Modular Golomb rulers and almost difference sets
Daniel M. Gordon

TL;DR
This paper investigates the existence and construction of modular Golomb rulers and almost difference sets, extending previous results and exploring specific cases such as octic residues for certain primes.
Contribution
It extends the theory of difference sets by constructing new almost difference sets through element addition/removal and analyzing octic residues for primes.
Findings
Conditions for existence of almost difference sets identified
New constructions of almost difference sets provided
Octic residues form almost difference sets for specific primes
Abstract
A -difference set in a group of order is a subset of such that in the group ring satisfies where . In other words, the nonzero elements of all occur exactly times as differences of elements in . A -almost difference set has nonzero elements of occurring times, and the other occurring times. When , this is equivalent to a modular Golomb ruler. In this paper we investigate existence questions on these objects, and extend previous results constructing almost difference sets by adding or removing an element from a difference set. We also show for which primes the octic residues, with or without zero, form an almost difference set.
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Taxonomy
Topicsgraph theory and CDMA systems
