Codes from difference sets

Includes index

Includes bibliography (p. 329-338)and references

1.Mathematical Foundations 1.1.Group Actions 1.2.The Rings Zn 1.3.Finite Fields 1.3.1.Introduction to finite fields 1.3.2.Traces, norms, and bases 1.3.3.Polynomials over finite fields 1.3.4.Additive and multiplicative characters 1.3.5.Character sums 1.4.Cyclotomy in GF(r) 1.4.1.Cyclotomy 1.4.2.Cyclotomy in GF(r) 1.5.Generalized Cyclotomy in Zn1n2 1.6.Finite Geometries 1.6.1.Projective planes 1.6.2.Affine planes 1.6.3.Projective spaces PG(m, GF(q)) 1.6.4.Affine spaces AG(m, GF(q)) 1.7.Planar Functions 1.7.1.Definitions and properties 1.7.2.Some known planar functions 1.7.3.Planar functions from semifields 1.7.4.Affine planes from planar functions 1.8.Periodic Sequences 1.8.1.The linear span 1.8.2.Correlation functions 2.Linear Codes over Finite Fields 2.1.Linear Codes 2.1.1.Linear codes over GF(q) 2.1.2.Equivalences of linear codes Contents note continued: 2.1.3.Hamming and simplex codes 2.1.4.Subfield subcodes 2.1.5.ReedMuller 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) 2.4.1.Factorization of xn - 1 over GF(q) 2.4.2.Generator and parity check polynomials 2.4.3.Idempotents of cyclic codes 2.4.4.Zeros of cyclic codes 2.4.5.Lower bounds on the minimum distance 2.4.6.BCH codes 2.4.7.Quadratic residue codes 2.4.8.Duadic codes 2.4.9.Bounds on weights in irreducible cyclic codes 2.5.A Combinatorial Approach to Cyclic Codes 3.Designs and Their Codes 3.1.Incidence Structures 3.1.1.Definitions 3.1.2.Incidence matrices 3.1.3.Isomorphisms and automorphisms 3.1.4.Linear codes of 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 Contents note continued: 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 4.9.1.Properties of skew Hadamard difference sets 4.9.2.Construction with planar functions and presemifields 4.9.3.Construction with Ree-Tifts slice sympletic spreads 4.9.4.Cyclotomic constructions 4.9.5.Construction with Dickson polynomials of order 7 4.9.6.Equivalence of skew Hadamard difference sets 4.9.7.Other constructions 4.10.Difference Sets with Twin-Prime Power Parameters 4.11.Difference Sets with Singer Parameters 4.11.1.The Singer construction 4.11.2.The HKM and Lin constructions 4.11.3.The Maschietti construction 4.11.4.Another construction 4.11.5.The DillonDobbertin construction Contents note continued: 4.11.6.The GordonMillsWelch construction 4.11.7.The No construction 4.11.8.Other constructions 4.12.Planar Difference Sets 5.Almost Difference Sets 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 6.2.1.Bent functions and Hadamard difference sets 6.2.2.Constructions of Hadamard difference sets 6.2.3.Linear codes from Hadamard difference sets Contents note continued: 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 6.5.1.Basic notations and results 6.5.2.The cyclic codes of the (almost) difference sets 6.6.Cyclic Codes of the Two-Prime (Almost) Difference Sets 6.6.1.The cyclic codes of the two-prime sets 6.6.2.Properties of the generalized cyclotomy of order 2 6.6.3.Parameters of the cyclic codes CGF(q) (D)c 6.6.4.Minimum weight of the codes CcGF(q)(D) 6.7.Cyclic Codes of Singer Difference Sets 6.8.Cyclic Codes of the Hyperoval Difference Sets 6.8.1.The Segre Case 6.8.2.The Glynn I Case 6.8.3.The Glynn II Case 6.9.Cyclic Codes of the Nk Type of Difference Sets 6.9.1.The first class of cyclic codes 6.9.2.The second class of cyclic codes 6.9.3.The third class of cyclic codes 6.9.4.The fourth class of cyclic codes 6.9.5.The fifth class of cyclic codes Contents note continued: 6.10.Cyclic Codes of DillonDobbertin Difference Sets 6.11.Cyclic Codes 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 8.3.1.Codebooks from the two-prime almost difference sets 8.3.2.Codebooks from other almost difference sets Appendix A Tables of Best Binary and Ternary Cyclic Codes A.1.Basic Notation and Symbols in the Tables A.2.Tables of Best Binary Cyclic Codes of Length up to 125 A.3.Tables of Best Ternary Cyclic Codes of Length up to 79.

This is the first monograph on codebooks and linear codes from difference sets and almost difference sets.