A search for affine difference sets of even order

The Prime Power Conjecture asserts that the order of an affine difference set is an integral power of a prime number. K.T. Arasu and D. Jungnickel have shown that if the order of an affine difference set is even, then the order is either two or four or is divisible by eight. This paper extends this direction of research by studying cyclic affine difference sets whose orders are congruent to eight modulo sixteen. In particular, we give several numerical constraints that support the Prime Power Conjecture in this case. We conclude with a computer search for cyclic affine difference sets of order eight modulo sixteen that satisfy these new conditions.     

Partitioned difference families and almost difference sets

Partitioned difference families (PDFs) were first studied by Ding and Yin in conjunction with the construction of constant composition codes (CCCs). In 2008, Yin et al. presented the constructions of a number of infinite classes of PDFs based on known difference sets in GF(q). In this paper, we further investigate the constructions of PDFs by using known almost difference sets in GF(q), and establish some recursive constructions of PDFs. As their applications, we also get a number of perfect difference systems of sets (DSSs) over Z_q² with q odd prime.     

Nested Hadamard difference sets

A Hadamard difference set (HDS) has the parameters (4N², 2N² − N, N² − N). In the abelian case it is equivalent to a perfect binary array, which is a multidimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. We show that if a group of the form H × Z²_p^r contains a (hp^2r, √hp^r(2√hp^r − 1), √hp^r(√hp^r − 1)) HDS (HDS), p a prime not dividing |H| = h and p^j ≡ −1 (mod exp(H)) for some j, then H × Z²_p^t has a (hp^2t, √hp^t(2√hp^t − 1), √hp^t(√hp^t − 1)) HDS for every 0⩽t⩽r. Thus, if these families do not exist, we simply need to show that H × Z²_p does not support a HDS. We give two examples of families that are ruled out by this procedure.     

Diffuse difference triangle sets

A particular kind of difference triangle sets (DTSs) called diffuse DTS (DDTS) are considered. Their combinatorial structure is underlying the construction of all known types of self-orthogonal diffuse codes. A number of constructions of DDTS are described, and lower and upper bounds on the maximal element of an optimal DDTS are given. The asymptotic behaviour of the maximal element is studied. Finally, tables of DDTS and of their minimal possible maximal elements are presented.     

A class of non-abelian 2-groups containing Menon difference sets

We show that every group in a certain class of 2-groups contains a Menon difference set. This provides further positive evidence for a conjecture of Dillon concerning 2-groups of order 2^2l which contain a normal subgroup isomorphic to Z^l₂. The conjecture, however, remains open.     

A conjecture on the existence of cyclic Hadamard difference sets

If there exists a cyclic Hadamard difference set of length v, then v = 4n − 1 is conjectured to be either a prime, or a product of “twin primes”, or one less than a power of 2.     

On Whiteman's and Storer's difference sets

In this paper, we prove that all Storer's difference sets are cyclic. We also prove that for p<1.8×10²⁵, Whiteman's difference sets exist if and only if (p,q)=(17,53) and (46817,140453).     

Optimal designs,supplementary difference sets and multipliers

We investigate multipliers of 2 - {v; q², q²; λ} supplementary difference sets where cyclotomy has been used to construct D-optimal designs.     

A modified random-effect procedure for combining risk difference in sets of 2×2 tables from clinical trials

Summary Meta-analyses of sets of clinical trials often combine risk differences from several 2×2 tables according to a random-effects model. The DerSimonian-Laird random-effects procedure, widely used for estimating the populaton mean risk difference, weights the risk difference from each primary study inversely proportional to an estimate of its variance (the sum of the between-study variance and the conditional within-study variance). Because those weights are not independent of the risk differences, however, the procedure sometimes exhibits bias and unnatural behavior. The present paper proposes a modified weighting scheme that uses the unconditional within-study variance to avoid this source of bias. The modified procedure has variance closer to that available from weighting by ideal weights when such weights are known. We studied the modified procedure in extensive simulation experiments using situations whose parameters resemble those of actual studies in medical research. For comparison we also included two unbiased procedures, the unweighted mean and a sample-size-weighted mean; their relative variability depends on the extent of heterogeneity among the primary studies. An example illustrates the application of the procedures to actual data and the differences among the results. This research was supported by Grant HS 05936 from the Agency for Health Care Policy and Research to Harvard University.     

Some necessary conditions on the parameters of partial difference sets

In this paper, we study some necessary conditions on the parameters of nontrivial regular (υ, κ, λ, μ)-partial difference sets in abelian groups. In particular, we settle some undecided cases in Ma's table [Designs, Codes Cryptography, 4 (1994)]. Also, the case when λ ⩽ 1 is studied. Nonexistence results are obtained when λ = 0 and a complete characterization is given when λ = 1. Finally, parameters of partial difference sets with an odd μ are determined.     

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

The 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.     

An application of difference sets to a problem concerning graphical codes

By using difference sets, we give an answer to the following problem concerning graphical codes: When is the binary code generated by the complete graph K_n contained in some binary Hamming code? It turns out that this holds if and only if n is one of the numbers 2, 3 and 6.     

A general inversion theorem for cointegration

A generalization of the Granger and the Johansen Representation Theorems valid for any (possibly fractional) order of integration is presented. This Representation Theorem is based on inversion results that characterize the order of the pole and the coefficients of the Laurent series representation of the inverse of a matrix function around a singular point. Explicit expressions of the matrix coefficients of the (polynomial) cointegrating relations, of the Common Trends and of the Triangular representations are provided, either starting from the Moving Average or the Auto Regressive form. This contribution unifies different approaches in the literature and extends them to an arbitrary order of integration. The role of deterministic terms is discussed in detail.     

A further note on a theorem on the difference of the generalized inverses of two nonnegative definite matrices

Andrews and Phillips (1986) gave a simplified proof for the result that established the nonnegative definiteness of the difference of the Moore-Penrose inverses of two nonoegative definite matrices, a result originally due to Milliken and Akdeniz (1977), The purpose of this paper is to offer a simple proof for a generalization of this result,     

A central limit theorem for generalized discounting

A central limit theorem is derived for the discounted sum of a sequence of i.i.d. random variables with finite first and second moment, where the rate of interest may depend on time. For the special case of constant rate of interest GERBER (1971) obtains a Berry-Esséen result assuming finite third moment. The same result is shown to hold in the non-const ant case.     

A lagrange multiplier test for the error components model with incomplete panels

This Paper extends the Breusch and Pagan(1980) lagrange Multiplier test for the random effects model to the incomplete panel data case. It is shown that this test retains the simple additive structure observed in the complete panel data case. It should prove useful for practitioners facing incomplete panel data applications.     

A Lusin theorem for a class of Choquet capacities

We extend the Lusin Theorem to supermodular Dempster capacities. Received: July 2000; revised version: May 2001