Supplementary Difference Sets Research Articles

Square (1,−1)-matrices with large determinants and near-Hadamard matrices

We give explicit constructions of square (1,−1)-matrices of order congruent to 2 modulo 4 with large determinants. In particular, we construct nine families of near-Hadamard matrices based on almost supplementary difference sets with two blocks, and compute their determinants. As consequences, we obtain many (1,−1)-matrices which are almost D-optimal or have at least 80% D-efficiency.

Modified second order slope rotatable designs using supplementary difference sets

In this paper, following the methods of constructions of Mutiso et al. [7-8], Chiranjeevi and Victorbabu [2-3], a new method of construction of modified second order slope rotatable designs using supplementary difference sets is suggested. Some illustrative examples are also presented.

Almost supplementary difference sets and quaternary sequences with optimal autocorrelation

We introduce almost supplementary difference sets (ASDS). For odd m, certain ASDS in $\mathbb Z_{m}$ that have amicable incidence matrices are equivalent to quaternary sequences of odd length m with optimal autocorrelation. As one consequence, if 2m − 1 is a prime power, or m ≡ 1 mod 4 is prime, then ASDS of this kind exist. We also explore connections to optimal binary sequences and group cohomology.

Statistical Measure of Second Order Response Surface Rotatability Using an Infinite Class of Supplementary Difference Sets

Rotatability is a desirable feature of a response surface experimental design. In case a design is non rotatable or exhibit surface of constant prediction variances that are nearly spherical then an attempt is made to make the design rotatable. In this paper, a measure of rotatability of five level second order rotatable designs using an infinite class of supplementary difference sets is suggested. The variance function of a second-order response design and an infinite class of supplementary difference sets is used in coming up with the design.

Measure of rotatability of modified five-level second-order rotatable design using supplementary difference sets

ABSTRACTRotatability is a desirable quality of fitting response surface experimental designs. The property states that the variance of the estimated response made from the Taylor’s series expansion are constant on circles, spheres and hyper-spheres about the centre of the design. In this article, a measure of rotatability of modified second-order rotatable design is presented. The variance function of a second-order response design and an infinite class of supplementary difference sets is used in coming up with the design.

Construction and analysis of edge designs from skew-symmetric supplementary difference sets

ABSTRACTThe purpose of screening experiments is to identify the dominant variables from a set of many potentially active variables which may affect some characteristic y. Edge designs were recently introduced in the literature and are constructed by using conferences matrices and were proved to be robust. We introduce a new class of edge designs which are constructed from skew-symmetric supplementary difference sets. These designs are particularly useful since they can be applied for experiments with an even number of factors and they may exist for orders where conference matrices do not exist. Using this methodology, examples of new edge designs for 6, 14, 22, 26, 38, 42, 46, 58, and 62 factors are constructed. Of special interest are the new edge designs for studying 22 and 58 factors because edge designs with these parameters have not been constructed in the literature since conference matrices of the corresponding order do not exist. The suggested new edge designs achieve the same model-robustness as the traditional edge designs. We also suggest the use of a mirror edge method as a test for the linearity of the true underlying model. We give the details of the methodology and provide some illustrating examples for this new approach. We also show that the new designs have good D-efficiencies when applied to first order models.

Constructing near-Hadamard designs with (almost) D-optimality by general supplementary difference sets

Constructing near-Hadamard designs with (almost) D-optimality by general supplementary difference sets

A General Construction Method for Five-Level Second-Order Rotatable Designs

Response surface methodology is widely used for developing, improving, and optimizing processes in various fields. In this article, we present a method for constructing second-order rotatable designs in order to explore and optimize response surfaces based on an infinite class of supplementary difference sets. The produced designs achieve both properties of rotatability and estimation efficiency. Also, they possess good predictive properties.

Compression of periodic complementary sequences and applications

A collection of complex sequences of length v is complementary if the sum of their periodic autocorrelation function values at all non-zero shifts is constant. For a complex sequence A=[a_0,a_1,...,a_{v-1}] of length v=dm we define the m-compressed sequence A^{(d)} of length d whose terms are the sums a_i + a_{i+d} + ... + a_{i+(m-1)d}. We prove that the m-compression of a complementary collection of sequences is also complementary. The compression procedure can be used to simplify the construction of complementary {+1,-1}-sequences of composite length. In particular, we construct several supplementary difference sets (v;r,s;lambda) with v even and lambda=(r+s)-v/2, given here for the first time. There are 15 normalized parameter sets (v;r,s;lambda) with v <= 50 for which the existence question was open. We resolve all but one of these cases.

Some New Orders of Hadamard and Skew‐Hadamard Matrices

AbstractWe construct Hadamard matrices of orders and , and skew‐Hadamard matrices of orders and . As far as we know, such matrices have not been constructed previously. The constructions use the Goethals–Seidel array, suitable supplementary difference sets on a cyclic group and a new efficient matching algorithm based on hashing techniques.

New Results on D‐Optimal Matrices

AbstractWe construct a number of new supplementary difference sets (SDS) with v odd and . In particular, these give rise to D‐optimal matrices of the four new orders 206, 242, 262, 482, constructed here for the first time.

A general construction of E(f NOD) -optimal multi-level supersaturated designs

The present paper deals with E ( f NOD ) - optimal multi-level supersaturated designs. We present a new technique for the construction of supplementary difference sets. Based on the new supplementary difference sets, we also provide E ( f NOD ) - optimal multi-level supersaturated designs with a large number of columns when compared with other designs. Moreover, these designs retain the equal occurrence property.

An Algorithmic Construction ofE(s2)-Optimal Supersaturated Designs

Abstract A general method based on an effective algorithm for construction of E(s 2)-optimal, two-level supersaturated designs with the equal occurrence property, from supplementary difference sets is introduced. Supersaturated designs constructed in this way are E(s 2)-optimal. Comparisons are made with previous works and it is shown that the proposed method gives promising results for the construction of supersaturated designs with good properties.

Supplementary difference sets with symmetry for Hadamard matrices

Supplementary difference sets with symmetry for Hadamard matrices

A general construction of E(s 2)-optimal supersaturated designs via supplementary difference sets

Supplementary difference sets can be used to construct a variety of combinatorial designs. In this work, we construct supplementary difference sets that provide E(s 2)-optimal, two-level supersaturated designs with the equal occurrence property. Comparisons are made with previous works and it is shown that the proposed method gives promising results for the construction of supersaturated designs with good properties.

Heuristic algorithms for Hadamard matrices with two circulant cores

We design heuristic algorithms to construct Hadamard matrices with two circulant cores. This hard combinatorial problem can be formulated in terms of objective functions of several binary variables, so that heuristic methodologies can be used. Our algorithms are based on local and tabu search and they use information on the geometry of the objective function landscapes. In addition, we use the supplementary difference sets formalism to detect when solutions of a special structure exist. Using these algorithms we have computed at least one Hadamard matrix with two circulant cores of the sixteen orders 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116. In particular, the Hadamard matrix with two circulant cores of order 116 is constructed here for the first time, indeed it was accidentally reported as known in an earlier paper.

A general construction of E(s 2)-optimal large supersaturated designs

A general method for construction of E(s2)-optimal, two-level supersaturated designs (SSDs) with the equal occurrence property, from supplementary difference sets is introduced. It is proved that SSDs constructed in this way are E(s2)-optimal. Comparisons are made with previous works and it is shown that the proposed method gives promising results for the construction of E(s2)-optimal large SSDs.

Values of minors of some infinite families of matrices constructed from supplementary difference sets and their application to the growth problem

We obtain explicit formulae for the values of the 2 v − j minors, j = 0, 1, 2 of some infinite families of matrices of order 2 v constructed from supplementary difference sets of the form 2 − { v; k 1, k 2; λ}. This allows us to obtain information on the growth problem for families of matrices with moderate growth. Several examples specifying the pivot pattern of these matrices are presented and a conjecture about their pivot structure is posed.

Values of Minors of an Infinite Family of D-Optimal Designs and Their Application to the Growth Problem: II

We obtain explicit formulae for the values of the 2v-j minors, j=0,1,2 of D-optimal designs of order 2v = x2 + y2, v odd, where the design is constructed using two circulant or type 1 incidence matrices of $2-\{s^2+s+1;\frac{s(s-1)}{2},\frac{s(s+1)}{2};\frac{s(s-1)}{2}\}$ supplementary difference sets (SDS). This allows us to obtain information on the growth problem for families of matrices which have moderately large growth. Some of our theoretical formulae suggest that growth greater than 2v may occur, but experimentation has not yet supported this result. An open problem remains to establish whether the (1,-1) completely pivoted (CP) incidence matrices of $2-\{s^2+s+1;\frac{s(s-1)}{2},\frac{s(s+1)}{2};\frac{s(s-1)}{2}\}$ SDS, which yield D-optimal designs, can have growth greater than 2v.

On good matrices, skew Hadamard matrices and optimal designs

In two-level factorial experiments and in other linear models the coefficients of the unknown parameters can take one out of two values. When the number of observations is a multiple of four, the D-optimal design is a Hadamard matrix. Skew Hadamard matrices are of special interest due to their use, among others, in constructing D-optimal weighing designs for n≡3 ( mod 4) . A method is given for constructing skew Hadamard matrices which is based on the construction of good matrices. The construction is achieved through an algorithm which is also presented and relies on the discrete Fourier transform. It is known that good matrices of order n, exist for all odd n⩽35 and n=127. In this paper, we give for the first time all non-equivalent circulant good matrices of odd order 33⩽ n⩽39. We note that no good matrices were previously known for orders 37 and 39. These are presented in a table in the form of the corresponding non-equivalent supplementary difference sets. In the sequel we use good matrices to construct some skew Hadamard matrices and orthogonal designs.