Abstract
In this article, some row-cyclic error-correcting codes are shown to be ideals in group rings in which the underlying group is metacyclic. For a given underlying group, several nonequivalent codes with this structure may be generated. Each is related to a cyclic code generated in response, to the metrics associated with the underlying metacyclic group. Such codes in the same group ring are isomorphic as vector spaces but may vary greatly in weight distributions and so are nonequivalent. If the associated cyclic code is irreducible, examining the structure of its isomorphic finite field yields all nonequivalent codes with the desired structure. Several such codes have been found to have minimum distances equalling those of the best known linear codes of the same length and dimension.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brouwer, A.E. and Tom Verhoeff, An Updated Table of Minimum-Distance Bounds for Binary Linear Codes,IEEE Trans. IT, Vol. 36, pp. 662–677, (1993).
P. Delsarte, Automorphisms of Abelian Codes,Philips Res. Repts. Vol. 25, pp. 389–403, (1970).
M. Hall,The Theory of Groups, Macmillan: New York, NY, (1959).
T.W. Hungerford,Algebra, Springer-Verlag; New York, NY, (1974).
N. Jacobson,Basic Algebra I, Freeman: San Francisco, CA, (1974).
R. Keown,An Introduction to Group Representation Theory, Academic Press: Newark, NJ, (1975).
F.J. MacWilliams and N.J.A. Sloane,The Theory of Error-Correcting codes, North-Holland: Amsterdam, (1977).
W.W. Peterson and E.J. Walden, Jr.,Error-Correcting Codes, MIT Press: Cambridge, MA, (1972).
P. Piret, Good Block Codes Derived from Cyclic Codes,Electronics Letters, Vol. 10, pp. 391–392, (1974).
R.E. Sabin, Metacyclic Error-Correcting Codes, Ph.D. dissertation., University of Maryland, Baltimore, MD, (1990).
R.E. Sabin, On Determining All Codes in Semi-Simple Group Rings, inLecture Notes in Comp. Sc., 673, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer-Verlag: Berlin, pp. 279–290, (1993).
R.E. Sabin and S.J. Lomonaco, Metacyclic Error-Correcting Codes, to appear AAECC, 1984.
H.J. Zassenhaus,The Theory of Groups, Chalice, New York, (1949).
Author information
Authors and Affiliations
Additional information
Communicated by S. Vanstone
Rights and permissions
About this article
Cite this article
Sabin, R.E. On row-cyclic codes with algebraic structure. Des Codes Crypt 4, 145–155 (1994). https://doi.org/10.1007/BF01578868
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01578868