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This monograph contributes to the existence theory of difference sets, cyclic irreducible codes and similar objects. The new method of field descent for cyclotomic integers of presribed absolute value is developed. Applications include the first substantial progress towards the Circulant Hadamard Matrix Conjecture and Ryser`s conjecture since decades. It is shown that there is no Barker sequence of length l with 13<1<4x10^(12). Finally, a conjecturally complete classification of all irreducible cyclic two-weight codes is obtained.

Cyclotomy. --- Difference sets. --- Finite geometries. --- Difference sets --- Cyclotomy --- Finite geometries --- Algebra --- Mathematical Theory --- Mathematics --- Physical Sciences & Mathematics --- Combinatorics. --- Combinatorics --- Mathematical analysis

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Geometry --- Difference sets --- Eindige meetkunden --- Ensembles de differences --- Finite geometries --- Fourier [Transformations de ] --- Fourier transformaties --- Fourier transformations --- Geometries finies --- Sets [Difference] --- Transformations [Fourier ] --- Transformations de Fourier --- Transforms [Fourier ] --- Verschilverzamelingen --- Verzamelingen [Verschil ] --- Difference sets. --- Finite geometries. --- Fourier transformations.

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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

Combinatorial designs and configurations. --- Coding theory. --- Difference sets. --- Sets, Difference --- Combinatorial analysis --- Data compression (Telecommunication) --- Digital electronics --- Information theory --- Machine theory --- Signal theory (Telecommunication) --- Computer programming --- Configurations and designs, Combinatorial --- Designs and configurations, Combinatorial

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The explanation of the formal duality of Kerdock and Preparata codes is one of the outstanding results in the field of applied algebra in the last few years. This result is related to the discovery of large sets of quad­ riphase sequences over Z4 whose correlation properties are better than those of the best binary sequences. Moreover, the correlation properties of sequences are closely related to difference properties of certain sets in (cyclic) groups. It is the purpose of this book to illustrate the connection between these three topics. Most articles grew out of lectures given at the NATO Ad­ vanced Study Institute on "Difference sets, sequences and their correlation properties". This workshop took place in Bad Windsheim (Germany) in August 1998. The editors thank the NATO Scientific Affairs Division for the generous support of this workshop. Without this support, the present collection of articles would not have been realized.

Difference sets --- Sequences (Mathematics) --- Congresses --- Combinatorics. --- Electrical engineering. --- Computer science—Mathematics. --- Signal processing. --- Image processing. --- Speech processing systems. --- Algebra. --- Field theory (Physics). --- Electrical Engineering. --- Discrete Mathematics in Computer Science. --- Signal, Image and Speech Processing. --- Field Theory and Polynomials. --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Mathematical analysis --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Electric engineering --- Engineering --- Combinatorics --- Algebra --- Difference sets. --- Sets, Difference --- Combinatorial analysis