2024/08/29 by Daniel M. Gordon, Gordon, Daniel M.
Engineering · #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2408.16721
A (v,k,λ)-difference set in a group G of order v is a subset \d1, d2, …,dk\ of G such that D=∑ di in the group ring \mathbb Z[G] satisfies D D-1 = n + λG, where n=k-λ. In other words, the nonzero elements of G all occur exactly λ times as differences of elements in D. A (v,k,λ,t)-almost difference set has t nonzero elements of G occurring λ times, and the other v-1-t occurring 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.