doi: 10.3934/amc.2021031
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A decoding algorithm for 2D convolutional codes over the erasure channel

1. 

Institute of Mathematics, University of Zurich

2. 

Department of Mathematics, University of Aveiro

* Corresponding author: Julia Lieb

Received  June 2020 Revised  March 2021 Early access August 2021

Two-dimensional (2D) convolutional codes are a generalization of (1D) convolutional codes, which are suitable for transmission over an erasure channel. In this paper, we present a decoding algorithm for 2D convolutional codes over such a channel by reducing the decoding process to several decoding steps applied to 1D convolutional codes. Moreover, we provide constructions of 2D convolutional codes that are specially tailored to our decoding algorithm.

Citation: Julia Lieb, Raquel Pinto. A decoding algorithm for 2D convolutional codes over the erasure channel. Advances in Mathematics of Communications, doi: 10.3934/amc.2021031
References:
[1]

P. AlmeidaD. Napp and R. Pinto, MDS 2D convolutional codes with optimal 1D horizontal projections, Des. Codes Cryptogr., 86 (2018), 285-302.  doi: 10.1007/s10623-017-0357-1.  Google Scholar

[2]

P. J. AlmeidaD. Napp and R. Pinto, Superregular matrices and applications to convolutional codes, Linear Algebra Appl., 499 (2016), 1-25.  doi: 10.1016/j.laa.2016.02.034.  Google Scholar

[3]

P. J. Almeida and J. Lieb, Complete j-MDP convolutional codes, IEEE Trans. Inform. Theory, 66 (2020), 7348-7359.  doi: 10.1109/TIT.2020.3015698.  Google Scholar

[4]

J. ClimentD. NappC. Perea and R. Pinto, A construction of MDS 2D convolutional codes of rate 1/n based on superregular matrices, Linear Algebra Appl., 437 (2012), 766-780.  doi: 10.1016/j.laa.2012.02.032.  Google Scholar

[5]

J. ClimentD. NappC. Perea and R. Pinto, Maximum distance seperable 2D convolutional codes, IEEE Trans. Inform. Theory, 62 (2016), 669-680.  doi: 10.1109/TIT.2015.2509075.  Google Scholar

[6]

J. ClimentD. NappR. Pinto and R. Simoes, Decoding of 2D convolutional codes over an erasure channel, Adv. Math. Commun., 10 (2016), 179-193.  doi: 10.3934/amc.2016.10.179.  Google Scholar

[7]

E. Fornasini and R. Pinto, Matrix fraction descriptions in convolutional coding, Linear Algebra Appl., 392 (2004), 119-158.  doi: 10.1016/j.laa.2004.06.007.  Google Scholar

[8]

E. Fornasini and M. E. Valcher, Algebraic aspects of two-dimensional convolutional codes, IEEE Trans. Inform. Theory, 40 (1994), 1068-1082.  doi: 10.1109/18.335967.  Google Scholar

[9]

H. Gluesing-LuerssenJ. Rosenthal and R. Smarandache, Strongly-MDS convolutional codes, IEEE Trans. Inform. Theory, 52 (2006), 584-598.  doi: 10.1109/TIT.2005.862100.  Google Scholar

[10]

R. HutchinsonJ. Rosenthal and R. Smarandache, Convolutional codes with maximum distance profile, Systems Control Lett., 54 (2005), 53-63.  doi: 10.1016/j.sysconle.2004.06.005.  Google Scholar

[11]

P. Jangisarakul and C. Charoenlarpnopparut, Decoding of $2$-d convolutional codes based on algebraic approach, International Journal of Pure and Applied Mathematics, 97 (2014), 21-30.  doi: 10.12732/ijpam.v97i1.3.  Google Scholar

[12]

J. Lieb, Complete MDP convolutional codes, J. Algebra Appl., 18 (2019), 1950105. doi: 10.1142/S0219498819501056.  Google Scholar

[13]

J. Lieb, R. Pinto and J. Rosenthal, Convolutional codes, in Concise Encyclopedia of Coding Theory (eds. C. Huffman, J. Kim, P. Sole), CRC Press, 2021. Google Scholar

[14]

R. LoboD. L. Blitzer and M. A. Vouk, Locally invertible multidimensional convolutional encoders, IEEE Trans. Inform. Theory, 58 (2012), 1774-1782.  doi: 10.1109/TIT.2011.2178129.  Google Scholar

[15]

D. NappC. Perea and R. Pinto, Input-state-output representations and constructions of finite-support 2d convolutional codes, Adv. Math. Commun., 4 (2010), 533-545.  doi: 10.3934/amc.2010.4.533.  Google Scholar

[16]

V. Strassen, Gaussian elimination is not optimal, Numer. Math., 13 (1969) 354–356. doi: 10.1007/BF02165411.  Google Scholar

[17]

V. TomasJ. Rosenthal and R. Smarandache, Decoding of convolutional codes Over the erasure channel, IEEE Trans. Inform. Theory, 58 (2012), 90-108.  doi: 10.1109/TIT.2011.2171530.  Google Scholar

[18]

P. A. Weiner, Multidimensional Convolutional Codes, Thesis (Ph.D.) University of Notre Dame. 1998.  Google Scholar

[19]

E. V. York, Algebraic Description and Construction of Error Correcting Codes: A Linear Systems Point of View, Thesis (Ph.D.) University of Notre Dame. 1997.  Google Scholar

show all references

References:
[1]

P. AlmeidaD. Napp and R. Pinto, MDS 2D convolutional codes with optimal 1D horizontal projections, Des. Codes Cryptogr., 86 (2018), 285-302.  doi: 10.1007/s10623-017-0357-1.  Google Scholar

[2]

P. J. AlmeidaD. Napp and R. Pinto, Superregular matrices and applications to convolutional codes, Linear Algebra Appl., 499 (2016), 1-25.  doi: 10.1016/j.laa.2016.02.034.  Google Scholar

[3]

P. J. Almeida and J. Lieb, Complete j-MDP convolutional codes, IEEE Trans. Inform. Theory, 66 (2020), 7348-7359.  doi: 10.1109/TIT.2020.3015698.  Google Scholar

[4]

J. ClimentD. NappC. Perea and R. Pinto, A construction of MDS 2D convolutional codes of rate 1/n based on superregular matrices, Linear Algebra Appl., 437 (2012), 766-780.  doi: 10.1016/j.laa.2012.02.032.  Google Scholar

[5]

J. ClimentD. NappC. Perea and R. Pinto, Maximum distance seperable 2D convolutional codes, IEEE Trans. Inform. Theory, 62 (2016), 669-680.  doi: 10.1109/TIT.2015.2509075.  Google Scholar

[6]

J. ClimentD. NappR. Pinto and R. Simoes, Decoding of 2D convolutional codes over an erasure channel, Adv. Math. Commun., 10 (2016), 179-193.  doi: 10.3934/amc.2016.10.179.  Google Scholar

[7]

E. Fornasini and R. Pinto, Matrix fraction descriptions in convolutional coding, Linear Algebra Appl., 392 (2004), 119-158.  doi: 10.1016/j.laa.2004.06.007.  Google Scholar

[8]

E. Fornasini and M. E. Valcher, Algebraic aspects of two-dimensional convolutional codes, IEEE Trans. Inform. Theory, 40 (1994), 1068-1082.  doi: 10.1109/18.335967.  Google Scholar

[9]

H. Gluesing-LuerssenJ. Rosenthal and R. Smarandache, Strongly-MDS convolutional codes, IEEE Trans. Inform. Theory, 52 (2006), 584-598.  doi: 10.1109/TIT.2005.862100.  Google Scholar

[10]

R. HutchinsonJ. Rosenthal and R. Smarandache, Convolutional codes with maximum distance profile, Systems Control Lett., 54 (2005), 53-63.  doi: 10.1016/j.sysconle.2004.06.005.  Google Scholar

[11]

P. Jangisarakul and C. Charoenlarpnopparut, Decoding of $2$-d convolutional codes based on algebraic approach, International Journal of Pure and Applied Mathematics, 97 (2014), 21-30.  doi: 10.12732/ijpam.v97i1.3.  Google Scholar

[12]

J. Lieb, Complete MDP convolutional codes, J. Algebra Appl., 18 (2019), 1950105. doi: 10.1142/S0219498819501056.  Google Scholar

[13]

J. Lieb, R. Pinto and J. Rosenthal, Convolutional codes, in Concise Encyclopedia of Coding Theory (eds. C. Huffman, J. Kim, P. Sole), CRC Press, 2021. Google Scholar

[14]

R. LoboD. L. Blitzer and M. A. Vouk, Locally invertible multidimensional convolutional encoders, IEEE Trans. Inform. Theory, 58 (2012), 1774-1782.  doi: 10.1109/TIT.2011.2178129.  Google Scholar

[15]

D. NappC. Perea and R. Pinto, Input-state-output representations and constructions of finite-support 2d convolutional codes, Adv. Math. Commun., 4 (2010), 533-545.  doi: 10.3934/amc.2010.4.533.  Google Scholar

[16]

V. Strassen, Gaussian elimination is not optimal, Numer. Math., 13 (1969) 354–356. doi: 10.1007/BF02165411.  Google Scholar

[17]

V. TomasJ. Rosenthal and R. Smarandache, Decoding of convolutional codes Over the erasure channel, IEEE Trans. Inform. Theory, 58 (2012), 90-108.  doi: 10.1109/TIT.2011.2171530.  Google Scholar

[18]

P. A. Weiner, Multidimensional Convolutional Codes, Thesis (Ph.D.) University of Notre Dame. 1998.  Google Scholar

[19]

E. V. York, Algebraic Description and Construction of Error Correcting Codes: A Linear Systems Point of View, Thesis (Ph.D.) University of Notre Dame. 1997.  Google Scholar

Table1 
$ \hat{v}_{ij} $ $ j=0 $ $ j=1 $ $ j=2 $ $ j=3 $ $ j=4 $
$ i=0 $ $ \ast $ $ v_{01,1} $ $ \ast $ $ \ast $ $ \ast $
$ \ast $ $ v_{01,2} $ $ \ast $ $ \ast $ $ \ast $
$ \ast $ $ v_{01,3} $ $ \ast $ $ v_{03,3} $ $ \ast $
$ i=1 $ $ \ast $ $ \ast $ $ v_{12,1} $ $ \ast $ $ v_{14,1} $
$ \ast $ $ \ast $ $ v_{12,2} $ $ \ast $ $ v_{14,2} $
$ \ast $ $ \ast $ $ v_{12,3} $ $ v_{13,3} $ $ v_{14,3} $
$ i=2 $ $ v_{20,1} $ $ \ast $ $ \ast $ $ \ast $ $ v_{24,1} $
$ v_{20,2} $ $ \ast $ $ \ast $ $ \ast $ $ v_{24,2} $
$ v_{20,3} $ $ v_{21,3} $ $ \ast $ $ v_{23,3} $ $ v_{24,3} $
$ i=3 $ $ \ast $ $ v_{31,1} $ $ \ast $ $ \ast $ $ v_{34,1} $
$ \ast $ $ v_{31,2} $ $ \ast $ $ \ast $ $ v_{34,2} $
$ \ast $ $ v_{31,3} $ $ \ast $ $ v_{33,3} $ $ v_{34,3} $
$ i=4 $ $ \ast $ $ v_{41,1} $ $ \ast $ $ \ast $ $ v_{44,1} $
$ \ast $ $ v_{41,2} $ $ \ast $ $ \ast $ $ v_{44,2} $
$ \ast $ $ v_{41,3} $ $ \ast $ $ v_{43,3} $ $ v_{44,3} $
$ \hat{v}_{ij} $ $ j=0 $ $ j=1 $ $ j=2 $ $ j=3 $ $ j=4 $
$ i=0 $ $ \ast $ $ v_{01,1} $ $ \ast $ $ \ast $ $ \ast $
$ \ast $ $ v_{01,2} $ $ \ast $ $ \ast $ $ \ast $
$ \ast $ $ v_{01,3} $ $ \ast $ $ v_{03,3} $ $ \ast $
$ i=1 $ $ \ast $ $ \ast $ $ v_{12,1} $ $ \ast $ $ v_{14,1} $
$ \ast $ $ \ast $ $ v_{12,2} $ $ \ast $ $ v_{14,2} $
$ \ast $ $ \ast $ $ v_{12,3} $ $ v_{13,3} $ $ v_{14,3} $
$ i=2 $ $ v_{20,1} $ $ \ast $ $ \ast $ $ \ast $ $ v_{24,1} $
$ v_{20,2} $ $ \ast $ $ \ast $ $ \ast $ $ v_{24,2} $
$ v_{20,3} $ $ v_{21,3} $ $ \ast $ $ v_{23,3} $ $ v_{24,3} $
$ i=3 $ $ \ast $ $ v_{31,1} $ $ \ast $ $ \ast $ $ v_{34,1} $
$ \ast $ $ v_{31,2} $ $ \ast $ $ \ast $ $ v_{34,2} $
$ \ast $ $ v_{31,3} $ $ \ast $ $ v_{33,3} $ $ v_{34,3} $
$ i=4 $ $ \ast $ $ v_{41,1} $ $ \ast $ $ \ast $ $ v_{44,1} $
$ \ast $ $ v_{41,2} $ $ \ast $ $ \ast $ $ v_{44,2} $
$ \ast $ $ v_{41,3} $ $ \ast $ $ v_{43,3} $ $ v_{44,3} $
Table2 
$ \hat{v}_{ij} $ $ j=0 $ $ j=1 $ $ j=2 $ $ j=3 $ $ j=4 $ $ j=5 $ $ j=6 $
$ i=0 $ $ v_{00,1} $ $ \ast $ $ v_{02,1} $ $ \ast $ $ v_{04,1} $ $ v_{05,1} $ $ v_{06,1} $
$ v_{00,2} $ $ \ast $ $ v_{02,2} $ $ \ast $ $ \ast $ $ v_{05,2} $ $ v_{06,2} $
$ i=1 $ $ v_{10,1} $ $ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{15,1} $ $ v_{16,1} $
$ v_{10,2} $ $ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{15,2} $ $ v_{16,2} $
$ i=2 $ $ \ast $ $ v_{21,1} $ $ v_{22,1} $ $ \ast $ $ \ast $ $ v_{25,1} $ $ v_{26,1} $
$ \ast $ $ v_{21,2} $ $ v_{22,2} $ $ \ast $ $ \ast $ $ v_{25,2} $ $ v_{26,2} $
$ i=3 $ $ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{34,1} $ $ v_{35,1} $ $ \ast $
$ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{34,2} $ $ v_{35,2} $ $ \ast $
$ i=4 $ $ v_{40,1} $ $ v_{41,1} $ $ v_{42,1} $ $ \ast $ $ \ast $ $ v_{45,1} $ $ v_{46,1} $
$ v_{40,2} $ $ v_{41,2} $ $ v_{42,2} $ $ \ast $ $ \ast $ $ v_{45,2} $ $ v_{46,2} $
$ i=5 $ $ v_{50,1} $ $ v_{51,1} $ $ v_{52,1} $ $ \ast $ $ \ast $ $ v_{55,1} $ $ v_{56,1} $
$ v_{50,2} $ $ v_{51,2} $ $ v_{52,2} $ $ \ast $ $ \ast $ $ v_{55,2} $ $ v_{56,2} $
$ i=6 $ $ v_{60,1} $ $ v_{61,1} $ $ v_{62,1} $ $ \ast $ $ \ast $ $ v_{65,1} $ $ v_{66,1} $
$ v_{60,2} $ $ v_{61,2} $ $ v_{62,2} $ $ \ast $ $ \ast $ $ v_{65,2} $ $ v_{66,2} $
$ \hat{v}_{ij} $ $ j=0 $ $ j=1 $ $ j=2 $ $ j=3 $ $ j=4 $ $ j=5 $ $ j=6 $
$ i=0 $ $ v_{00,1} $ $ \ast $ $ v_{02,1} $ $ \ast $ $ v_{04,1} $ $ v_{05,1} $ $ v_{06,1} $
$ v_{00,2} $ $ \ast $ $ v_{02,2} $ $ \ast $ $ \ast $ $ v_{05,2} $ $ v_{06,2} $
$ i=1 $ $ v_{10,1} $ $ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{15,1} $ $ v_{16,1} $
$ v_{10,2} $ $ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{15,2} $ $ v_{16,2} $
$ i=2 $ $ \ast $ $ v_{21,1} $ $ v_{22,1} $ $ \ast $ $ \ast $ $ v_{25,1} $ $ v_{26,1} $
$ \ast $ $ v_{21,2} $ $ v_{22,2} $ $ \ast $ $ \ast $ $ v_{25,2} $ $ v_{26,2} $
$ i=3 $ $ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{34,1} $ $ v_{35,1} $ $ \ast $
$ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{34,2} $ $ v_{35,2} $ $ \ast $
$ i=4 $ $ v_{40,1} $ $ v_{41,1} $ $ v_{42,1} $ $ \ast $ $ \ast $ $ v_{45,1} $ $ v_{46,1} $
$ v_{40,2} $ $ v_{41,2} $ $ v_{42,2} $ $ \ast $ $ \ast $ $ v_{45,2} $ $ v_{46,2} $
$ i=5 $ $ v_{50,1} $ $ v_{51,1} $ $ v_{52,1} $ $ \ast $ $ \ast $ $ v_{55,1} $ $ v_{56,1} $
$ v_{50,2} $ $ v_{51,2} $ $ v_{52,2} $ $ \ast $ $ \ast $ $ v_{55,2} $ $ v_{56,2} $
$ i=6 $ $ v_{60,1} $ $ v_{61,1} $ $ v_{62,1} $ $ \ast $ $ \ast $ $ v_{65,1} $ $ v_{66,1} $
$ v_{60,2} $ $ v_{61,2} $ $ v_{62,2} $ $ \ast $ $ \ast $ $ v_{65,2} $ $ v_{66,2} $
Table3 
$ \hat{v}_{ij} $ $ j=0 $ $ \cdots $ $ j=\deg_{z_1}(v(z_1,z_2)) $
$ i=0 $ $ \ast $ $ \cdots $ $ \ast $
$ \vdots $ $ \vdots $ $ \vdots $
$ i=(L_1+1)(n-k) $ $ \ast $ $ \cdots $ $ \ast $
$ i=(L_1+1)(n-k)+1 $ $ v_{(L_1+1)(n-k)+1,0} $ $ \cdots $ $ v_{(L_1+1)(n-k)+1,\deg_{z_1}(v(z_1,z_2))} $
$ \vdots $ $ \vdots $ $ \vdots $
$ i=(L_1+1)n $ $ v_{(L_1+1)n,0} $ $ \cdots $ $ v_{(L_1+1)n,\deg_{z_1}(v(z_1,z_2))} $
$ \hat{v}_{ij} $ $ j=0 $ $ \cdots $ $ j=\deg_{z_1}(v(z_1,z_2)) $
$ i=0 $ $ \ast $ $ \cdots $ $ \ast $
$ \vdots $ $ \vdots $ $ \vdots $
$ i=(L_1+1)(n-k) $ $ \ast $ $ \cdots $ $ \ast $
$ i=(L_1+1)(n-k)+1 $ $ v_{(L_1+1)(n-k)+1,0} $ $ \cdots $ $ v_{(L_1+1)(n-k)+1,\deg_{z_1}(v(z_1,z_2))} $
$ \vdots $ $ \vdots $ $ \vdots $
$ i=(L_1+1)n $ $ v_{(L_1+1)n,0} $ $ \cdots $ $ v_{(L_1+1)n,\deg_{z_1}(v(z_1,z_2))} $
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