doi: 10.3934/amc.2022028
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Convolutional codes over finite chain rings, MDP codes and their characterization

1. 

Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

2. 

Department of Mathematics and Computer Science, Eindhoven University of Technology, De Groene Loper 5, 5612 AZ Eindhoven, the Netherlands

* Corresponding author: Gianira N. Alfarano

Received  May 2021 Revised  March 2022 Early access April 2022

In this paper, we develop the theory of convolutional codes over finite commutative chain rings. In particular, we focus on maximum distance profile (MDP) convolutional codes and we provide a characterization of these codes, generalizing the one known for fields. Moreover, we relate (reverse) MDP convolutional codes over a finite chain ring with (reverse) MDP convolutional codes over its residue field. Finally, we provide a construction of (reverse) MDP convolutional codes over finite chain rings generalizing the notion of (reverse) superregular matrices.

Citation: Gianira N. Alfarano, Anina Gruica, Julia Lieb, Joachim Rosenthal. Convolutional codes over finite chain rings, MDP codes and their characterization. Advances in Mathematics of Communications, doi: 10.3934/amc.2022028
References:
[1]

G. N. Alfarano, D. Napp, A. Neri and V. Requena, Weighted Reed-Solomon convolutional codes, preprint, (2020), arXiv: 2012.11417.

[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.

[3]

R. Baldini Filho and P. G. Farrell, Coded modulation with convolutional codes over rings, International Symposium on Coding Theory and Applications (held in Europe), (1990), 271-280.  doi: 10.1007/3-540-54303-1_138.

[4]

R. Baldini, A. Pessoa and D. Arantes, Systematic linear codes over a ring for encoded phase modulation, Int. Symposium on Inform. and Coding Theory (ISICT 87), (1987).

[5]

G. Bini and F. Flamini, Finite Commutative Rings and Their Applications, The Kluwer International Series in Engineering and Computer Science, 680. Kluwer Academic Publishers, Boston, MA, 2002. doi: 10.1007/978-1-4615-0957-8.

[6]

I. F. Blake, Codes over certain rings, Inf. Control., 20 (1972), 396-404.  doi: 10.1016/S0019-9958(72)90223-9.

[7]

I. F. Blake, Codes over integer residue rings, Inf. Control., 29 (1975), 295-300.  doi: 10.1016/S0019-9958(75)80001-5.

[8]

A. R. Calderbank and N. J. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Cryptogr., 6 (1995), 21-35.  doi: 10.1007/BF01390768.

[9]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.

[10]

M. El Oued and P. Solé, MDS convolutional codes over a finite ring, IEEE Trans. Inform. Theory, 59 (2013), 7305-7313.  doi: 10.1109/TIT.2013.2277721.

[11]

F. Fagnani and S. Zampieri, System-theoretic properties of convolutional codes over rings, IEEE Trans. Inform. Theory, 47 (2001), 2256-2274.  doi: 10.1109/18.945247.

[12]

C. FengR. W. NóbregaF. R. Kschischang and D. Silva, Communication over finite-chain-ring matrix channels, IEEE Trans. Inform. Theory, 60 (2014), 5899-5917.  doi: 10.1109/TIT.2014.2346079.

[13]

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.

[14]

A. Gruica, MDP Convolutional Codes Over $\mathbb{Z}_{p^r}$, Master's thesis, University of Zurich, 2020, https://www.math.uzh.ch/index.php?id=pmastertheses&key1=604.

[15]

T. Honold and I. Landjev, Linear codes over finite chain rings, Electron. J. Comb., 7 (2000), Research Paper 11, 22 pp.

[16]

R. Hutchinson, The existence of strongly MDS convolutional codes, SIAM J Control Optim., 47 (2008), 2812-2826.  doi: 10.1137/050638977.

[17]

M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Inform. Theory, 55 (2009), 4890-4897.  doi: 10.1109/TIT.2009.2030486.

[18]

M. KuijperR. Pinto and J. W. Polderman, The predictable degree property and row reducedness for systems over a finite ring, Linear Algebra Appl., 425 (2007), 776-796.  doi: 10.1016/j.laa.2007.04.015.

[19]

J. Lieb, Complete MDP convolutional codes, J. Algebra its Appl., 18 (2019), 1950105, 13 pp. doi: 10.1142/S0219498819501056.

[20]

J. Massey, Convolutional Codes Over Rings, Fourth Joint Swedish-Soviet International Workshop on Information Theory, 1989.

[21]

B. R. McDonald, Finite Rings with Identity, Pure and Applied Mathematics, Vol. 28. Marcel Dekker, Inc., New York, 1974.

[22]

D. NappR. Pinto and C. Rocha, Noncatastrophic convolutional codes over a finite ring, J. Algebra Appl., (2021), 2350029.  doi: 10.1142/S0219498823500299.

[23]

D. NappR. Pinto and M. Toste., On MDS convolutional codes over $\mathbb{Z}_{p^r}$, Des. Codes Cryptogr., 83 (2017), 101-114.  doi: 10.1007/s10623-016-0204-9.

[24]

D. NappR. Pinto and M. Toste, Column distances of convolutional codes over $\mathbb{Z}_{p^r}$, IEEE Trans. Inform. Theory, 65 (2019), 1063-1071.  doi: 10.1109/TIT.2018.2870436.

[25]

A. A. Nechaev, Finite rings with applications, Handbook of Algebra, Handb. Algebr., Elsevier/North-Holland, Amsterdam, 5 (2008), 213-320.  doi: 10.1016/S1570-7954(07)05005-X.

[26]

G. H. Norton and A. Sălăgean, On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Inform. Theory, 46 (2000), 1060-1067.  doi: 10.1109/18.841186.

[27]

G. H. Norton and A. Sălăgean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebra Eng. Commun. Comput., 10 (2000), 489-506.  doi: 10.1007/PL00012382.

[28]

J. RennerA. Neri and S. Puchinger, Low-rank parity-check codes over Galois rings, Des. Codes Cryptogr., 89 (2021), 351-386.  doi: 10.1007/s10623-020-00825-9.

[29]

E. Spiegel, Codes over $\mathbb{Z}_m$, revisited, Inf. Control., 37 (1978), 100-104.  doi: 10.1016/S0019-9958(78)90461-8.

[30]

V. TomásJ. 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.

[31]

M. Toste, Distance Properties of Convolutional Codes over $\mathbb{Z}_{p^r}$, PhD thesis, Universidade de Aveiro (Portugal), 2016.

show all references

References:
[1]

G. N. Alfarano, D. Napp, A. Neri and V. Requena, Weighted Reed-Solomon convolutional codes, preprint, (2020), arXiv: 2012.11417.

[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.

[3]

R. Baldini Filho and P. G. Farrell, Coded modulation with convolutional codes over rings, International Symposium on Coding Theory and Applications (held in Europe), (1990), 271-280.  doi: 10.1007/3-540-54303-1_138.

[4]

R. Baldini, A. Pessoa and D. Arantes, Systematic linear codes over a ring for encoded phase modulation, Int. Symposium on Inform. and Coding Theory (ISICT 87), (1987).

[5]

G. Bini and F. Flamini, Finite Commutative Rings and Their Applications, The Kluwer International Series in Engineering and Computer Science, 680. Kluwer Academic Publishers, Boston, MA, 2002. doi: 10.1007/978-1-4615-0957-8.

[6]

I. F. Blake, Codes over certain rings, Inf. Control., 20 (1972), 396-404.  doi: 10.1016/S0019-9958(72)90223-9.

[7]

I. F. Blake, Codes over integer residue rings, Inf. Control., 29 (1975), 295-300.  doi: 10.1016/S0019-9958(75)80001-5.

[8]

A. R. Calderbank and N. J. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Cryptogr., 6 (1995), 21-35.  doi: 10.1007/BF01390768.

[9]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.

[10]

M. El Oued and P. Solé, MDS convolutional codes over a finite ring, IEEE Trans. Inform. Theory, 59 (2013), 7305-7313.  doi: 10.1109/TIT.2013.2277721.

[11]

F. Fagnani and S. Zampieri, System-theoretic properties of convolutional codes over rings, IEEE Trans. Inform. Theory, 47 (2001), 2256-2274.  doi: 10.1109/18.945247.

[12]

C. FengR. W. NóbregaF. R. Kschischang and D. Silva, Communication over finite-chain-ring matrix channels, IEEE Trans. Inform. Theory, 60 (2014), 5899-5917.  doi: 10.1109/TIT.2014.2346079.

[13]

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.

[14]

A. Gruica, MDP Convolutional Codes Over $\mathbb{Z}_{p^r}$, Master's thesis, University of Zurich, 2020, https://www.math.uzh.ch/index.php?id=pmastertheses&key1=604.

[15]

T. Honold and I. Landjev, Linear codes over finite chain rings, Electron. J. Comb., 7 (2000), Research Paper 11, 22 pp.

[16]

R. Hutchinson, The existence of strongly MDS convolutional codes, SIAM J Control Optim., 47 (2008), 2812-2826.  doi: 10.1137/050638977.

[17]

M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Inform. Theory, 55 (2009), 4890-4897.  doi: 10.1109/TIT.2009.2030486.

[18]

M. KuijperR. Pinto and J. W. Polderman, The predictable degree property and row reducedness for systems over a finite ring, Linear Algebra Appl., 425 (2007), 776-796.  doi: 10.1016/j.laa.2007.04.015.

[19]

J. Lieb, Complete MDP convolutional codes, J. Algebra its Appl., 18 (2019), 1950105, 13 pp. doi: 10.1142/S0219498819501056.

[20]

J. Massey, Convolutional Codes Over Rings, Fourth Joint Swedish-Soviet International Workshop on Information Theory, 1989.

[21]

B. R. McDonald, Finite Rings with Identity, Pure and Applied Mathematics, Vol. 28. Marcel Dekker, Inc., New York, 1974.

[22]

D. NappR. Pinto and C. Rocha, Noncatastrophic convolutional codes over a finite ring, J. Algebra Appl., (2021), 2350029.  doi: 10.1142/S0219498823500299.

[23]

D. NappR. Pinto and M. Toste., On MDS convolutional codes over $\mathbb{Z}_{p^r}$, Des. Codes Cryptogr., 83 (2017), 101-114.  doi: 10.1007/s10623-016-0204-9.

[24]

D. NappR. Pinto and M. Toste, Column distances of convolutional codes over $\mathbb{Z}_{p^r}$, IEEE Trans. Inform. Theory, 65 (2019), 1063-1071.  doi: 10.1109/TIT.2018.2870436.

[25]

A. A. Nechaev, Finite rings with applications, Handbook of Algebra, Handb. Algebr., Elsevier/North-Holland, Amsterdam, 5 (2008), 213-320.  doi: 10.1016/S1570-7954(07)05005-X.

[26]

G. H. Norton and A. Sălăgean, On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Inform. Theory, 46 (2000), 1060-1067.  doi: 10.1109/18.841186.

[27]

G. H. Norton and A. Sălăgean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebra Eng. Commun. Comput., 10 (2000), 489-506.  doi: 10.1007/PL00012382.

[28]

J. RennerA. Neri and S. Puchinger, Low-rank parity-check codes over Galois rings, Des. Codes Cryptogr., 89 (2021), 351-386.  doi: 10.1007/s10623-020-00825-9.

[29]

E. Spiegel, Codes over $\mathbb{Z}_m$, revisited, Inf. Control., 37 (1978), 100-104.  doi: 10.1016/S0019-9958(78)90461-8.

[30]

V. TomásJ. 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.

[31]

M. Toste, Distance Properties of Convolutional Codes over $\mathbb{Z}_{p^r}$, PhD thesis, Universidade de Aveiro (Portugal), 2016.

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