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 n.

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…