Modular Golomb Rulers and Almost Difference Sets
Daniel M. Gordon · IEEE Transactions on Information Theory · 2025
A (v, k, λ)-difference set in a groupGof ordervis a subset {d1,d2, . . . ,dk} ofGsuch thatD= Σdiin the group ring Z[G] satisfiesDD−1=n+ λG, wheren = k− λ. In other words, the nonzero elements ofGall occur exactly λ times as differences of elements inD. A (v, k, λ, t)-almost difference set has t nonzero elements ofGoccurring λ times, and the otherv− 1 −toccurring λ + 1 times. When λ = 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.