2,401,684 research outputs found
On some new constructions of difference sets
Difference sets are mathematical structures which arise in algebra and combinatorics, with applications in coding theory. The fundamental question is when and how one can construct difference sets. This largely expository paper looks at standard construction methods and describes recent findings that resulted in new families of difference sets. This paper provides explicit examples of difference sets that arise from the recent constructions. By gaining a thorough understanding of these new techniques, it may be possible to generalize the results to find additional new families of difference sets. The paper also introduces partial and relative difference sets and discusses how the three types of difference sets relate to other combinatorial structures such as block designs and certain strongly regular graphs
A Generalization of Combinatorial Designs Related to Almost Difference Sets
In this paper we study a certain generalization of combinatorial designs
related to almost difference sets, namely the -adesign, which was coined by
Cunsheng Ding in 2015. It is clear that -adesigns are a kind of partially
balanced incomplete block design which naturally arise in many combinatorial
and statistical problems. We discuss some of their basic properties and give
several constructions of -adesigns (some of which correspond to new almost
difference sets, and others of which correspond to new almost difference
families), as well as two constructions of -adesigns. We also discuss some
basic properties of their incidence matrices and codes
Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets
In this paper, we give a new lifting construction of "hyperbolic" type of
strongly regular Cayley graphs. Also we give new constructions of strongly
regular Cayley graphs over the additive groups of finite fields based on
partitions of subdifference sets of the Singer difference sets. Our results
unify some recent constructions of strongly regular Cayley graphs related to
-ovoids and -tight sets in finite geometry. Furthermore, some of the
strongly regular Cayley graphs obtained in this paper are new or nonisomorphic
to known strongly regular graphs with the same parameters.Comment: 19page
New Partial Geometric Difference Sets and Partial Geometric Difference Families
Olmez, in "Symmetric -Designs and -Difference
Sets" (2014), introduced the concept of a partial geometric difference set
(also referred to as a -design), and showed that partial
geometric difference sets give partial geometric designs. Nowak et al., in
"Partial Geometric Difference Families" (2014), introduced the concept of a
partial difference family, and showed that these also give partial geometric
designs. It was shown by Brouwer et al. in "Directed strongly regular graphs
from -designs" (2012) that directed strongly regular graphs can
be obtained from partial geometric designs. In this correspondence we construct
several families of partial geometric difference sets and partial difference
families with new parameters, thereby giving directed strongly regular graphs
with new parameters. We also discuss some of the links between partially
balanced designs, -adesigns (which were recently coined by Cunsheng Ding in
"Codes from Difference Sets" (2015)), and partial geometric designs, and make
an investigation into when a -adesign is partial geometric
Almost Difference Sets in Nonabelian Groups
We give two new constructions of almost difference sets. The first is a
generic construction of almost
difference sets in certain groups of order ( is an odd prime
power) having ( as a subgroup. The construction occurs in
any group of order ( is an odd prime) having
( as an additive subgroup. This construction yields
several infinite families of almost difference sets, many of which occur in
nonabelian groups. The second construction yields almost
difference sets in dihedral groups of order where is a prime. Moreover, it turns out that some of the infinite families of
almost difference sets obtained have Cayley graphs which are Ramanujan graphs.
\keywords{Difference set \and Almost difference set \and Nonabelian group
The weighted difference substitutions and Nonnegativity Decision of Forms
In this paper, we study the weighted difference substitutions from
geometrical views. First, we give the geometric meanings of the weighted
difference substitutions, and introduce the concept of convergence of the
sequence of substitution sets. Then it is proven that the sequence of the
successive weighted difference substitution sets is convergent. Based on the
convergence of the sequence of the successive weighted difference sets, a new,
simpler method to prove that if the form F is positive definite on T_n, then
the sequence of sets {SDS^m(F)} is positively terminating is presented, which
is different from the one given in [11]. That is, we can decide the
nonnegativity of a positive definite form by successively running the weighted
difference substitutions finite times. Finally, an algorithm for deciding an
indefinite form with a counter-example is obtained, and some examples are
listed by using the obtained algorithm.Comment: 10 pages, 1 figure
Frequency hopping sequences with optimal partial Hamming correlation
Frequency hopping sequences (FHSs) with favorable partial Hamming correlation
properties have important applications in many synchronization and
multiple-access systems. In this paper, we investigate constructions of FHSs
and FHS sets with optimal partial Hamming correlation. We first establish a
correspondence between FHS sets with optimal partial Hamming correlation and
multiple partition-type balanced nested cyclic difference packings with a
special property. By virtue of this correspondence, some FHSs and FHS sets with
optimal partial Hamming correlation are constructed from various combinatorial
structures such as cyclic difference packings, and cyclic relative difference
families. We also describe a direct construction and two recursive
constructions for FHS sets with optimal partial Hamming correlation. As a
consequence, our constructions yield new FHSs and FHS sets with optimal partial
Hamming correlation.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1506.0737
New families of optimal frequency hopping sequence sets
Frequency hopping sequences (FHSs) are employed to mitigate the interferences
caused by the hits of frequencies in frequency hopping spread spectrum systems.
In this paper, we present some new algebraic and combinatorial constructions
for FHS sets, including an algebraic construction via the linear mapping, two
direct constructions by using cyclotomic classes and recursive constructions
based on cyclic difference matrices. By these constructions, a number of series
of new FHS sets are then produced. These FHS sets are optimal with respect to
the Peng-Fan bounds.Comment: 10 page
Sets of Zero-Difference Balanced Functions and Their Applications
Zero-difference balanced (ZDB) functions can be employed in many
applications, e.g., optimal constant composition codes, optimal and perfect
difference systems of sets, optimal frequency hopping sequences, etc. In this
paper, two results are summarized to characterize ZDB functions, among which a
lower bound is used to achieve optimality in applications and determine the
size of preimage sets of ZDB functions. As the main contribution, a generic
construction of ZDB functions is presented, and many new classes of ZDB
functions can be generated. This construction is then extended to construct a
set of ZDB functions, in which any two ZDB functions are related uniformly.
Furthermore, some applications of such sets of ZDB functions are also
introduced.Comment: 20 page
New constructions of strongly regular Cayley graphs on abelian groups
Davis and Jedwab (1997) established a great construction theory unifying many
previously known constructions of difference sets, relative difference sets and
divisible difference sets. They introduced the concept of building blocks,
which played an important role in the theory. On the other hand, Polhill (2010)
gave a construction of Paley type partial difference sets (conference graphs)
based on a special system of building blocks, called a covering extended
building set, and proved that there exists a Paley type partial difference set
in an abelian group of order for any odd positive integer and
any . His result covers all orders of nonelementary abelian groups in
which Paley type partial difference sets exist. In this paper, we give new
constructions of strongly regular Cayley graphs on abelian groups by extending
the theory of building blocks. The constructions are large generalizations of
Polhill's construction. In particular, we show that for a positive integer
and elementary abelian groups , , of order such
that , there exists a decomposition of the complete graph on the
abelian group by strongly regular
Cayley graphs with negative Latin square type parameters , where and . Such strongly
regular decompositions were previously known only when or is a
-group. Moreover, we find one more new infinite family of decompositions of
the complete graphs by Latin square type strongly regular Cayley graphs. Thus,
we obtain many strongly regular graphs with new parameters.Comment: 14 pages; Some typos are fixed, and Abst and Intro are rewritte
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