Modular Golomb rulers and almost difference sets

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 $(v,k,\lambda)$-difference set in a group $G$ of order $v$ is a subset $\{d_1, d_2, \ldots,d_k\}$ of $G$ such that $D=\sum d_i$ in the group ring ${\mathbb Z}[G]$ satisfies $$D D^{-1} = n + \lambda G,$$ where $n=k-\lambda$. In other words, the nonzero elements of $G$ all occur exactly $\lambda$ times as differences of elements in $D$. A $(v,k,\lambda,t)$-almost difference set has $t$ nonzero elements of $G$ occurring $\lambda$ times, and the other $v-1-t$ occurring $\lambda+1$ times. When $\lambda=0$, 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.