A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $Δ[G]$. The fraction $ð[G]:=\frac{Δ[G]}{\sqrt{|G|}}$ is called the difference characteristic of $G$. Using properies of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $p\ge 11$, any finite Abelian $p$-group $G$ has difference characteristic $ð[G]<\frac{\sqrt{p}-1}{\sqrt{p}-3}\cdot\sup_{k\in\mathbb N}ð[C_{p^k}]<\sqrt{2}\cdot\frac{\sqrt{p}-1}{\sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality $<96$.
A difference set is said to have classical parameters if $ (v,k, λ) = (\frac{q^d-1}{q-1}, \frac{q^{d-1}-1}{q-1}, \frac{q^{d-2}-1}{q-1}).$ The case $d=3$ corresponds to planar difference sets. We focus here on the family of abelian difference sets with $d=4$. The only known examples of such difference sets correspond to the projective geometries $PG(3,q)$. We consider an arbitrary difference set with the parameters of $PG(3,q)$ in an abelian group and establish constraints on its structure. In particular, we discern embedded substructures.
In this paper, we consider regular automorphism groups of graphs in the RT$2$ family and the Davis-Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non-abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT$2$ family and Davis-Xiang family and their amorphic abelian Cayley schemes to produce amorphic non-abelian Cayley schemes.
Strong external difference families (SEDFs) have applications to cryptography and are rich combinatorial structures in their own right; until now, all SEDFs have been in abelian groups. In this paper, we consider SEDFs in both abelian and non-abelian groups. We characterize the order of groups possessing admissible parameters for non-trivial SEDFs, develop non-existence and existence results, several of which extend known results, and present the first family of non-abelian SEDFs. We introduce the concept of equivalence for EDFs and SEDFs, and begin the task of enumerating SEDFs. Complete resu…
Partial difference sets with parameters $(v,k,λ,μ)=(v, (v-1)/2, (v-5)/4, (v-1)/4)$ are called Paley type partial difference sets. In this note we prove that if there exists a Paley type partial difference set in an abelian group $G$ of an order not a prime power, then $|G|=n^4$ or $9n^4$, where $n>1$ is an odd integer. In 2010, Polhill \cite{Polhill} constructed Paley type partial difference sets in abelian groups with those orders. Thus, combining with the constructions of Polhill and the classical Paley construction using non-zero squares of a finite field, we completely answer the following…
Combinatorial transfer: a new method for constructing infinite families of nonabelian difference sets, partial difference sets, and relative difference sets
In this paper we prove that an abelian group contains $(2^{2m+1}(2^{m-1}+1), 2^m(2^m+1), 2^m)$-difference sets with $m\geqslant 3$ if and only if it contains an elementary abelian 2-group of order $2^{2m}$. Our proof shows that the method of constructing such difference sets is essentially unique.
A difference matrix over a group is a discrete structure that is intimately related to many other combinatorial designs, including mutually orthogonal Latin squares, orthogonal arrays, and transversal designs. Interest in constructing difference matrices over $2$-groups has been renewed by the recent discovery that these matrices can be used to construct large linking systems of difference sets, which in turn provide examples of systems of linked symmetric designs and association schemes. We survey the main constructive and nonexistence results for difference matrices, beginning with a classic…
Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the special structure of the group ring of an abelian group to establish a one-to one correspondence of the class of difference sets with specific parameters in that group with the set of all complex solutions of a specified system of polynomial equations. The correspondence also develops some tests for a Boolean function to be a bent function.
Let $v$ be a product of at most three not necessarily distinct primes. We prove that there exists no strong external difference family with more than two subsets in abelian group $G$ of order $v$, except possibly when $G=C_p^3$ and $p$ is a prime greater than $3 \times 10^{12}$.
Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A \subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible cardinality of the difference set $A-A$, which is always greater than or equal to the smallest possible cardinality of $A+A$ and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that $G$ is a cyclic group or a vector space over a finite field. This resolves a conjecture of Bajnok and Matzke on signed sum…
Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters $(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We prove that there does not exist a $(2p,p,2p,2)$ relative difference set in any group of order $2p^2$, and an abelian $(4p,p,4p,4)$ relative difference set can only exist in the group $\Bbb{Z}_2^2\times \Bbb{Z}_3^2$. On the other hand, we construct a family of non-abelian relative difference sets with parameters $(4q,q,4q,4)$, where $q$ is an odd prime power greater th…
For a compact abelian group $G$, a corner in $G \times G$ is a triple of points $(x,y)$, $(x,y+d)$, $(x+d,y)$. The classical corners theorem of Ajtai and Szemerédi implies that for every $α> 0$, there is some $δ> 0$ such that every subset $A \subset G \times G$ of density $α$ contains a $δ$ fraction of all corners in $G \times G$, as $x,y,d$ range over $G$. Recently, Mandache proved a "popular differences" version of this result in the finite field case $G = \mathbb F_p^n$, showing that for any subset $A \subset G \times G$ of density $α$, one can fix $d \neq 0$ such that $A$ contains a large …
Partial difference sets (for short, PDSs) with parameters ($n^2$, $r(n-ε)$, $εn+r^2-3εr$, $r^2-εr$) are called Latin square type (respectively negative Latin square type) PDSs if $ε=1$ (respectively $ε=-1$). In this paper, we will give restrictions on the parameter $r$ of a (negative) Latin square type partial difference set in an abelian group of non-prime power order. As far as we know no previous general restrictions on $r$ were known. Our restrictions are particularly useful when $a$ is much larger than $b$. As an application, we show that if there exists an abelian negative Latin square t…
In this paper we prove that if there is a regular Paley type partial difference set in an Abelian group $G$ of order $v$, where $v=p_1^{2k_1}p_2^{2k_2}\cdots p_n^{2k_n}$, $n\ge 2$, $p_1$, $p_2$, $\cdots$, $p_n$ are distinct odd prime numbers, then for any $1 \le i \le n$, $p_i$ is congruent to 3 modulo 4 whenever $k_i$ is odd. These new necessary conditions further limit the specific order of an Abelian group $G$ in which there can exist a Paley type partial difference set. Our result is similar to a result on Abelian Hadamard (Menon) difference sets proved by Ray-Chaudhuri and Xiang in 1997.
A partial difference set $S$ in a finite group $G$ satisfying $1 \notin S$ and $S = S^{-1}$ corresponds to an undirected strongly regular Cayley graph ${\rm Cay}(G,S)$. While the case when $G$ is abelian has been thoroughly studied, there are comparatively few results when $G$ is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups and are especially effective in groups with a nontrivial center. In particular, these results apply to $p$-groups, and we are able to rule out the existence of partial diffe…
Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, we construct the first known family of partial difference sets with negative Latin square type parameters in nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all $k < \ell$ when $\ell$ is even. Similarly, we construct partial difference sets with Latin square type parameters in the same groups for all $k$ when $\ell$ is even and for all $k<\ell$ when $\ell$ is odd. These constructions provide the first examp…
In this note we prove the non-existence of two types of partial difference sets in Abelian groups of order 216. This finalizes the classification of parameters for which a partial difference set of size at most 100 exists in an Abelian group.
This paper addresses the question whether triple arrays can be constructed from Youden squares developed from difference sets. We prove that if the difference set is abelian, then having $-1$ as multiplier is both a necessary and sufficient condition for the construction to work. Using this, we are able to give a new infinite family of triple arrays. We also give an alternative and more direct version of the construction, leaving out the intermediate step via Youden squares. This is used when we analyse the case of non-abelian difference sets, for which we prove a sufficient condition for givi…
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 $9^iv^4$ for any odd positiv…
Strongly regular Cayley graphs with Paley parameters over abelian groups of rank 2 were studied in [J.A Davis, Partial difference sets in p-groups, Arch.Math.63 (1994) 103-110; K.H Leung, S.L. Ma, Partial difference sets with Paley parameters, Bull. London Math Soc. 27 (1995) 553-564]. It was shown that such graphs exist iff the corresponding group is isomorphic to ${\mathbb Z}_{p^n} \oplus {\mathbb Z}_{p^n}$, where $p$ is an odd prime. In this paper we classify all strongly regular Cayley graphs over this group using Schur rings method. As a consequence we obtain a complete classification of …
A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over many years, although often only implicitly. We consider packings of certain Latin square type partial difference sets in abelian groups having identical parameters, the size of the collection being either the maximum possible or one smaller. We unify and extend numerous previous results in a common framework, recognizing that a particular subgroup reveals imp…
In an abelian group G, a more sums than differences (MSTD) set is a subset A of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets in finite abelian groups, extending previous results of Nathanson. The proof contains an application of a recently resolved conjecture of Alon and Kahn on the number of independent sets in a regular graph.
For a fixed prime $p$, $\mathbb F_{p}$ denotes the field with $p$ elements, and $\mathbb F_{p}^ω$ denotes the countable direct sum $\bigoplus_{n=1}^{\infty} \mathbb F_{p}$. Viewing $\mathbb F_{p}^ω$ as a countable abelian group, we construct a set $A\subseteq \mathbb F_{p}^ω$ having positive upper Banach density while the difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any $c\in \mathbb F_{p}^ω$. For $p=2$ we obtain a stronger conclusion: $A-A$ does not contain a set of the form $g+(B-B)$, where $B$ is piecewise syndetic. This construction answers negatively a va…
This chapter investigates the properties of (linear) codes in $ A_n $ lattices, the practical motivation for which is found in several communication scenarios, such as asymmetric channels, sticky-insertion channels, bit-shift channels, and permutation channels. In particular, a connection between these codes and notions of difference sets and Sidon sets in Abelian groups is demonstrated. It is shown that the $ A_n $ lattice admits a linear perfect code of radius $ 1 $ if and only if there exists an Abelian planar difference set of cardinality $ n + 1 $. Similarly, a direct link is given betwee…
This paper continues an earlier work on the structure of solutions to two classes of functional equation. Let $Z$ be a compact Abelian group and $U_1$, \ldots, $U_k \leq Z$ be closed subgroups. Given $f:Z\to\mathbb{T}$ and $w \in Z$, one defines the differenced function \[d_wf(z) := f(z+w) - f(z).\] In this notation, we shall study solutions to the system of difference equations \[d_{u_1}\cdots d_{u_k}f \equiv 0 \quad \forall (u_1,\ldots,u_k) \in \prod_{i\leq k}U_i,\] and to the zero-sum problem \[f_1 + \cdots + f_k = 0\] for functions $f_i:Z\to \mathbb{T}$ that are $U_i$-invariant for each $i…
In this paper, we present constructions of abelian Paley type group schemes by using multiplicative characters of finite fields and Arasu-Dillon-Player difference sets. The constructions produce many new Paley type group schemes that were previous unknown in our classification of Paley type group schemes in finite fields of small orders.
A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=a-b$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $Δ[G]$. We prove that for every $n\in\mathbb N$ the cyclic group $C_n$ of order $n$ has difference size $\frac{1+\sqrt{4|n|-3}}2\le Δ[C_n]\le\frac32\sqrt{n}$. If $n\ge 9$ (and $n\ge 2\cdot 10^{15}$), then $Δ[C_n]\le\frac{12}{\sqrt{73}}\sqrt{n}$ (and $Δ[C_n]<\frac2{\sqrt{3}}\sqrt{n}$). Also we calculate the dif…
Let $C(d,k)$ and $AC(d,k)$ be the largest order of a Cayley graph and a Cayley graph based on an abelian group, respectively, of degree $d$ and diameter $k$. When $k=2$, it is well-known that $C(d,2)\le d^2+1$ with equality if and only if the graph is a Moore graph. In the abelian case, we have $AC(d,2)\le \frac{d^2}{2}+d+1$. The best currently lower bound on $AC(d,2)$ is $\frac{3}{8}d^2-1.45 d^{1.525}$ for all sufficiently large $d$. In this paper, we consider the construction of large graphs of diameter $2$ using generalized difference sets. We show that $AC(d,2)\ge \frac{25}{64}d^2-2.1 d^{1…
Consider a compact Abelian group $Z$ and closed subgroups $U_1$, \ldots, $U_k \leq Z$. Let $\mathbb{T} := \mathbb{R}/\mathbb{Z}$. This paper examines two kinds of functional equation for measurable functions $Z\to \mathbb{T}$. First, given $f:Z\to \mathbb{T}$ and $w \in Z$, the resulting differenced function is \[d_wf(z) := f(z-w) - f(z).\] In this notation, we study solutions to the system of difference equations \[d_{u_1}\cdots d_{u_k}f \equiv 0 \quad \forall u_1 \in U_1,\ u_2 \in U_2,\ \ldots\ u_k \in U_k.\] Second, we study tuples of measurable functions $f_i:Z\to \mathbb{T}$ such that $f_…
Let $G$ be a finite group and $f:G \to {\mathbb C}$ be a function. For a non-empty finite subset $Y\subset G$, let $I_Y(f)$ denote the average of $f$ over $Y$. Then, $I_G(f)$ is the average of $f$ over $G$. Using the decomposition of $f$ into irreducible components of ${\mathbb C}^G$ as a representation of $G\times G$, we define non-negative real numbers $V(f)$ and $D(Y)$, each depending only on $f$, $Y$, respectively, such that an inequality of the form $|I_G(f)-I_Y(f)|\leq V(f)\cdot D(Y)$ holds. We give a lower bound of $D(Y)$ depending only on $\#Y$ and $\#G$. We show that the lower bound i…
We introduce and study the notion of the $G$-Tutte polynomial for a list $\mathcal{A}$ of elements in a finitely generated abelian group $Γ$ and an abelian group $G$, which is defined by counting the number of homomorphisms from associated finite abelian groups to $G$. The $G$-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Brändén-Moci's arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy…
The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let $M_1$, $M_2$ be $k\times k$ integer matrices, $G$ be a finite abelian group of order $N$, and $A\subseteq G^k$ with $|A|\geαN^k$. If $M_1$, $M_2$, $M_1-M_2$, and $M_1+M_2$ are automorphisms of $G^k$, is it true that there exists a popular difference $d \in G^k\setminus\{0\}$ such that \[\#\{x \in G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d \in A\} \ge (α^4-o(1))N^k.\] We show that this conjecture is false in general, but holds for $G = \mathbb{F}_p^n$ with $p$ an odd prim…
Let A be a finite subset of the integers or, more generally, of any abelian group, written additively. The set A has "more sums than differences" if |A+A|>|A-A|. A set with this property is called an MSTD set. This paper gives explicit constructions of families of MSTD sets of integers.
Strong external difference family (SEDF) and its generalizations GSEDF, BGSEDF in a finite abelian group $G$ are combinatorial designs raised by Paterson and Stinson [7] in 2016 and have applications in communication theory to construct optimal strong algebraic manipulation detection codes. In this paper we firstly present some general constructions of these combinatorial designs by using difference sets and partial difference sets in $G$. Then, as applications of the general constructions, we construct series of SEDF, GSEDF and BGSEDF in finite fields by using cyclotomic classes.
Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. One is a structure theorm for sets which are almost sum-free. If A is a subset of [N] which contains just o(N^2) triples (x,y,z) such that x + y = z then A may be written as the union of B and C, where B is sum-free and |C| = o(N). Another answers a question of Bergelson, Host and Kra. If alpha, epsilon > 0, if N > N_0(alp…
We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-γ)|S||A|$ with a real $γ\in(0,1]$, then $|A| \ge 4^{(1-1/d)γ|S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible. As a corollary, we derive an upper bound for the size of the largest independent subset of the set of popular differences of a finite subset of an abelian group. For …
Let $G$, $H$ be finite groups and let $X$ be a finite $G$-set. $G$-perfect nonlinear functions from $X$ to $H$ have been studied in several papers. They have more interesting properties than perfect nonlinear functions from $G$ itself to $H$. By introducing the concept of a $(G, H)$-related difference family of $X$, we obtain a characterization of $G$-perfect nonlinear functions on $X$. When $G$ is abelian, we characterize a $G$-difference set of $X$ by the Fourier transform on a normalized $G$-dual set $\widehat X$. We will also investigate the existence and constructions of $G$-perfect nonli…
Over the past few years, a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by Ruzsa, Tao and others, motivated by analogous results in additive combinatorics. The present work extends these earlier results to the case of random variables taking values in $\mathbb{R}^n$ or, more generally, in arbitrary locally compact and Polish abelian groups. We isolate and study a key quantity, the Ruzsa divergence between two probability distributions, and we show that its properties can be used to extend the earlier inequalities to the…
