Tiling Groups with Difference Sets

Ante Ćustić,Vedran Krčadinac,Yue Zhou

doi:10.37236/5157

Abstract

We study tilings of groups with mutually disjoint difference sets. Some necessary existence conditions are proved and shown not to be sufficient. In the case of tilings with two difference sets we show the equivalence to skew Hadamard difference sets, and prove that they must be normalized if the group is abelian. Furthermore, we present some constructions of tilings based on cyclotomy and investigate tilings consisting of Singer difference sets.

Highlights

Let G be an additively written group of order v
A (v, k, λ) difference set in G is a ksubset D ⊆ G such that every nonzero element of G can be expressed as a difference x − y
It is not possible to partition the whole group G into disjoint (v, k, λ) difference sets. This follows from the necessary existence condition λ(v − 1) = k(k − 1) when v > k > λ 1, which is assumed throughout the paper

Summary

Introduction

Let G be an additively written group of order v. It is not possible to partition the whole group G into disjoint (v, k, λ) difference sets. This follows from the necessary existence condition λ(v − 1) = k(k − 1) when v > k > λ 1, which is assumed throughout the paper. The following five difference sets are a (31, 6, 1) tiling of the cyclic group Z31: D1 = {1, 5, 11, 24, 25, 27}, D2 = {2, 10, 17, 19, 22, 23}, D3 = {3, 4, 7, 13, 15, 20}, D4 = {6, 8, 9, 14, 26, 30}, D5 = {12, 16, 18, 21, 28, 29}. Disjoint difference sets have been used to design hopping sequences for multichannel wireless networks [13].

Necessary existence conditions and small examples

Tiling with two difference sets

Tilings based on cyclotomy

Tilings using Singer difference sets