Denniston \cite{D1969} constructed partial difference sets (PDS) with parameters $(2^{3m}, (2^{m+r}-2^m+2^r)(2^m-1), 2^m-2^r+(2^{m+r}-2^m+2^r)(2^r-2), (2^{m+r}-2^m+2^r)(2^r-1))$ in elementary abelian groups of order $2^{3m}$ for all $m\geq 2$ and $1 \leq r < m$. ā¦
Denniston \cite{D1969} constructed partial difference sets (PDS) with parameters $(2^{3m}, (2^{m+r}-2^m+2^r)(2^m-1), 2^m-2^r+(2^{m+r}-2^m+2^r)(2^r-2), (2^{m+r}-2^m+2^r)(2^r-1))$ in elementary abelian groups of order $2^{3m}$ for all $m\geq 2$ and $1 \leq r < m$. These PDS correspond to maximal arcs in the Desarguesian projective planes PG$(2, 2^m)$. Davis et al. \cite{DHJP2024} and also De Winter \cite{dewinter23} presented constructions of PDS with Denniston parameters $(p^{3m}, (p^{m+r}-p^m+p^r)(p^m-1), p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2), (p^{m+r}-p^m+p^r)(p^r-1))$ in elementary abelian groups of order $p^{3m}$ for all $m \geq 2$ and $r \in \{1, m-1\}$, where $p$ is an odd prime. The constructions in \cite{DHJP2024, dewinter23} are particularly intriguing, as it was shown by Ball, Blokhuis, and Mazzocca \cite{BBM1997} that no nontrivial maximal arcs in PG$(2, q^m)$ exist for any odd prime power $q$. In this paper, we show that PDS with Denniston parameters $(q^{3m}, (q^{m+r}-q^m+q^r)(q^m-1), q^m-q^r+(q^{m+r}-q^m+q^r)(q^r-2), (q^{m+r}-q^m+q^r)(q^r-1))$ exist in elementary abelian groups of order $q^{3m}$ for all $m \geq 2$ and $1 \leq r < m$, where $q$ is an arbitrary prime power.
Partial difference sets with parameters $(v,k,\lambda,\mu)=(v, (v-1)/2, (v-5)/4, (v-1)/4)$ are called Paley type partial difference sets. In this note we prove that if there exists a Paley type partial difference ā¦
Partial difference sets with parameters $(v,k,\lambda,\mu)=(v, (v-1)/2, (v-5)/4, (v-1)/4)$ 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 question: "For which odd positive integer $v > 1$, can we find a Paley type partial difference set in an abelian group of order $v$?"
Abstract Partial difference sets with parameters are called Paley type partial difference sets. In this note, we prove that if there exists a Paley type partial difference set in an ā¦
Abstract Partial difference sets with parameters 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 of order v , where v is not a prime power, then or , an odd integer. In 2010, 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 nonzero squares of a finite field, we completely answer the following question: āFor which odd positive integers , can we find a Paley type partial difference set in an abelian group of order ?ā
In this paper, when the order of $Īø$ is even, we prove that there exists no central difference sets in $A_2(m,Īø)$ and establish some non-existence results of central partial difference ā¦
In this paper, when the order of $Īø$ is even, we prove that there exists no central difference sets in $A_2(m,Īø)$ and establish some non-existence results of central partial difference sets in $A_p(m,Īø)$ with $p>2$. When the order of $Īø$ is odd, we construct central difference sets in $A_2(m,Īø)$. Furthermore, we give some reduced linking systems of difference sets in $A_2(m,Īø)$ by using the difference sets we constructed. In the case $p>2$, we construct Latin square type central partial difference sets in $A_p(m,Īø)$ by a similar method.
A partial difference set having parameters $(n^2, r(n-1), n+r^2-3r,r^2-r)$ is called a Latin square type partial difference set, while a partial difference set having parameters $(n^2, r(n+1), -n+r^2+3r,r^2+r)$ is called ā¦
A partial difference set having parameters $(n^2, r(n-1), n+r^2-3r,r^2-r)$ is called a Latin square type partial difference set, while a partial difference set having parameters $(n^2, r(n+1), -n+r^2+3r,r^2+r)$ is called a negative Latin square type partial difference set. In this paper, we generalize well-known negative Latin square type partial difference sets derived from the theory of cyclotomy. We use the partial difference sets in elementary abelian groups to generate analogous partial difference sets in nonelementary abelian groups of the form $(Z_p)^{4s} \times (Z_{p^s})^4$. It is believed that this is the first construction of negative Latin square type partial difference sets in nonelementary abelian $p$-groups where the $p$ can be any prime number. We also give a generalization of subsets of Type Q, partial difference sets consisting of one fourth of the nonidentity elements from the group, to nonelementary abelian groups. Finally, we give a similar product construction of negative Latin square type partial difference sets in the additive groups of $(F_q)^{4t+2}$ for an integer $t \geq 1$. This construction results in some new parameters of strongly regular graphs.
In this paper we prove non-existence of nontrivial partial difference sets in Abelian groups of order 8p^3, where p \geq 3 is a prime number.
In this paper we prove non-existence of nontrivial partial difference sets in Abelian groups of order 8p^3, where p \geq 3 is a prime number.
We review some existence and nonexistence results ā new and old ā on abelian difference sets. Recent surveys on difference sets can be found in Arasu (1990), Jungnickel (1992a, b), ā¦
We review some existence and nonexistence results ā new and old ā on abelian difference sets. Recent surveys on difference sets can be found in Arasu (1990), Jungnickel (1992a, b), Pott (1995), Jungnickel and Schmidt (1997), and Davis and Jedwab (1996). Standard references for difference sets are Baumert (1971), Beth et al. (1998), and Lander (1983). This article presents a flavour of the subject, by discussing some selected topics.
We review some existence and nonexistence results ā new and old ā on abelian difference sets. Recent surveys on difference sets can be found in Arasu (1990), Jungnickel (1992a, b)Pott ā¦
We review some existence and nonexistence results ā new and old ā on abelian difference sets. Recent surveys on difference sets can be found in Arasu (1990), Jungnickel (1992a, b)Pott (1995), Jungnickel and Schmidt (1997), and Davis and Jedwab (1996). Standard references for difference sets are Baumert (1971), Beth et al. (1998), and Lander (1983). This article presents a flavour of the subject, by discussing some selected topics.
Abstract As a generalization of plateaued functions on finite fields and bent functions (perfect nonlinear functions) on finite abelian groups, plateaued functions on finite abelian groups were introduced in [B. ā¦
Abstract As a generalization of plateaued functions on finite fields and bent functions (perfect nonlinear functions) on finite abelian groups, plateaued functions on finite abelian groups were introduced in [B. Xu, Plateaued functions, partial geometric difference sets, and partial geometric designs, J. Combin. Des. 27 (2019), 756ā783]. In this paper, we continue the research in the paper mentioned above. We will obtain various characterizations of plateaued functions; these characterizations establish close connections between plateaued functions and some combinatorial objects: partial geometric difference sets and related partial geometric difference families. Then we introduce the complementary matrix and Cayley matrix for a subset of a finite group and use them to characterize partial geometric difference sets. As applications, we will show how to construct directed strongly regular graphs from partial geometric difference sets and establish a natural relation between partial geometric difference sets and partial geometric designs. The tensor product of a group algebra and a cyclotomic field is an important tool for our discussions.
We extend the notion of free $p$-central groups for odd primes $p$ to the case $p=2$ by introducing a variant of the lower $p$-central series. This enables us to calculate ā¦
We extend the notion of free $p$-central groups for odd primes $p$ to the case $p=2$ by introducing a variant of the lower $p$-central series. This enables us to calculate Schur multipliers of free $p$-central groups. We also prove that for any $p$-central group the exponent of its Schur multiplier divides the exponent of the group, and determine its exponential rank.
We extend the notion of free $p$-central groups for odd primes $p$ to the case $p=2$ by introducing a variant of the lower $p$-central series. This enables us to calculate ā¦
We extend the notion of free $p$-central groups for odd primes $p$ to the case $p=2$ by introducing a variant of the lower $p$-central series. This enables us to calculate Schur multipliers of free $p$-central groups. We also prove that for any $p$-central group the exponent of its Schur multiplier divides the exponent of the group, and determine its exponential rank.
Let $G$ be a finite group. For maximal cyclic subgroups $M, N$ of $G$, we denote by $\text{d}(M,N)$ the minimum of number of elements in the set differences $M {\setminus} ā¦
Let $G$ be a finite group. For maximal cyclic subgroups $M, N$ of $G$, we denote by $\text{d}(M,N)$ the minimum of number of elements in the set differences $M {\setminus} N$ and $N {\setminus} M$. The difference number $\delta(G)$ of $G$ is defined as the maximum of $\text{d}(M,N)$ as $M$ and $N$ vary over every pair of maximal cyclic subgroups of $G$. Whereas, the power graph $\Gamma_G$ of $G$ is the undirected simple graph with vertex set $G$ and two distinct vertices are adjacent if one of them is a positive power of the other. A connected graph $\Gamma$ is said to be cyclically separable if it has a vertex set whose deletion results in a disconnected subgraph with at least two components containing cycles. In this paper, we derive a relationship between the difference number and the power graph of a group. We prove that for a finite $p$-group $G$, $\delta(G) \geq 3$ if and only if $\Gamma_G$ is cyclically separable.
In this note, we construct the irreducible characters of Suzuki p-groups of types A p (m, Īø) and C p (m, Īø, Ļµ).
In this note, we construct the irreducible characters of Suzuki p-groups of types A p (m, Īø) and C p (m, Īø, Ļµ).
Linked systems of symmetric designs are equivalent to 3-class Q-antipodal association schemes. Only one infinite family of examples is known, and this family has interesting origins and is connected to ā¦
Linked systems of symmetric designs are equivalent to 3-class Q-antipodal association schemes. Only one infinite family of examples is known, and this family has interesting origins and is connected to important applications. In this paper, we define linking systems, collections of difference sets that correspond to systems of linked designs, and we construct linking systems in a variety of nonelementary abelian groups using Galois rings, partial difference sets, and a product construction. We include some partial results in the final section.
This paper concerns difference sets in finite groups.The approach is as follows: if D is a difference set in a group G, and Ļ any character of G, Ļ(D) -Ī£Ļ>x(ĪÆ7) ā¦
This paper concerns difference sets in finite groups.The approach is as follows: if D is a difference set in a group G, and Ļ any character of G, Ļ(D) -Ī£Ļ>x(ĪÆ7) is an algebraic integer of absolute value V~n in the field of mth roots of 1, where m is the order of Ļ.Known facts about such integers and the relations which the Ļ(D) must satisfy (as Ļ varies) may yield information about D by the Fourier inversion formula.In particular, if Ļ(D) is necessarily divisible by a relatively large integer, the number of elements g of D for which Ļ(g) takes on any given value must be large; this yields some nonexistence theorems.Another theorem, which does not depend on a magnitude argument, states that if n and v are both even and Ī±, the power of 2 in v, is at least half of that in n, then G cannot have a character of order 2 Ī± , and thus G cannot be cyclic.A difference set with v = An gives rise to an Īadamard matrix; it has been conjectured that no such cyclic sets exist with v > 4.This is proved for n even by the above theorem, and is proved for various odd n by the theorems which depend on magnitude arguments.In the last section, two classes of abelian, but not cyclic, difference sets with v = in are exhibited.A subset D of a finite group G is called a difference set if every element Ī¦e of G can be represented in precisely Ī» ways as d x d^, d { e D. If Ļ is any nonprincipal character of G, we must then have | ^jdeD Ļ(d) \ -Ī/ĪĪŖ, n -k -Ī», where k is the order of D. We shall write Ļ(D) for Ī£idĪ²D lid) (as in [8]).If G is abelian and | Ļ{D) \ = V~n for some subset D and all nonprincipal characters of G, D is a difference set in G.This work originated in a search for difference sets with G cyclic of order v, and the parameters related by v = in.Because in this case every divisor of n is a divisor of v, Hall's theorem on multipliers, [5], one of the main tools in the study of difference sets, cannot be applied.The method presented here is particularly suitable for computation of difference sets if v and n have common factors.It is roughly as follows: the numbers X(D) are algebraic integers of absolute value V~n in the field of mth roots of 1, where m is the order of Ļ (as an
A point in a finite projective plane PG(2, pn), may be denoted by the symbol (xi, x2, x3), where the coordinates xi, x2, x3 are marks of a Galois field ā¦
A point in a finite projective plane PG(2, pn), may be denoted by the symbol (xi, x2, x3), where the coordinates xi, x2, x3 are marks of a Galois field of order pn, GF(pn).The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (xi, Xi, x3) and (kxh kx2, kx3) are to be thought of as the same point.The totality of points whose coordinates satisfy the equation uiXi+u2x2+u3x3 = 0, where ui, w2, u3 are marks of the GF(pn), not all zero, is called a line.The plane then consists of p2n+pn+l = q points and q lines; each line contains pn + \ points, j A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian ; it exists for every prime p and positive integer Ā», and there is only one such PG(2, pn) for a given p and Ā» (VB, p. 247, VY, p. 151).Let Ao be a point of a given PG(2, pn), and let C be a collineation of the points of the plane.(A collineation is a 1-1 transformation carrying points into points and lines into lines.)Suppose C carries A o into A\, Ax into * Presented to the Society, October 27, 1934, under a different title;
Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type ā¦
Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type parameters is constructed in nonelementary abelian groups, the groups Z 4 2 k Ć Z 4 4 l ā 4 k for all k when ā is odd and for all k < ā when ā is even. Similarly, partial difference sets with Latin square type parameters are constructed in the same groups for all k when ā is even and for all k<ā when ā is odd. These constructions provide the first example where the non-homomorphic bijection approach outlined by Hagita and Schmidt can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the partial difference sets are not isomorphic to the known graphs, and it is conjectured that the family of strongly regular graphs will be new.
Abstract Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the ā¦
Abstract Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in where ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of is a new skew Hadamard difference set in with m odd, where denotes the first kind of Dickson polynomials of order n and . The key observation in the proof is that is a planar function from to for m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all , the set is a skew Hadamard difference set in , where m is odd and . The proof is more complicated and different than that of DingāYuan skew Hadamard difference sets since is not planar in . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for by comparing the triple intersection numbers.
Abstract A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked ā¦
Abstract A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2āgroups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in nonā2āgroups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2āgroups. We give a new construction for linking systems of difference sets in 2āgroups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2āgroups, new linking systems in other 2āgroups for which no system was previously known, and the first known examples in nonabelian groups.
Abstract In this paper, we consider regular automorphism groups of graphs in the RT2 family and the DavisāXiang family and amorphic abelian Cayley schemes from these graphs. We derive general ā¦
Abstract In this paper, we consider regular automorphism groups of graphs in the RT2 family and the DavisāXiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of nonāabelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and DavisāXiang family and their amorphic abelian Cayley schemes to produce amorphic nonāabelian Cayley schemes.