A note on planar difference sets

It is proven that if G is an abelian regular automorphism group of a projective plane of order n and if p is a prime dividing n exactly once, then a certain identity holds in the group algebra $\text{F}$_p|G|. As a corollary we obtain that n = p if p = 2 or 3.     

A note on new semi-regular divisible difference sets

We give a construction for new families of semi-regular divisible difference sets. The construction is a variation of McFarland's scheme [5] for noncyclic difference sets.This work is partially supported by NSA grant # MDA 904-92-H-3067.     

A note on difference sets in dihedral groups

It is proved that there are no nontrivial difference sets in dihedral groups of order 4p^t, where p is a prime, t > 0 is a positive integer. Received: 22 February 2002     

A note on almost difference sets in nonabelian groups

We present a new construction of almost difference sets. The construction occurs in nonabelian groups of order 4N with a subgroup H of order N so that H has an (N,\(\frac{N-1}{2}\),\(\frac{N-3}{4}\)) difference set (and hence N must be an integer that is 3 (mod 4)).     

A note on maximized likelihood sets

The purpose of this note is to show that the posterior measure under a partial prior information, which is constructed on the maximized likelihood function,is compatible with the Bayesian properties of the likelihood sets.     

A note on DOL length sets

Length sets of polynomially bounded DOL systems are considered by analyzing sets K ∩ {n : n ? q} where K is a DOL length set. It turns out that DOL and PDOL systems behave in different ways with respect to these sets. Especially, it follows that all DOL length sets— contrary to PDOL length sets—are not generated unambiguously, i.e. without repetitions, by DOL systems. Moreover it is proved that the number of axioms used determines an infinite hierarchy of length set families both in DOL and PDOL case.     

A note on maximal progression-free sets

Erd?s et al [Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math. 200 (1999) 119-135.] asked whether there exists a maximal set of positive integers containing no three-term arithmetic progression and such that the difference of its adjacent elements approaches infinity. This note answers the question affirmatively by presenting such a set in which the difference of adjacent elements is strictly increasing. The construction generalizes to arithmetic progressions of any finite length.     

A note on orthogonal sets of hypercubes

Mutually orthogonal sets of hypercubes are higher dimensional generalizations of mutually orthogonal sets of Latin squares. For Latin squares, it is well known that the Cayley table of a group of order n is a Latin square, which has no orthogonal mate if n is congruent to 2 modulo 4. We will prove an analogous result for hypercubes. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 231–233, 1997