Abstract
Given any finite fieldF q , an (N, K) quasi cyclic code is defined as aK dimensional linear subspace ofF N q which is invariant underT nfor some integern, 0 <n ≦N, and whereT is the cyclic shift operator. Quasi cyclic codes are shown to be isomorphic to theF q [λ]-submodules ofF N q where the productμ(gl)·ν is naturally defined asμ 0 ν+μ 1νTn +...+μ m νT mnifμ(λ)= μ 0 +μ 1 +...+μ m λ m .In the case where (N/n, q)=1, all quasi cyclic codes are shown to be decomposable into the direct sum of a fixed number of indecomposable components called irreducible cyclicF q [λ]-submodules providing for the complete characterisation and enumeration of some subclasses of quasi cyclic codes including the cyclic codes, the quasi cyclic codes with a cyclic basis, the maximal and the irreducible ones. Finally a general procedure is presented which allows for the determination and characterisation of the dual of any quasi cyclic code.
Similar content being viewed by others
References
Bourbaki, N.: Modules sur les anneaux principaux. In: Algèbre, Chap. VII, Paris: Hermann 1959
Camion, P.: Codes correcteurs d'erreurs. Revue du CETHEDEC, Numéro spécial, 3ième trimestre 1966
Curtis, C. W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. New York: John Wiley 1962
Drolet, G.: Sur les codes quasi-cycliques à base cyclique. Thèse de Ph.D., Université Laval, Québec, Canada, June 1990
Jacobson, N.: Lectures in Abstract Algebra, Vol II. New York: D. Van Nostrand 1953
Kasami, T.: A Gilbert-Varshamov Bound for Quasi Cyclic Codes of Rate 1/2. IEEE Trans. Inf. Theory, p. 679, 1974
MacWilliams, F. J. Sloane, N. J. A.: The Theory of Error-Correcting Codes. New York: North Holland 1978
Massey, J. L.: Error Bounds for Tree Codes, Trellis Codes and Convolutional Codes with Encoding and Decoding Procedures. In: Coding and Complexity, Longo G. (ed) Berlin, Heidelberg New York: Springer 1976
Séguin, G. E., Huynh, H. T.: Quasi-Cyclic Codes: A Study. Report published by the Laboratoire de Radiocommunications et de Traitement du Signal, Université Laval, Québec, Canada, 1985
Solomon, G., van Tilborg, H. C. A.: A Connection between Block and Convolutional Codes. SIAM J. Appl. Math.,37, No(2), 358–369 (1989)
Viterbi, A. J.: Convolutional Codes and their Performance in Communication Systems. IEEE Trans. Comm. Tech.COM-19, 751–772 (1971)
Zigangirov, K. SR.: Some Sequential Decoding Procedures. Prob. Peredachi Inform.2, 13–25 (1966)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Conan, J., Séguin, G. Structural properties and enumeration of quasi cyclic codes. AAECC 4, 25–39 (1993). https://doi.org/10.1007/BF01270398
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01270398