arXiv subjects

Laura Johnson

Publications and source records attributed to Laura Johnson.

6 recordsLinked to original sources

arXiv2025

On relative simple Heffter spaces

In this paper, we introduce the concept of a relative Heffter space which simultaneously generalizes those of relative Heffter arrays and Heffter spaces. Given a subgroup $J$ of an abelian group $G$, a relative Heffter space is a resolvable configuration whose points form a half-set of $G\setminus{J}$ and whose blocks are all zero-sum in $G$. Here we present two infinite families of relative Heffter spaces satisfying the additional condition of being simple. As a consequence, we get new results on globally simple relative Heffter arrays, on mutually orthogonal cycle decompositions and on biembeddings of cyclic cycle decompositions of the complete multipartite graph into an orientable surface.

math.CO↗

arXiv2025

Strongly Regular Graphs with Generalized Denniston and Dual Generalized Denniston Parameters

We construct two families of strongly regular Cayley graphs, or equivalently, partial difference sets, based on elementary abelian groups. The parameters of these two families are generalizations of the Denniston and the dual Denniston parameters, in contrast to the well known Latin square type and negative Latin square type parameters. The two families unify and subsume a number of existing constructions which have been presented in various contexts such as strongly regular graphs, partial difference sets, projective sets, and projective two-weight codes, notably including Denniston's seminal construction concerning maximal arcs in classical projective planes with even order. Our construction generates further momentum in this area, which recently saw exciting progress on the construction of the analogue of the famous Denniston partial difference sets in odd characteristic.

math.CO↗

arXiv2024

Beyond uniform cyclotomy

Cyclotomy, the study of cyclotomic classes and cyclotomic numbers, is an area of number theory first studied by Gauss. It has natural applications in discrete mathematics and information theory. Despite this long history, there are significant limitations to what is known explicitly about cyclotomic numbers, which limits the use of cyclotomy in applications. The main explicit tool available is that of uniform cyclotomy, introduced by Baumert, Mills and Ward in 1982. In this paper, we present an extension of uniform cyclotomy which gives a direct method for evaluating all cyclotomic numbers over $GF(q^n)$ of order dividing $(q^n-1)/(q-1)$, for any prime power $q$ and $n \geq 2$, which does not use character theory nor direct calculation in the field. This allows the straightforward evaluation of many cyclotomic numbers for which other methods are unknown or impractical. Our methods exploit connections between cyclotomy, Singer difference sets and finite geometry.

math.NT↗

arXiv2024

Additive triples in groups of odd prime order

Let $p$ be an odd prime. For nontrivial proper subsets $A,B$ of $\mathbb{Z}_p$ of cardinality $s,t$, respectively, we count the number $r(A,B,B)$ of additive triples, namely elements of the form $(a, b, a+b)$ in $A \times B \times B$. For given $s,t$, what is the spectrum of possible values for $r(A,B,B)$? In the special case $A=B$, the additive triple is called a Schur triple. Various authors have given bounds on the number $r(A,A,A)$ of Schur triples, and shown that the lower and upper bound can each be attained by a set $A$ that is an interval of $s$ consecutive elements of $\mathbb{Z}_p$. However, there are values of $p,s$ for which not every value between the lower and upper bounds is attainable. We consider here the general case where $A,B$ can be distinct. We use Pollard's generalization of the Cauchy-Davenport Theorem to derive bounds on the number $r(A,B,B)$ of additive triples. In contrast to the case $A=B$, we show that every value of $r(A,B,B)$ from the lower bound to the upper bound is attainable: each such value can be attained when $B$ is an interval of $t$ consecutive elements of $\mathbb{Z}_p$.

math.CO↗

arXiv2023

Denniston partial difference sets exist in the odd prime case

Denniston constructed partial difference sets (PDSs) with the 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 $m \geq 2, 1 \leq r < m$. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters $(p^{3m}, (p^{m+r} - p^m + p^r)(p^m-1), p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2), (p^{m+r}-p^m+p^r)(p^r-1))$ exist in all elementary abelian groups of order $p^{3m}$ for all $m \geq 2, r \in \{1, m-1\}$ where $p$ is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes.

math.CO↗

arXiv2022

Internal and external partial difference families and cyclotomy

We introduce the concept of a disjoint partial difference family (DPDF) and an external partial difference family (EPDF), a natural generalisation of the much-studied structures of disjoint difference family (DDF), external difference family (EDF) and partial difference set (PDS). We establish properties and constructions, and indicate connections to other recently-studied combinatorial structures. We show how DPDFs and EPDFs may be formed from collections of PDSs, and also present cyclotomic methods yielding DPDFs and EPDFs whose component sets are not in general PDSs. As part of this, we develop a unified framework encompassing various known constructions for cyclotomic difference structures, which also yields new results on DDFs and EDFs.

math.CO↗