Difference sets in abelian groups and their generalizations
Date
2017
Authors
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Publisher
University of Delaware
Abstract
Difference sets exist at the intersection of algebra and combinatorics, and are motivated by a practical and efficient constructive method for symmetric block designs. Many of the seminal papers on the subject, such as those by Bose [10], Hall [40], Singer [83], and Bruck [13], deal with this explicitly. More specific uses include the construction of complex vector codes satisfying the Welch bound [93]. A short 1979 treatise by Camion [14] frames the entire study of difference sets in terms of linear projective codes, and entire books on this subject have since been written as well [32]. For uses of difference sets and combinatorial designs in computer science, the paper of Colbourn and van Oorschot [20] is fairly comprehensive, and difference sets have played a large role in the field of cosmology, where they are used for special kinds of imaging [103]. More recently, applications of difference sets to signal processing [101] and quantum information and computing [78] have become active areas of research. Moore and Pollatsek [72] note that member nations of the North Atlantic Treaty Organization (NATO) have sponsored advanced study on difference sets as well. ☐ In this work we define difference sets, give several standard results on abelian difference sets, and discuss the uses of tools such as multipliers, the integral group rings, and characters to prove existence and nonexistence of various possible difference sets. We close with a brief survey of some recent work on some open conjectures (we take `recent' to mean `since the last major surveys were published', i.e., the 1990's) and new results in the last few years. ☐ We assume a basic knowledge of some requisite algebra and comfort with combinatorial manipulation, but all results and terminology specifically connected to difference sets are made explicit, regardless of their level of sophistication. We prove results which deal with difference sets explicitly or are particularly canonical, and provide sources for proofs in other cases.
Description
Keywords
Pure sciences, Combinatorial designs, Design theory, Difference sets