A Systems Theory Approach to Periodically Time-Varying Convolutional Codes by Means of Their Invariant Equivalent

  • Joan-Josep Climent
  • Victoria Herranz
  • Carmen Perea
  • Virtudes Tomás
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5527)


In this paper we construct (n,k,δ) time-variant convolutional codes of period τ. We use the systems theory to represent our codes by the input-state-output representation instead of using the generator matrix. The obtained code is controllable and observable. This construction generalizes the one proposed by Ogasahara, Kobayashi, and Hirasawa (2007). We also develop and study the properties of the time-invariant equivalent convolutional code and we show a lower bound for the free distance in the particular case of MDS block codes.


Time-varying convolutional codes controllability observability free distance 


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  1. 1.
    Allen, B.M.: Linear Systems Analysis and Decoding of Convolutional Codes. Ph.D thesis, Department of Mathematics, University of Notre Dame, Indiana, USA (June 1999)Google Scholar
  2. 2.
    Antsaklis, P.J., Michel, A.N.: Linear Systems. McGraw-Hill, New York (1997)Google Scholar
  3. 3.
    Balakirsky, V.B.: A necessary and sufficient condition for time-variant convolutional encoders to be noncatastrophic. In: Cohen, G.D., Litsyn, S., Lobstein, A., Zémor, G. (eds.) Algebraic Coding 1993. LNCS, vol. 781, pp. 1–10. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  4. 4.
    Climent, J.J., Herranz, V., Perea, C.: A first approximation of concatenated convolutional codes from linear systems theory viewpoint. Linear Algebra and its Applications 425, 673–699 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Climent, J.J., Herranz, V., Perea, C.: Linear system modelization of concatenated block and convolutional codes. Linear Algebra and its Applications 429, 1191–1212 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dholakia, A.: Introduction to Convolutional Codes with Applications. Kluwer Academic Publishers, Boston (1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    Grasselli, O.M., Longhi, S.: Finite zero structure of linear periodic discrete-time systems. International Journal of Systems Science 22(10), 1785–1806 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hautus, M.L.J.: Controllability and observability condition for linear autonomous systems. Proceedings of Nederlandse Akademie voor Wetenschappen (Series A) 72, 443–448 (1969)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Hutchinson, R., Rosenthal, J., Smarandache, R.: Convolutional codes with maximum distance profile. Systems & Control Letters 54(1), 53–63 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Johannesson, R., Zigangirov, K.S.: Fundamentals of Convolutional Coding. IEEE Press, New York (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Justesen, J.: New convolutional code constructions and a class of asymptotically good time-varying codes. IEEE Transactions on Information Theory 19(2), 220–225 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Levy, Y., Costello Jr., D.J.: An algebraic approach to constructing convolutional codes from quasicyclic codes. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 14, 189–198 (1993)CrossRefzbMATHGoogle Scholar
  13. 13.
    Massey, J.L., Costello, D.J., Justesen, J.: Polynomial weights and code constructions. IEEE Transactions on Information Theory 19(1), 101–110 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    McEliece, R.J.: The algebraic theory of convolutional codes. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, pp. 1065–1138. Elsevier, North-Holland, Amsterdam (1998)Google Scholar
  15. 15.
    Mooser, M.: Some periodic convolutional codes better than any fixed code. IEEE Transactions on Information Theory 29(5), 750–751 (1983)CrossRefzbMATHGoogle Scholar
  16. 16.
    O’Donoghue, C., Burkley, C.: Catastrophicity test for time-varying convolutional encoders. In: Walker, M. (ed.) Cryptography and Coding 1999. LNCS, vol. 1746, pp. 153–162. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  17. 17.
    Ogasahara, N., Kobayashi, M., Hirasawa, S.: The construction of periodically time-variant convolutional codes using binary linear block codes. Electronics and Communications in Japan, Part 3 90(9), 31–40 (2007)CrossRefGoogle Scholar
  18. 18.
    Rosenthal, J.: An algebraic decoding algorithm for convolutional codes. Progress in Systems and Control Theory 25, 343–360 (1999)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Rosenthal, J.: Connections between linear systems and convolutional codes. In: Marcus, B., Rosenthal, J. (eds.) Codes, Systems and Graphical Models. The IMA Volumes in Mathematics and its Applications, vol. 123, pp. 39–66. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    Rosenthal, J., Smarandache, R.: Maximum distance separable convolutional codes. Applicable Algebra in Engineering, Communication and Computing 10, 15–32 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rosenthal, J., Schumacher, J., York, E.V.: On behaviors and convolutional codes. IEEE Transactions on Information Theory 42(6), 1881–1891 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rosenthal, J., York, E.V.: BCH convolutional codes. IEEE Transactions on Information Theory 45(6), 1833–1844 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tanner, R.M.: Convolutional codes from quasicyclic codes: A link between the theories of block and convolutional codes. Technical Report USC-CRL-87-21 (November 1987)Google Scholar
  24. 24.
    Thommesen, C., Justesen, J.: Bounds on distances and error exponents of unit memory codes. IEEE Transactions on Information Theory 29(5), 637–649 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    York, E.V.: Algebraic Description and Construction of Error Correcting Codes: A Linear Systems Point of View. Ph.D thesis, Department of Mathematics, University of Notre Dame, Indiana, USA (May 1997)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Joan-Josep Climent
    • 1
  • Victoria Herranz
    • 2
  • Carmen Perea
    • 2
  • Virtudes Tomás
    • 1
  1. 1.Departament de Ciència de la Computació i Intel·ligència ArtificialUniversitat d’AlacantAlacantSpain
  2. 2.Centro de Investigación Operativa Departamento de Estadística, Matemáticas e InformáticaUniversidad Miguel HernándezAlicanteSpain

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