# Topics in Algebraic Coding Theory

• W. Heise
Chapter
Part of the International Centre for Mechanical Sciences book series (CISM, volume 313)

## 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.

## Keywords

Linear Code Cyclic Code Dual Code Perfect Code Laguerre Plane
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Assmus, E. F. Jr., Mattson, H. F. Jr.: New 5-Designs. J. Comb. Theory 6 (1969) 122–151.
2. Haider, H.-R., Heise, W.: Einführung in die Kombinatorik. München, Wien: Hanser 1976 and Berlin: Akademie 1977.Google Scholar
3. Heise, W., Quattrocchi, P.: Informations-und Codierungstheorie. 2. Aufl., Berlin, Heidelberg, New York: Springer 1989.
4. Heise, W., Zehendner,: Quadratsummen in GF(p). Mitt. Math. Sem. Giessen 164 (1984) 185–189.
5. Heise, W., Kellerer, H.: Eine Verschärfung der Quadratwurzel-Schranke für Quadratische-Rest-Codes einer Länge n=-1 mod 4. J. Inf. Process. Cybern. EIK 23 (1987) 113–124.
6. Heise, W., Kellerer, H.: Über die Quadratwurzel-Schranke für Quadratische-RestCodes. J. of Geometry 31 (1988) 96–99.
7. Kaplan, M.: A Class of Cosets of Second Order Reed-Muller-Codes. To appear in Atti Sem. Mat. Fis. Univ. Modena.Google Scholar
8. Klemm, M.: Über die Wurzelschranke für das Minimalgewicht von Codes. J. Comb. Theory A36 (1984) 364–372.
9. MacWilliams, F. J., Sloane, N. J. A.: The Theory of Error-Correcting Codes. Amsterdam, New York, Oxford: North Holland 1977.
10. McEliece, R. J.: The Theory of Information and Coding. London, Don Mills, Sydney, Tokyo: Addison-Wesley 1977.
11. McEliece, R. J.: A Public-Key Cryptosystern Based on Algebraic Coding Theory. DSN Progress Report 42–44 (1978) 114–116Google Scholar
12. Newhart, D. W.: On Minimum Weight Codewords in QR Codes. J. Comb. Theory A 48 (1988) 104–119
13. Staiger, L.: On the Square-Root Bound for QR-Codes. J. of Geometry 31 (1988) 172–178.