Abstract
We present an intrinsecal characterization of when a linear code C is a (left) group code, i.e. the ambient space can be identified with a group algebra in which the standard basis is the group basis such that C is a (left) ideal in this group algebra. As application we obtain a class containing properly the class of metacyclic groups such that every group code is an abelian group code. We also use the characterization to describe all the possible group structures on some classes of generalized Reed-Solomon codes.
Research supported by D.G.I. of Spain and Fundación Séneca of Murcia.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bernal, J.J., del Río, Á., Simón, J.J.: An intrinsical description of group codes in preparation
Dür, A.: The automorphism groups of Reed-Solomon codes. J. Combin. Theory Ser. A. 44, 69–82 (1987)
Evans Sabin, R., Lomonaco, S.J.: Metacyclic Error-Correcting Codes. AAECC 6, 191–210 (1995)
Huffman, W.C.: Codes and groups. In: Pless, V.S., Huffman, W.C., Brualdi, R.A. (eds.) Handbook of coding theory, vol. II, pp. 1345–1440. North-Holland, Amsterdam (1998)
Landrock, P., Manz, O.: Classical codes as ideals in group algebras. Des. Codes Cryptogr. 2, 273–285 (1992)
Kelarev, A.V., Solé, P.: Error-correcting codes as ideals in group rings. Abelian groups, rings and modules (Perth, 2000), Contemp. Math. 273, 11–18 (2001)
Phelps, K.T., Rifà, J.: On binary 1-perfect additive codes: Some structural properties. IEEE Trans. Inform. Theory 48, 2587–2592 (2002)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bernal, J.J., del Río, Á., Simón, J.J. (2008). How to Know if a Linear Code Is a Group Code?. In: Barbero, Á. (eds) Coding Theory and Applications. Lecture Notes in Computer Science, vol 5228. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87448-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-87448-5_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87447-8
Online ISBN: 978-3-540-87448-5
eBook Packages: Computer ScienceComputer Science (R0)