experiments involve two or more factors and it is desirable to study the main effects and the factor interactions. In this paper, four circulant matrices are used for the construction of orthogonal designs. In particular, it is shown that all 26 two-variable full orthogonal designs OD ( 52 ; s 1 , 52 - s 1 ) exist and they can be constructed by four circulants. From these new orthogonal designs one can obtain many inequivalent Hadamard matrices and new saturated two-level factorial designs, with 52 runs, suitable for screening and weighing experiments. Some of these designs are directly applicable in linear models for estimating all the main effects.

Projection Properties of Hadamard Matrices of Order 36 Obtained from Paley’s Constructions

In this paper we study the projection properties of 12 inequivalent Hadamard matrices of order 36 obtained from Paley’s constructions, using several statistical criteria. We also present generalized minimum aberration designs with 36 runs and up to 7 columns that are embedded into these Hadamard matrices.

Some orthogonal arrays with 32 runs and their projection properties

Screening designs are useful for situations where a large number of factors are examined but only a few, k, of them are expected to be important. Traditionally orthogonal arrays such as Hadamard matrices and Plackett Burman designs have been studied for this purpose. It is therefore of practical interest for a given k to know all the classes of inequivalent projections of the design into the k dimensions that have certain statistical properties. In this paper we present 15 inequivalent Hadamard matrices of order n=32 constructed from circulant cores. We study their projection properties using several well-known statistical criteria and we provide minimum generalized aberration 2 level designs with 32 runs and up to seven factors that are embedded into these Hadamard matrices. A concept of generalized projectivity and design selection of such designs is also discussed.

Evaluation of inequivalent projections of Hadamard matrices of order 24

Screening designs are useful for situations where a large number of factors (q) is examined but only few (k) of these are expected to be important. It is of practical interest for a given k to know all the inequivalent projections of the design into the k dimensions. In this paper we give all the (combinatorially) inequivalent projections of inequivalent Hadamard matrices of order 24 into k=3,4 and 5 dimensions, as well as their frequencies. Then, we sort these projections according to their generalized resolution, generalized aberration and centered L2-discrepancy measure of uniformity. Then, we study the hidden projection properties of these designs as they are introduced by Wang and Wu (1995). The hidden projection property suggests that complex aliasing allows some interactions to be estimated without making additional runs.

Inequivalent Hadamard matrices with buckets

We give an optimal algorithm to locate all inequivalent Hadamard matrices, given any finite set of Hadamard matrices of the same dimension. When the cardinality of the original set is big, it is advantageous to proceed in an incremental manner and therefore we supply a distributed version of the buckets algorithm. The distributed buckets algorithm comprises a multi-stage merging phase, during which the intermediate results obtained in the previous phases are combined to yield the final result. The buckets algorithm for Hadamard inequivalence is independent of the particular method utilized to check Hadamard inequivalence. When the Hadamard matrices in the given set exhibit a certain structure, any corresponding methods for checking Hadamard inequivalence, tailored for this structure, may be used. Such structured Hadamard matrices arise for instance as the solution sets of Hadamard ideals for matrices with one and two circulant cores. The serial and distributed buckets algorithms, are implemented in the Computational Algebra systems Magma and Maple and are applied to a number of cases.

On the Self-Dual Codes over GF (7) Generated by Hadamard Matrices of Order 28

It is known that there are exactly 487 inequivalent Hadamard matrices of order 28. In this note, we show that the 487 self-dual codes over GF (7) generated by the 487 inequivalent Hadamard matrices of order 28 are inequivalent.

A note on inequivalent Hadamard matrices.

A note on inequivalent Hadamard matrices.

Some notes on integral equivalence of combinatorial matrices

Recent work on integral equivalence of Hadamard matrices and block designs is generalized in two directions. We first determine the two greatest invariants under integral equivalence of the incidence matrix of a symmetric balanced incomplete block design. This enables us to write down all the invariants in the case wherek −λ is square-fre. Some other results on the sequence of invariants are presented. Secondly we consider the existence of inequivalent Hadamard matrices under integral equivalence. We show that if there is a skew-Hadamard matrix of order 8m then there are two inequivalent Hadamard matrices of order 16m, that and there are precisely eleven inequivalent Hadamard matrices of order 32.

Integral equivalence of hadamard matrices

SupposeA is a non-singular matrix with entries 0 and 1, the zero and identity elements of a Euclidean domain. We obtain a “best-possible” lower bound for the number of equivalence invariants ofA (over the domain) which equal 1. From this it is proven that the sequence of invariants under integral equivalence of an Hadamard matrix must obey certain conditions. Finally, lower bounds are found for the number of inequivalent Hadamard matrices of order a power of 2, and consequently for the number of Hadamard-inequivalent Hadamard matrices of those orders.