Nonexistence of Certain Perfect Binary Arrays

A perfect binary array (PBA) is an r-dimensional matrix with elements ±I such that all out-of-phase periodic autocorrelation coefficients are zero. The two smallest sizes for which the existence of a PBA is undecided, 2 x 2 x 3 x 3 x 9 and 4 x 3 x 3 x 9, are ruled out using computer search and a combinatorial argument

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Rely to Comment on \u27Nonexistence of Certain Perfect Binary Arrays\u27 and \u27Nonexistence of Perfect Binary Arrays\u27

Yang\u27s comment [C] is based on a lemma which claims to construct an s0 x s1 x s2 x ... x s, perfect binary array (PBA) from an s0s1 x s2 x ... x sr PBA

The combinatorics of binary arrays

This paper gives an account of the combinatorics of binary arrays, mainly concerning their randomness properties. In many cases the problem reduces to the investigation on difference sets.postprin

A Survey of Hadamard Difference Sets

A (v, k, λ) difference set is a k-element subset D of a group G of order v for which the multiset {d1d2-1 : d1, d2 ∈ D, d1 ≠ d2} contains each nonidentity element of G exactly λ times. A difference set is called abelian, nonabelian or cyclic according to the properties of the underlying group. Difference sets are important in design theory because they are equivalent to symmetric (v, k, λ) designs with a regular automorphism group [L]

A Nonexistence Result for Abelian Menon Difference Sets Using Perfect Binary Arrays

A Menon difference set has the parameters (4N2, 2N2-N, N2-N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group H×K×Zpα contains a Menon difference set, where p is an odd prime, |K|=pα, and pj≡−1 (mod exp (H)) for some j. Using the viewpoint of perfect binary arrays we prove that K must be cyclic. A corollary is that there exists a Menon difference set in the abelian group H×K×Z3α, where exp (H)=2 or 4 and |K|=3α, if and only if K is cyclic

A Summary of Menon Difference Sets

A (v, k, λ) difference set is a k-element subset D of a group G of order v for which the multiset {d1d2-1 : d1,d2 ∈ D, d1 ≠ d2} contains each nonidentity element of G exactly λ times. A difference set is called abelian, nonabelian or cyclic if the underlying group is. Difference sets a.re important in design theory because they a.re equivalent to symmetric (v, k, λ) designs with a regular automorphism group. Abelian difference sets arise naturally in the solution of many problems of signal design in digital communications, including synchronization, radar, coded aperture imaging and optical image alignment. A Menon difference set (MDS) has para.meters of the form (v,k,λ) = (4N2,2N2 - N,N2 - N); alternative names used by some authors are Hadamard difference set or H-set. The Menon para.meters provide the richest source of known examples of difference sets. The central research question is: for each integer N, which groups of order 4N2 support a MDS? This question remains open, for abelian and nonabelian groups, despite a large literature spanning thirty years. The techniques so far used include algebraic number theory, character theory, representation theory, finite geometry and graph theory as well as elementary methods and computer search. Considerable progress has been made recently, both in terms of constructive and nonexistence results. Indeed some of the most surprising advances currently exist only in preprint form, so one intention of this survey is to clarify the status of the subject and to identify future research directions. Another intention is to show the interplay between the study of MDSs and several diverse branches of discrete mathematics. It is intended that a more detailed version of this survey will appear in a future publication