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Designs, Codes and Cryptography

, Volume 86, Issue 2, pp 285–302 | Cite as

MDS 2D convolutional codes with optimal 1D horizontal projections

  • Paulo Almeida
  • Diego Napp
  • Raquel Pinto
Article
Part of the following topical collections:
  1. Special Issue on Network Coding and Designs

Abstract

Two dimensional (2D) convolutional codes is a class of codes that generalizes standard one-dimensional (1D) convolutional codes in order to treat two dimensional data. In this paper we present a novel and concrete construction of 2D convolutional codes with the particular property that their projection onto the horizontal lines yield optimal [in the sense of Almeida et al. (Linear Algebra Appl 499:1–25, 2016)] 1D convolutional codes with a certain rate and certain Forney indices. Moreover, using this property we show that the proposed constructions are indeed maximum distance separable, i.e., are 2D convolutional codes having the maximum possible distance among all 2D convolutional codes with the same parameters. The key idea is to use a particular type of superregular matrices to build the generator matrix.

Keywords

2D convolutional codes Optimal codes MDS codes Superregular matrices 

Mathematics Subject Classification

94B10 15B33 11T71 

Notes

Acknowledgements

The authors are grateful to the anonymous referees for the many insightful comments. This work was 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-Fundação para a Ciência e a Tecnologia), within Project UID/MAT/04106/2013.

References

  1. 1.
    Almeida P., Napp D., Pinto R.: A new class of superregular matrices and MDP convolutional codes. Linear Algebra Appl. 439(7), 2145–2157 (2013).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Almeida P., Napp D., Pinto R.: Superregular matrices and applications to convolutional codes. Linear Algebra Appl. 499, 1–25 (2016).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Charoenlarpnopparut C., Bose N.K.: Grobner bases for problem solving in multidimensional systems. Multidimens. Syst. Signal Process. 12(3), 365–376 (2001).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Climent J.J., Napp D., Perea C., Pinto R.: A construction of MDS \(2\)D convolutional codes of rate \(1/n\) based on superregular matrices. Linear Algebra Appl. 437, 766–780 (2012).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Climent J.J., Napp D., Perea C., Pinto R.: Maximum distance separable 2D convolutional codes. IEEE Trans. Inf. Theory 62(2), 669–680 (2016).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Climent J.J., Napp D., Pinto R., Simões R.: Decoding of 2D convolutional codes over the erasure channel. Adv. Math. Commun. 10(1), 179–193 (2016).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    El Oued M., Sole P.: MDS convolutional codes over a finite ring. IEEE Trans. Inf. Theory 59(11), 7305–7313 (2013).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fornasini E., Valcher M.E.: Algebraic aspects of two-dimensional convolutional codes. IEEE Trans. Inf. Theory 40(4), 1068–1082 (1994).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gluesing-Luerssen H., Rosenthal J., Smarandache R.: Strongly MDS convolutional codes. IEEE Trans. Inf. Theory 52(2), 584–598 (2006).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hansen J., Østergaard J., Kudahl J., Madsen J.: On the construction of jointly superregular lower triangular Toeplitz matrices. International Symposium on Information Theory (ISIT) (2016).Google Scholar
  11. 11.
    Hutchinson R.: The existence of strongly MDS convolutional codes. SIAM J. Control Optim. 47(6), 2812–2826 (2008).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hutchinson R., Smarandache R., Trumpf J.: On superregular matrices and MDP convolutional codes. Linear Algebra Appl. 428, 2585–2596 (2008).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Justesen J., Forchhammer S.: Two Dimensional Information Theory and Coding. With Applications to Graphics Data and High-Density Storage Media. Cambridge University Press, Cambridge (2010).MATHGoogle Scholar
  14. 14.
    La Guardia G.: On classical and quantum MDS-convolutional BCH codes. IEEE Trans. Inf. Theory 60(1), 304–312 (2013).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lobo R.G., Bitzer D.L., Vouk M.A.: On locally invertible encoders and muldimensional convolutional codes. IEEE Trans. Inf. Theory 58(3), 1774–1782 (2012).CrossRefMATHGoogle Scholar
  16. 16.
    Mahmood R., Badr A., Khisti A.: Convolutional codes with maximum column sum rank for network streaming. IEEE Trans. Inf. Theory 62(6), 3039–3052 (2016).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    McEliece R.J.: The algebraic theory of convolutional codes. In: Pless V., Huffman W.C. (eds.) Handbook of Coding Theory, vol. 1, pp. 1065–1138. Elsevier Science Publishers, Amsterdam (1998).Google Scholar
  18. 18.
    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).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Napp D., Pinto R., Toste T.: On MDS convolutional codes over \({\mathbb{Z}}_{p^r}\). Des. Codes Cryptogr. 83, 101–114 (2017).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Norton G.: On minimal realization over a finite chain ring. Des. Codes Cryptogr. 16(2), 161–178 (1999).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ozbudak F., Ozkaya B.: A minimum distance bound for quasi-nd-cyclic codes. Finite Fields Appl. 41, 193–222 (2016).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pinho T., Pinto R., Rocha P.: Realization of 2D convolutional codes of rate \(\frac{1}{n}\) by separable Roesser models. Des. Codes Cryptogr. 70(1), 241–250 (2014).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rosenthal J., York E.V.: BCH convolutional codes. IEEE Trans. Inf. Theory 45(6), 1833–1844 (1999).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Roth R.M., Lempel A.: On MDS codes via Cauchy matrices. IEEE Trans. Inf. Theory 35(6), 1314–1319 (1989).MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Smarandache R., Gluesing-Luerssen H., Rosenthal J.: Constructions of MDS-convolutional codes. IEEE Trans. Automat. Control 47(5), 2045–2049 (2001).MathSciNetMATHGoogle Scholar
  26. 26.
    Tomás V.: Complete-MDP Convolutional Codes over the Erasure Channel. PhD thesis, Departamento de Ciencia de la Computación e Inteligencia Artificial, Universidad de Alicante, Alicante, España (2010).Google Scholar
  27. 27.
    Tomás V., Rosenthal J., Smarandache R.: Decoding of MDP convolutional codes over the erasure channel. In: Proceedings of the 2009 IEEE International Symposium on Information Theory (ISIT 2009), pp. 556–560, Seoul, Korea (2009). IEEE.Google Scholar
  28. 28.
    Tomas V., Rosenthal J., Smarandache R.: Decoding of convolutional codes over the erasure channel. IEEE Trans. Inf. Theory 58(1), 90–108 (2012).MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Valcher M.E., Fornasini E.: On 2D finite support convolutional codes: an algebraic approach. Multidimens. Syst. Signal Process. 5, 231–243 (1994).CrossRefMATHGoogle Scholar
  30. 30.
    Weiner P.: Muldimensional Convolutional Codes. PhD dissertation, University of Notre Dame, USA (1998).Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.CIDMA - Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal

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