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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 diffe...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
New Jersey :
World Scientific Publishing,
2014.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface; Contents; 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; 1.3.3.1 Permutation Polynomials; 1.3.3.2 Dickson Polynomials; 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 Functions1.7.1 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; 2.1.3 Hamming and simplex codes; 2.1.4 Subfield subcodes; 2.1.5 Reed-Muller 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 Codes4. 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; 4.9.1 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 constructions4.9.5 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 Dillon-Dobbert in construction; 4.11.6 The Gordon-Mills-Welch construction; 4.11.7 The No construction; 4.11.8 Other constructions; 4.12 Planar Difference Sets; 5. Almost Difference Sets.