Codes from Difference Sets

This is the first monograph on code-books 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. Book jacket 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 Characteristic sequences of almost difference sets in Zn. 5.3. Characteristic functions of almost difference sets. 5.4. Cyclotomic constructions. 5.5. Generalized cyclotomic constructions. 5.6. Constructions with difference sets. 5.7. Generic constructions with planar functions. 5.8. Planar almost difference sets. 5.9. A generic construction of almost difference sets. 5.10. Comments and open problems -- 6. Linear codes of difference sets. 6.1. Basic theory of linear codes of difference sets. 6.2. Linear codes from Hadamard difference sets. 6.3. Cyclic codes of paley (almost) difference sets. 6.4. Linear codes of Hall difference sets. 6.5. Cyclic codes of cyclotomic (almost) difference sets of order 4. 6.6. Cyclic codes of the two-prime (almost) differenceSets. 6.7. Cyclic codes of Singer difference sets. 6.8. Cyclic codes of the hyperoval difference sets. 6.9. Cyclic codes of the Nk type of difference sets. 6.10. Cyclic codes of Dillon-Dobbertin difference sets. 6.11. Cyclic odes of GMW difference sets. 6.12. Cyclic codes of the HKM and Lin difference sets. 6.13. Two more constructions of codes with difference sets -- 7. Linear codes of almost difference sets. 7.1. Definitions and fundamentals. 7.2. Cyclic codes of DHL almost difference sets. 7.3. Linear codes of planar almost difference sets -- 8. Codebooks from (almost) difference sets. 8.1. A generic construction of complex codebooks. 8.2. Optimal codebooks from difference sets. 8.3. Codebooks from some almost difference sets