American Institute of Mathematical Sciences

Advanced
February  2016, 10(1): 29-43. doi: 10.3934/amc.2016.10.29

Convolutional codes with a matrix-algebra word-ambient

José Gómez-Torrecillas 1, , F. J. Lobillo 1, and  Gabriel Navarro 2,

1. 

Department of Algebra and CITIC-UGR, University of Granada, E18071 Granada
2. 

Department of Computer Sciences and AI, and CITIC, Universidad de Granada, E51001 Ceuta, Spain

Received  November 2014 Revised  July 2015 Published  March 2016

Let $\mathcal{M}_n(\mathbb{F})$be the algebra of \(n \times n\) matrices over the finite field $\mathbb{F}$. In this paper we prove that the dual code of each ideal convolutional code in the skew-polynomial ring $\mathcal{M}_n(\mathbb{F})[z;\sigma_U]$ which is a direct summand as a left ideal, is also an ideal convolutional code over $\mathcal{M}_n(\mathbb{F})[z;\sigma_UT]$ and a direct summand as a left ideal. Moreover we provide an algorithm to decide if \(\sigma_U\) is a separable automorphism and returns the corresponding separability element, when pertinent.
Keywords: skew polynomial., Convolutional code, dual code, Ideal code.
Mathematics Subject Classification: Primary: 94B10; Secondary: 94B15, 16S3.
Citation: José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Convolutional codes with a matrix-algebra word-ambient. Advances in Mathematics of Communications, 2016, 10 (1) : 29-43. doi: 10.3934/amc.2016.10.29
References:
[1]

S. Estrada, J. R. García-Rozas, J. Peralta and E. Sánchez-García, Group convolutional codes,, Adv. Math. Commun., (2008), 83. doi: 10.3934/amc.2008.2.83. Google Scholar

[2]

G. D. Forney Jr., Convolutional codes I: Algebraic structure,, IEEE Trans. Inf. Theory, 16 (1970), 720. doi: 10.1109/TIT.1970.1054541. Google Scholar

[3]

H. Gluesing-Luerssen and W. Schmale, On cyclic convolutional codes,, Acta Appl. Math., 82 (2004), 183. doi: 10.1023/B:ACAP.0000027534.61242.09. Google Scholar

[4]

J. Gómez-Torrecillas, F. J. Lobillo and G. Navarro, Ideal codes over separable ring extensions,, preprint, (). Google Scholar

[5]

J. Gómez-Torrecillas, F. J. Lobillo and G. Navarro, Cyclic convolutional codes over separable extensions,, in Coding Theory and Applications (eds. R. Pinto, (2015), 209. doi: 10.1007/978-3-319-17296-5_22. Google Scholar

[6]

K. Hirata and K. Sugano, On semisimple extensions and separable extensions over non commutative rings,, J. Math. Soc. Japan, 18 (1966), 360. doi: 10.2969/jmsj/01840360. Google Scholar

[7]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis,, Cambridge Univ. Press, (1994). doi: 10.1017/CBO9780511840371. Google Scholar

[8]

N. Jacobson, Basic Algebra: II,, W. H. Freeman Company, (1980). Google Scholar

[9]

S. R. López-Permouth and S. Szabo, Convolutional codes with additional algebraic structure,, J. Pure Appl. Algebra, 217 (2013), 958. doi: 10.1016/j.jpaa.2012.09.017. Google Scholar

[10]

R. Pierce, Associative Algebras,, Springer-Verlag, (1982). doi: 10.1007/978-1-4757-0163-0. Google Scholar

[11]

P. Piret, Structure and constructions of cyclic convolutional codes,, IEEE Trans. Inf. Theory, 22 (1976), 147. doi: 10.1109/TIT.1976.1055531. Google Scholar

show all references

References:
[1]

S. Estrada, J. R. García-Rozas, J. Peralta and E. Sánchez-García, Group convolutional codes,, Adv. Math. Commun., (2008), 83. doi: 10.3934/amc.2008.2.83. Google Scholar

[2]

G. D. Forney Jr., Convolutional codes I: Algebraic structure,, IEEE Trans. Inf. Theory, 16 (1970), 720. doi: 10.1109/TIT.1970.1054541. Google Scholar

[3]

H. Gluesing-Luerssen and W. Schmale, On cyclic convolutional codes,, Acta Appl. Math., 82 (2004), 183. doi: 10.1023/B:ACAP.0000027534.61242.09. Google Scholar

[4]

J. Gómez-Torrecillas, F. J. Lobillo and G. Navarro, Ideal codes over separable ring extensions,, preprint, (). Google Scholar

[5]

J. Gómez-Torrecillas, F. J. Lobillo and G. Navarro, Cyclic convolutional codes over separable extensions,, in Coding Theory and Applications (eds. R. Pinto, (2015), 209. doi: 10.1007/978-3-319-17296-5_22. Google Scholar

[6]

K. Hirata and K. Sugano, On semisimple extensions and separable extensions over non commutative rings,, J. Math. Soc. Japan, 18 (1966), 360. doi: 10.2969/jmsj/01840360. Google Scholar

[7]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis,, Cambridge Univ. Press, (1994). doi: 10.1017/CBO9780511840371. Google Scholar

[8]

N. Jacobson, Basic Algebra: II,, W. H. Freeman Company, (1980). Google Scholar

[9]

S. R. López-Permouth and S. Szabo, Convolutional codes with additional algebraic structure,, J. Pure Appl. Algebra, 217 (2013), 958. doi: 10.1016/j.jpaa.2012.09.017. Google Scholar

[10]

R. Pierce, Associative Algebras,, Springer-Verlag, (1982). doi: 10.1007/978-1-4757-0163-0. Google Scholar

[11]

P. Piret, Structure and constructions of cyclic convolutional codes,, IEEE Trans. Inf. Theory, 22 (1976), 147. doi: 10.1109/TIT.1976.1055531. Google Scholar

[1]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003
[2]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261
[3]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1
[4]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
[5]

M. De Boeck, P. Vandendriessche. On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$. Advances in Mathematics of Communications, 2014, 8 (3) : 281-296. doi: 10.3934/amc.2014.8.281
[6]

Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027
[7]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[8]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399
[9]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
[10]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407
[11]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83
[12]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Information--bit error rate and false positives in an MDS code. Advances in Mathematics of Communications, 2015, 9 (2) : 149-168. doi: 10.3934/amc.2015.9.149
[13]

Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503
[14]

Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275
[15]

Anna-Lena Horlemann-Trautmann, Kyle Marshall. New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 533-548. doi: 10.3934/amc.2017042
[16]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[17]

Jianying Fang. 5-SEEDs from the lifted Golay code of length 24 over Z4. Advances in Mathematics of Communications, 2017, 11 (1) : 259-266. doi: 10.3934/amc.2017017
[18]

Daniel Heinlein, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. A subspace code of size $ \bf{333} $ in the setting of a binary $ \bf{q} $-analog of the Fano plane. Advances in Mathematics of Communications, 2019, 13 (3) : 457-475. doi: 10.3934/amc.2019029
[19]

Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039
[20]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences