Burst Erasure Correction of 2D Convolutional Codes

  • Joan-Josep Climent
  • Diego Napp
  • Raquel PintoEmail author
  • Rita Simões
Conference paper
Part of the CIM Series in Mathematical Sciences book series (CIMSMS, volume 3)


In this paper we address the problem of decoding 2D convolutional codes over the erasure channel. In particular, we present a procedure to recover bursts of erasures that are distributed in a diagonal line. To this end we introduce the notion of balls around a burst of erasures which can be considered an analogue of the notion of sliding window in the context of 1D convolutional codes. The main result reduces the decoding problem of 2D convolutional codes to a problem of decoding a set of associated 1D convolutional codes.


2D convolutional codes Erasure channel 



This work was partially supported by Spanish grant MTM2011-24858 of the Ministerio de Ciencia e Innovacin of the Gobierno de Espaa. The work of D. Napp, R. Pinto, and R. Simes was partially supported by Portuguese funds through the CIDMA – Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundao para a Cincia e a Tecnologia”), within project UID/MAT/04106/2013.


  1. 1.
    Almeida, P., Napp, D., Pinto, R.: A new class of superregular matrices and MDP convolutional codes. Linear Algebra Appl. 439, 2145–2157 (2013). doi:10.1016/j.laa.2013.06.013 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arai, M., Yamamoto, A., Yamaguchi, A., Fukumoto, S., Iwasaki, K.: Analysis of using convolutional codes to recover packet losses over burst erasure channels. In: Proceedings of the 2001 Pacific Rim International Symposium on Dependable Computing, Seoul, pp. 258–265. IEEE (2001). doi:10.1109/PRDC.2001.992706
  3. 3.
    Charoenlarpnopparut, C.: Applications of Grbner bases to the structural description and realization of multidimensional convolutional code. Sci. Asia 35, 95–105 (2009). doi:10.2306/scienceasia1513-1874.2009.35.095 CrossRefGoogle Scholar
  4. 4.
    Climent, J.J., Napp, D., Perea, C., Pinto, R.: A construction of MDS 2D convolutional codes of rate 1∕n based on superregular matrices. Linear Algebra Appl. 437, 766–780 (2012). doi:10.1016/j.laa.2012.02.032 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fornasini, E., Valcher, M.E.: Algebraic aspects of two-dimensional convolutional codes. IEEE Trans. Inf. Theory 40(4), 1068–1082 (1994). doi:10.1109/18.335967 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jangisarakul, P., Charoenlarpnopparut, C.: Algebraic decoder of multidimensional convolutional code: constructive algorithms for determining syndrome decoder and decoder matrix based on Gröbner basis. Multidimens. Syst. Signal Process. 22(1–3), 67–81 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Napp, D., Perea, C., Pinto, R.: Input-state-output representations and constructions of finite support 2D convolutional codes. Adv. Math. Commun. 4(4), 533–545 (2010). doi:10.3934/amc.2010.4.533 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Tomás, V.: Complete-MDP convolutional codes over the erasure channel. Ph.D. thesis, Departamento de Ciencia de la Computación e Inteligencia Artificial, Universidad de Alicante, Alicante (2010)Google Scholar
  9. 9.
    Tomás, V., Rosenthal, J., Smarandache, R.: Reverse-maximum distance profile convolutional codes over the erasure channel. In: Edelmayer, A. (ed.) Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS2010), Budapest, pp. 2121–2127 (2010)Google Scholar
  10. 10.
    Tomás, V., Rosenthal, J., Smarandache, R.: Decoding of convolutional codes over the erasure channel. IEEE Trans. Inf. Theory 58(1), 90–108 (2012). doi:10.1109/TIT.2011.2171530 CrossRefGoogle Scholar
  11. 11.
    Weiner, P.A.: Multidimensional convolutional codes. Ph.D. thesis, Department of Mathematics, University of Notre Dame, Indiana (1998)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Joan-Josep Climent
    • 1
  • Diego Napp
    • 2
  • Raquel Pinto
    • 2
    Email author
  • Rita Simões
    • 2
  1. 1.Departament d’Estadística i Investigació OperativaUniversitat d’AlacantAlacantSpain
  2. 2.Department of Mathematics, CIDMA – Center for Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal

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