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The purpose of coding theory is the design of efficient systems for the transmission of information. The mathematical treatment leads to certain finite structures: the error-correcting codes. Surprisingly problems which are interesting for the design of codes turn out to be closely related to problems studied partly earlier and independently in pure mathematics. In this book, examples of such connections are presented. The relation between lattices studied in number theory and geometry and error-correcting codes is discussed. The book provides at the same time an introduction to the theory of integral lattices and modular forms and to coding theory.
In the 2nd edition numerous corrections have been made. More basic material has been included to make the text even more self-contained. A new section on the automorphism group of the Leech lattice has been added. Some hints to new results have been incorporated. Finally, several new exercises have been added.
Content Level »Upper undergraduate
Keywords »Codierungstheorie - Even Unimodular Lattices - Leech Lattice - Number theory - algebra - coding theory
1 Lattices and Codes.- 1.1 Lattices.- 1.2 Codes.- 1.3 From Codes to Lattices.- 1.4 Root Lattices.- 1.5 Highest Root and Weyl Vector.- 2 Theta Functions and Weight Enumerators.- 2.1 The Theta Function of a Lattice.- 2.2 Modular Forms.- 2.3 The Poisson Summation Formula.- 2.4 Theta Functions as Modular Forms.- 2.5 The Eisenstein Series.- 2.6 The Algebra of Modular Forms.- 2.7 The Weight Enumerator of a Code.- 2.8 The Golay Code and the Leech Lattice.- 2.9 The MacWilliams Identity and Gleason’s Theorem.- 2.10 Quadratic Residue Codes.- 3 Even Unimodular Lattices.- 3.1 Theta Functions with Spherical Coefficients.- 3.2 Root Systems in Even Unimodular Lattices.- 3.3 Overlattices and Codes.- 3.4 The Classification of Even Unimodular Lattices of Dimension 24.- 4 The Leech Lattice.- 4.1 The Uniqueness of the Leech Lattice.- 4.2 The Sphere Covering Determined by the Leech Lattice.- 4.3 Twenty-Three Constructions of the Leech Lattice.- 4.4 Embedding the Leech Lattice in a Hyperbolic Lattice.- 4.5 Automorphism Groups.- 5 Lattices over Integers of Number Fields and Self-Dual Codes.- 5.1 Lattices over Integers of Cyclotomic Fields.- 5.2 Construction of Lattices from Codes over 𝔽p.- 5.3 Theta Functions over Number Fields.- 5.4 The Case p = 3: Ternary Codes.- 5.5 The Equation of the Tetrahedron and the Cube.- 5.6 The Case p = 5: the Icosahedral Group.- 5.7 Theta Functions as Hilbert Modular Forms (by N.-P. Skoruppa).