Topics in Algebraic Coding Theory

Abstract

This article surveys some selected topics in algebraic coding theory and their links to geometry and cryptography.

In the first chapter the model of a discrete communication system with a noisy memoryless and stationary symmetric channel is described. It is shown, how memorylessness can approximately be achieved by the method of interleaving. To guarantee the security of data transmission through this noisy channel one uses an encoder, which concatenates every word m consisting of k information symbols with a word r consisting of k redundant symbols. Passing through the channel some of the n = k+r symbols of the codeword c = (m,r) will eventually be changed into other symbols. The maximum-likelihood-decoder receives a word y and tries to recover c by searching an admissible codeword, which differs from y in a minimum number of symbols. A good coding system uses a code with a high information rate k/n but which nevertheless is capable to correct the expected number of symbol errors. Some coding bounds show, which compromises one has to accept, when designing an optimal coding system. The implementation of the encoder and decoder is facilitated, if the used code bears a mathematical structure, i. e. if it is linear or a forteriori cyclic. The second chapter deals with some important classes of codes like Goppa-codes, Hamming- and Simplex-codes, MDS-codes and Reed-Muller-codes. In one section we show the connection between MDS-codes and Laguerre-geometry, another section describes the automorphism group of general linear codes.

The third chapter studies more en detail the quadratic residue codes.