Codes from difference sets / Cunsheng Ding.

Format:

Book

Author/Creator:

Ding, C. (Cunsheng), 1962- author.

Language:

English

Subjects (All):

Combinatorial designs and configurations.

Coding theory.

Difference sets.

Physical Description:

1 online resource (354 p.)

Place of Publication:

Singapore : World Scientific Publishing Company, 2015.

Language Note:

English

Summary:

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 n

Contents:

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)

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

5.1 Definitions and Properties

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

981-4619-36-1