Research Article
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Constructions of MDS convolutional codes using superregular matrices

Year 2020, Volume: 7 Issue: 1 (Special Issue in Algebraic Coding Theory: New Trends and Its Connections), 73 - 84, 29.02.2020
https://doi.org/10.13069/jacodesmath.645029

Abstract

Maximum distance separable convolutional codes are the codes that present best performance in error correction among all convolutional codes with certain rate and degree. In this paper, we show that taking the constant matrix coefficients of a polynomial matrix as submatrices of a superregular matrix, we obtain a column reduced generator matrix of an MDS convolutional code with a certain rate and a certain degree. We then present two novel constructions that fulfill these conditions by considering two types of superregular matrices.

Supporting Institution

Funda\c{c}\~ao para a Ci\^encia e a Tecnologia (FCT)

Project Number

UID/MAT/04106/2019

Thanks

This work was supported by Funda\c{c}\~ao para a Ci\^encia e a Tecnologia (FCT) within project UID/MAT/04106/2019 (CIDMA) and the German Research Foundation (DFG) within grant LI3103/1-1.

References

  • [1] P. J. Almeida, D. Napp, R. Pinto, A new class of superregular matrices and MDP convolutional codes, Linear Algebra Appl. 439(7) (2013) 2145–2157.
  • [2] P. J. Almeida, D. Napp, R. Pinto, Superregular matrices and applications to convolutional codes, Linear Algebra Appl. 499 (2016) 1–25.
  • [3] J. Climent, D. Napp, C. Perea, R. Pinto, A construction of MDS 2D convolutional codes of rate $1/n$ based on superregular matrices, Linear Algebra Appl. 437(3) (2012) 766–780.
  • [4] J. Climent, D. Napp, C. Perea, R. Pinto, Maximum distance seperable 2D convolutional codes, IEEE Trans. Inform. Theory 62(2) (2016) 669–680.
  • [5] G. Forney, Convolutional codes I: Algebraic structure, IEEE Transactions on Information Theory, 16(6) (1970) 720–738. Correction, Ibid., IT-17, (1971) 360.
  • [6] H. Gluesing–Luerssen, B. Langfeld, A class of one–dimensional MDS convolutional codes, J. Algebra Appl. 5(4) (2006) 505–520.
  • [7] H. Gluesing–Luerssen, J. Rosenthal, R. Smarandache, Strongly–MDS convolutional codes, IEEE Trans. Inform. Theory 52(2) (2006) 584–598.
  • [8] R. Hutchinson, J. Rosenthal, R. Smarandache, Convolutional codes with maximum distance profile, Systems & Control Letters 54 (2005) 53–63.
  • [9] J. Justesen, An algebraic construction of rate $1/{\nu}$ convolutional codes, IEEE Trans. Inform. Theory 21(5) (1975) 577–580.
  • [10] T. Kailath, Linear Systems, Englewood Cliffs, N.J.: Prentice Hall, 1980.
  • [11] J. Lieb, Complete MDP convolutional codes, J. Algebra Appl. 18(6) (2019) 1950105 (13 pages).
  • [12] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error–Correcting Codes, 6th ed. Amsterdam, The Netherlands: North–Holland, 1988.
  • [13] J. Rosenthal, R. Smarandache, Maximum distance separable convolutional codes, Appl. Algebra Engrg. Comm. Comput. 10(1) (1999) 15–32.
  • [14] R. Roth, A. Lempel, On MDS codes via Cauchy matrices, IEEE Trans. Inform. Theory 35(6) (1989) 1314–1319.
  • [15] R. Smarandache, H. Gluesing–Luerssen, J. Rosenthal, Constructions for MDS–convolutional codes, IEEE Trans. Inform. Theory 47(5) (2001) 2045–2049.
  • [16] R. Smarandache, J. Rosenthal, A state space approach for constructing MDS rate $1/n$ convolutional codes, Proceedings of the 1998 IEEE Information TheoryWorkshop on Information Theory, 116–117.
Year 2020, Volume: 7 Issue: 1 (Special Issue in Algebraic Coding Theory: New Trends and Its Connections), 73 - 84, 29.02.2020
https://doi.org/10.13069/jacodesmath.645029

Abstract

Project Number

UID/MAT/04106/2019

References

  • [1] P. J. Almeida, D. Napp, R. Pinto, A new class of superregular matrices and MDP convolutional codes, Linear Algebra Appl. 439(7) (2013) 2145–2157.
  • [2] P. J. Almeida, D. Napp, R. Pinto, Superregular matrices and applications to convolutional codes, Linear Algebra Appl. 499 (2016) 1–25.
  • [3] J. Climent, D. Napp, C. Perea, R. Pinto, A construction of MDS 2D convolutional codes of rate $1/n$ based on superregular matrices, Linear Algebra Appl. 437(3) (2012) 766–780.
  • [4] J. Climent, D. Napp, C. Perea, R. Pinto, Maximum distance seperable 2D convolutional codes, IEEE Trans. Inform. Theory 62(2) (2016) 669–680.
  • [5] G. Forney, Convolutional codes I: Algebraic structure, IEEE Transactions on Information Theory, 16(6) (1970) 720–738. Correction, Ibid., IT-17, (1971) 360.
  • [6] H. Gluesing–Luerssen, B. Langfeld, A class of one–dimensional MDS convolutional codes, J. Algebra Appl. 5(4) (2006) 505–520.
  • [7] H. Gluesing–Luerssen, J. Rosenthal, R. Smarandache, Strongly–MDS convolutional codes, IEEE Trans. Inform. Theory 52(2) (2006) 584–598.
  • [8] R. Hutchinson, J. Rosenthal, R. Smarandache, Convolutional codes with maximum distance profile, Systems & Control Letters 54 (2005) 53–63.
  • [9] J. Justesen, An algebraic construction of rate $1/{\nu}$ convolutional codes, IEEE Trans. Inform. Theory 21(5) (1975) 577–580.
  • [10] T. Kailath, Linear Systems, Englewood Cliffs, N.J.: Prentice Hall, 1980.
  • [11] J. Lieb, Complete MDP convolutional codes, J. Algebra Appl. 18(6) (2019) 1950105 (13 pages).
  • [12] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error–Correcting Codes, 6th ed. Amsterdam, The Netherlands: North–Holland, 1988.
  • [13] J. Rosenthal, R. Smarandache, Maximum distance separable convolutional codes, Appl. Algebra Engrg. Comm. Comput. 10(1) (1999) 15–32.
  • [14] R. Roth, A. Lempel, On MDS codes via Cauchy matrices, IEEE Trans. Inform. Theory 35(6) (1989) 1314–1319.
  • [15] R. Smarandache, H. Gluesing–Luerssen, J. Rosenthal, Constructions for MDS–convolutional codes, IEEE Trans. Inform. Theory 47(5) (2001) 2045–2049.
  • [16] R. Smarandache, J. Rosenthal, A state space approach for constructing MDS rate $1/n$ convolutional codes, Proceedings of the 1998 IEEE Information TheoryWorkshop on Information Theory, 116–117.
There are 16 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Julia Lieb This is me 0000-0003-4211-1596

Raquel Pinto This is me 0000-0002-8168-4023

Project Number UID/MAT/04106/2019
Publication Date February 29, 2020
Published in Issue Year 2020 Volume: 7 Issue: 1 (Special Issue in Algebraic Coding Theory: New Trends and Its Connections)

Cite

APA Lieb, J., & Pinto, R. (2020). Constructions of MDS convolutional codes using superregular matrices. Journal of Algebra Combinatorics Discrete Structures and Applications, 7(1), 73-84. https://doi.org/10.13069/jacodesmath.645029
AMA Lieb J, Pinto R. Constructions of MDS convolutional codes using superregular matrices. Journal of Algebra Combinatorics Discrete Structures and Applications. February 2020;7(1):73-84. doi:10.13069/jacodesmath.645029
Chicago Lieb, Julia, and Raquel Pinto. “Constructions of MDS Convolutional Codes Using Superregular Matrices”. Journal of Algebra Combinatorics Discrete Structures and Applications 7, no. 1 (February 2020): 73-84. https://doi.org/10.13069/jacodesmath.645029.
EndNote Lieb J, Pinto R (February 1, 2020) Constructions of MDS convolutional codes using superregular matrices. Journal of Algebra Combinatorics Discrete Structures and Applications 7 1 73–84.
IEEE J. Lieb and R. Pinto, “Constructions of MDS convolutional codes using superregular matrices”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 7, no. 1, pp. 73–84, 2020, doi: 10.13069/jacodesmath.645029.
ISNAD Lieb, Julia - Pinto, Raquel. “Constructions of MDS Convolutional Codes Using Superregular Matrices”. Journal of Algebra Combinatorics Discrete Structures and Applications 7/1 (February 2020), 73-84. https://doi.org/10.13069/jacodesmath.645029.
JAMA Lieb J, Pinto R. Constructions of MDS convolutional codes using superregular matrices. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7:73–84.
MLA Lieb, Julia and Raquel Pinto. “Constructions of MDS Convolutional Codes Using Superregular Matrices”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 7, no. 1, 2020, pp. 73-84, doi:10.13069/jacodesmath.645029.
Vancouver Lieb J, Pinto R. Constructions of MDS convolutional codes using superregular matrices. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7(1):73-84.