Year 2020, Volume 7 , Issue 1, Pages 73 - 84 2020-02-29

Constructions of MDS convolutional codes using superregular matrices

Julia LİEB [1] , Raquel PİNTO [2]


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.
Convolutional codes, MDS codes, Superregular matrices
  • [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.
Primary Language en
Subjects Engineering
Journal Section Articles
Authors

Orcid: 0000-0003-4211-1596
Author: Julia LİEB (Primary Author)

Orcid: 0000-0002-8168-4023
Author: Raquel PİNTO

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.
Dates

Publication Date : February 29, 2020

Bibtex @research article { jacodesmath645029, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2020}, volume = {7}, pages = {73 - 84}, doi = {10.13069/jacodesmath.645029}, title = {Constructions of MDS convolutional codes using superregular matrices}, key = {cite}, author = {Li̇eb, Julia and Pi̇nto, Raquel} }
APA Li̇eb, J , Pi̇nto, R . (2020). Constructions of MDS convolutional codes using superregular matrices . Journal of Algebra Combinatorics Discrete Structures and Applications , 7 (1) , 73-84 . DOI: 10.13069/jacodesmath.645029
MLA Li̇eb, J , Pi̇nto, R . "Constructions of MDS convolutional codes using superregular matrices" . Journal of Algebra Combinatorics Discrete Structures and Applications 7 (2020 ): 73-84 <https://dergipark.org.tr/en/pub/jacodesmath/issue/51990/645029>
Chicago Li̇eb, J , Pi̇nto, R . "Constructions of MDS convolutional codes using superregular matrices". Journal of Algebra Combinatorics Discrete Structures and Applications 7 (2020 ): 73-84
RIS TY - JOUR T1 - Constructions of MDS convolutional codes using superregular matrices AU - Julia Li̇eb , Raquel Pi̇nto Y1 - 2020 PY - 2020 N1 - doi: 10.13069/jacodesmath.645029 DO - 10.13069/jacodesmath.645029 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 73 EP - 84 VL - 7 IS - 1 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.645029 UR - https://doi.org/10.13069/jacodesmath.645029 Y2 - 2019 ER -
EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Constructions of MDS convolutional codes using superregular matrices %A Julia Li̇eb , Raquel Pi̇nto %T Constructions of MDS convolutional codes using superregular matrices %D 2020 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 7 %N 1 %R doi: 10.13069/jacodesmath.645029 %U 10.13069/jacodesmath.645029
ISNAD Li̇eb, Julia , Pi̇nto, 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
AMA Li̇eb J , Pi̇nto R . Constructions of MDS convolutional codes using superregular matrices. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020; 7(1): 73-84.
Vancouver Li̇eb J , Pi̇nto R . Constructions of MDS convolutional codes using superregular matrices. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020; 7(1): 73-84.