Input-State-Output Representation of Convolutional Product Codes

  • Joan-Josep ClimentEmail author
  • Victoria Herranz
  • Carmen Perea
Conference paper
Part of the CIM Series in Mathematical Sciences book series (CIMSMS, volume 3)


In this paper, we present an input-state-output representation of a convolutional product code; we show that this representation is non minimal. Moreover, we introduce a lower bound on the free distance of the convolutional product code in terms of the free distance of the constituent codes.


Convolutional code Product code ISO representation Free distance Kronecker product 



This work was partially supported by Spanish grant MTM2011-24858 of the Ministerio de Ciencia e Innovacin of the Gobierno de Espaa.


  1. 1.
    Bossert, M., Medina, C., Sidorenko, V.: Encoding and distance estimation of product convolutional codes. In: Proceedings of the 2005 IEEE International Symposium on Information Theory (ISIT 2005), Adelaide, pp. 1063–1066. IEEE (2005). doi:10.1109/ISIT.2005.1523502
  2. 2.
    Climent, J.J., Herranz, V., Perea, C.: A first approximation of concatenated convolutional codes from linear systems theory viewpoint. Linear Algebr. Appl. 425, 673–699 (2007). doi:10.1016/j.laa.2007.03.017 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Elias, P.: Error free coding. Trans. IRE Prof. Group Inf. Theory 4(4), 29–37 (1954)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Herranz, V.: Estudio y construccin de cdigos convolucionales: Cdigos perforados, cdigos concatenados desde el punto de vista de sistemas. Ph.D. thesis, Departamento de Estadstica, Matemticas e Informtica, Universidad Miguel Hernndez, Elche (2007)Google Scholar
  5. 5.
    Hutchinson, R., Rosenthal, J., Smarandache, R.: Convolutional codes with maximum distance profile. Syst. Control Lett. 54(1), 53–63 (2005). doi:10.1016/j.sysconle.2004.06.005 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Johannesson, R., Wan, Z.X.: A linear algebra approach to minimal convolutional encoders. IEEE Trans. Inf. Theory 39(4), 1219–1233 (1993). doi:10.1109/18.243440 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Justesen, J.: New convolutional code constructions and a class of asymptotically good time-varying codes. IEEE Trans. Inf. Theory 19(2), 220–225 (1973). doi:10.1109/TIT.1973.1054983 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Justesen, J.: An algebraic construction of rate 1∕ν convolutional codes. IEEE Trans. Inf. Theory 21(5), 577–580 (1975). doi:10.1109/TIT.1975.1055436 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kalman, R.E., Falb, P.L., Arbib, M.A.: Topics in Mathematical System Theory. McGraw-Hill, New York (1969)zbMATHGoogle Scholar
  10. 10.
    Massey, J.L., Costello, D.J., Justesen, J.: Polynomial weights and code constructions. IEEE Trans. Inf. Theory 19(1), 101–110 (1973). doi:10.1109/TIT.1973.1054936 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Medina, C., Sidorenko, V.R., Zyablov, V.V.: Error exponents for product convolutional codes. Probl. Inf. Transm. 42(3), 167–182 (2006). doi:10.1134/S003294600603001X MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rosenthal, J.: An algebraic decoding algorithm for convolutional codes. Prog. Syst. Control Theory 25, 343–360 (1999). doi:10.1007/978-3-0348-8970-4_16 Google Scholar
  13. 13.
    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, New York (2001). doi:10.1007/978-1-4613-0165-3_2 CrossRefGoogle Scholar
  14. 14.
    Rosenthal, J., Smarandache, R.: Construction of convolutional codes using methods from linear systems theory. In: Proccedings of the 35th Allerton Conference on Communications, Control and Computing, Urbana, pp. 953–960. Allerton House, Monticello (1997)Google Scholar
  15. 15.
    Rosenthal, J., Smarandache, R.: Maximum distance separable convolutional codes. Appl. Algebra Eng. Commun. Comput. 10(1), 15–32 (1999). doi:10.1007/s002000050120 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rosenthal, J., York, E.V.: BCH convolutional codes. IEEE Trans. Inf. Theory 45(6), 1833–1844 (1999). doi:10.1109/18.782104 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rosenthal, J., Schumacher, J.M., York, E.V.: On behaviors and convolutional codes. IEEE Trans. Inf. Theory 42(6), 1881–1891 (1996). doi:10.1109/18.556682 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sripimanwat, K. (ed.): Turbo Code Applications. A Journey from a Paper to Realization. Springer, Dordrecht (2005). doi:10.1007/1-4020-3685-X Google Scholar
  19. 19.
    Tanner, R.M.: Convolutional codes from quasicyclic codes: a link between the theories of block and convolutional codes. Technical Report USC-CRL-87-21, University of California, Santa Cruz (1987)Google Scholar
  20. 20.
    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, Notre Dame (1997)Google Scholar
  21. 21.
    Zyablov, V., Shavgulidze, S., Skopintsev, O., Hst, S., Johannesson, R.: On the error exponent for woven convolutional codes with outer warp. IEEE Trans. Inf. Theory 45(5), 1649–1653 (1999). doi:10.1109/18.771237 CrossRefzbMATHGoogle Scholar
  22. 22.
    Zyablov, V.V., Shavgulidze, S., Johannesson, R.: On the error exponent for woven convolutional codes with inner warp. IEEE Trans. Inf. Theory 47(3), 1195–1199 (2001). doi:10.1109/18.915681 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Joan-Josep Climent
    • 1
    Email author
  • Victoria Herranz
    • 2
  • Carmen Perea
    • 2
  1. 1.Departament d’Estadística i Investigació OperativaUniversitat d’AlacantAlacantSpain
  2. 2.Departamento de Estadística, Matemáticas e Informática, Centro de Investigación OperativaUniversidad Miguel Hernández de ElcheElcheSpain

Personalised recommendations