Codes from difference sets

This is the first monograph on codebooks and linear codes from difference sets and almost difference sets. It aims at providing a survey of constructions of difference sets and almost difference sets as well as an in-depth treatment of codebooks and linear codes from difference sets and almost difference sets. To be self-contained, this monograph covers necessary mathematical foundations and the basics of coding theory. It also contains tables of best BCH codes and best cyclic codes over GF(2) and GF(3) up to length 125 and 79, respectively. This repository of tables can be used to benchmark newly constructed cyclic codes. This monograph is intended to be a reference for postgraduates and researchers who work on combinatorics, or coding theory, or digital communications.

1. Mathematical foundations. 1.1. Group actions. 1.2. The rings Zn. 1.3. Finite fields. 1.4. Cyclotomy in GF(r). 1.5. Generalized cyclotomy in Zn[symbols]. 1.6. Finite geometries. 1.7. Planar functions. 1.8. Periodic sequences -- 2. Linear codes over finite fields. 2.1. Linear codes. 2.2. Bounds on the size of linear codes. 2.3. Bounds on the size of constant weight codes. 2.4. Cyclic codes over GF(q). 2.5. A combinatorial approach to cyclic codes -- 3. Designs and their codes. 3.1. Incidence structures. 3.2. t-designs and their codes. 3.3. t-adesigns and their codes -- 4. Difference sets. 4.1. Fundamentals of difference sets. 4.2 Divisible and relative difference sets. 4.3. Characteristic sequence of difference sets in Zn. 4.4. Characteristic functions of difference sets. 4.5. Cyclotomic difference sets. 4.6. Twin-prime difference sets. 4.7. McFarland difference sets and variations. 4.8. Menon difference sets. 4.9. Skew Hadamard difference sets. 4.10. Difference sets with twin-prime power parameters. 4.11. Difference sets with Singer parameters. 4.12. Planar difference sets -- 5. Almost difference sets. 5.1. Definitions and properties. 5.1. Definitions and properties. 5.2 Charact…