12 research outputs found
Frame difference families and resolvable balanced incomplete block designs
Frame difference families, which can be obtained via a careful use of
cyclotomic conditions attached to strong difference families, play an important
role in direct constructions for resolvable balanced incomplete block designs.
We establish asymptotic existences for several classes of frame difference
families. As corollaries new infinite families of 1-rotational
-RBIBDs over are
derived, and the existence of -RBIBDs is discussed. We construct
-RBIBDs for , whose
existence were previously in doubt. As applications, we establish asymptotic
existences for an infinite family of optimal constant composition codes and an
infinite family of strictly optimal frequency hopping sequences.Comment: arXiv admin note: text overlap with arXiv:1702.0750
Hadamard partitioned difference families and their descendants
If is a Hadamard difference set (HDS) in , then
is clearly a partitioned
difference family (PDF). Any -PDF will be said of Hadamard-type
if as the one above. We present a doubling construction which,
starting from any such PDF, leads to an infinite class of PDFs. As a special
consequence, we get a PDF in a group of order and three
block-sizes , and , whenever we have a
-HDS and the maximal prime power divisors of are
all greater than
New -designs from strong difference families
Strong difference families are an interesting class of discrete structures
which can be used to derive relative difference families. Relative difference
families are closely related to -designs, and have applications in
constructions for many significant codes, such as optical orthogonal codes and
optical orthogonal signature pattern codes. In this paper, with a careful use
of cyclotomic conditions attached to strong difference families, we improve the
lower bound on the asymptotic existence results of -DFs for .
We improve Buratti's existence results for - designs and
- designs, and establish the existence of seven new
- designs for
,
.Comment: Version 1 is named "Improved cyclotomic conditions leading to new
2-designs: the use of strong difference families". Major revision according
to the referees' comment
Partitioned difference families: the storm has not yet passed
Two years ago, we alarmed the scientific community about the large number of
bad papers in the literature on {\it zero difference balanced functions}, where
direct proofs of seemingly new results are presented in an unnecessarily
lengthy and convoluted way. Indeed, these results had been proved long before
and very easily in terms of difference families.
In spite of our report, papers of the same kind continue to proliferate.
Regrettably, a further attempt to put the topic in order seems unavoidable.
While some authors now follow our recommendation of using the terminology of
{\it partitioned difference families}, their methods are still the same and
their results are often trivial or even wrong. In this note, we show how a very
recent paper of this type can be easily dealt with
Partitioned difference families: the storm has not yet passed
Two years ago, we alarmed the scientific community about the large number of bad papers in the literature on zero difference balanced functions, where direct proofs of seemingly new results are presented in an unnecessarily lengthy and convoluted way. Indeed, these results had been proved long before and very easily in terms of difference families. In spite of our report, papers of the same kind continue to proliferate. Regrettably, a further attempt to put the topic in order seems unavoidable. While some authors now follow our recommendation of using the terminology of partitioned difference families, their methods are still the same and their results are often trivial or even wrong. In this note, we show how a very recent paper of this type can be easily dealt with
New Z-cyclic triplewhist frames and triplewhist tournament designs
AbstractTriplewhist tournaments are a specialization of whist tournament designs. The spectrum for triplewhist tournaments on v players is nearly complete. It is now known that triplewhist designs do not exist for v=5,9,12,13 and do exist for all other v≡0,1(mod4) except, possibly, v=17. Much less is known concerning the existence of Z-cyclic triplewhist tournaments. Indeed, there are many open questions related to the existence of Z-cyclic whist designs. A (triple)whist design is said to be Z-cyclic if the players are elements in Zm∪A where m=v, A=∅ when v≡1(mod4) and m=v-1, A={∞} when v≡0(mod4) and it is further required that the rounds also be cyclic in the sense that the rounds can be labelled, say, R1,R2,… in such a way that Rj+1 is obtained by adding +1(modm) to every element in Rj. The production of Z-cyclic triplewhist designs is particularly challenging when m is divisible by any of 5,9,11,13,17. Here we introduce several new triplewhist frames and use them to construct new infinite families of triplewhist designs, many for the case of m being divisible by at least one of 5,9,11,13,17