Galois rings and generalized Reed-Muller codes

This book contains two parts. Part I gives two descriptions of the Galois rings by using the algebraic number theory and the Witt ring. And some results on group ring over Galois ring are introduced. Part II presents a description of the Generalized Reed-Muller codes in a modular group algebra. A new proof of the famous result of Berman and Charpin is given: the Reed-Muller codes over a prime field can be identified with the radical powers of a modular group algebra.