Codes and groups. (English) Zbl 0926.94039
Pless, V. S. (ed.) et al., Handbook of coding theory. Vol. 1. Part 1: Algebraic coding. Vol. 2. Part 2: Connections, Part 3: Applications. Amsterdam: Elsevier. 1345-1440 (1998).
[For the entire collection see Zbl 0907.94001.]
The automorphism group of a code provides important information on the code. Chapter 17 of this Handbook comprises 9 sections and studies some of the relations between groups and group codes. This chapter is not self-contained, it uses the results of some previous chapters (Chapters 1, 9, 11, 16). The automorphism groups of Cauchy codes, generalized Reed-Solomon codes, extended cyclic affine-invariant codes, generalized Reed-Muller codes, extended primitive narrow sense BCH codes and generalized quadratic residue codes (as the Hamming, Kerdock and Preparata codes) are described. Given a group \(G\), find all codes whose automorphism group is related to \(G\) (equal to \(G\) or it is a subgroup of \(G\)). In particular, which codes have trivial automorphism group. Many results related to these problems can be found here. A detailed bibliography and a number of research problems are also given.
The automorphism group of a code provides important information on the code. Chapter 17 of this Handbook comprises 9 sections and studies some of the relations between groups and group codes. This chapter is not self-contained, it uses the results of some previous chapters (Chapters 1, 9, 11, 16). The automorphism groups of Cauchy codes, generalized Reed-Solomon codes, extended cyclic affine-invariant codes, generalized Reed-Muller codes, extended primitive narrow sense BCH codes and generalized quadratic residue codes (as the Hamming, Kerdock and Preparata codes) are described. Given a group \(G\), find all codes whose automorphism group is related to \(G\) (equal to \(G\) or it is a subgroup of \(G\)). In particular, which codes have trivial automorphism group. Many results related to these problems can be found here. A detailed bibliography and a number of research problems are also given.
Reviewer: P.Lakatos (Debrecen)
MSC:
| 94B60 | Other types of codes |
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
| 94-02 | Research exposition (monographs, survey articles) pertaining to information and communication theory |
| 05E20 | Group actions on designs, etc. (MSC2000) |
| 94B05 | Linear codes (general theory) |
| 94B15 | Cyclic codes |
