Convolutional codes with a matrix-algebra word-ambient

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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

Abstract
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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., 2 (2008), 83-94. doi: 10.3934/amc.2008.2.83.
[2]	G. D. Forney Jr., Convolutional codes I: Algebraic structure, IEEE Trans. Inf. Theory, 16 (1970), 720-738, doi: 10.1109/TIT.1970.1054541.
[3]	H. Gluesing-Luerssen and W. Schmale, On cyclic convolutional codes, Acta Appl. Math., 82 (2004), 183-237. doi: 10.1023/B:ACAP.0000027534.61242.09.
[4]	J. Gómez-Torrecillas, F. J. Lobillo and G. Navarro, Ideal codes over separable ring extensions, preprint, arXiv:1408.1546
[5]	J. Gómez-Torrecillas, F. J. Lobillo and G. Navarro, Cyclic convolutional codes over separable extensions, in Coding Theory and Applications (eds. R. Pinto, P. Rocha Malonek and P. Vettori), Springer, 2015, 209-215. doi: 10.1007/978-3-319-17296-5_22.
[6]	K. Hirata and K. Sugano, On semisimple extensions and separable extensions over non commutative rings, J. Math. Soc. Japan, 18 (1966), 360-373. doi: 10.2969/jmsj/01840360.
[7]	R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, 1994. doi: 10.1017/CBO9780511840371.
[8]	N. Jacobson, Basic Algebra: II, W. H. Freeman Company, 1980.
[9]	S. R. López-Permouth and S. Szabo, Convolutional codes with additional algebraic structure, J. Pure Appl. Algebra, 217 (2013), 958-972. doi: 10.1016/j.jpaa.2012.09.017.
[10]	R. Pierce, Associative Algebras, Springer-Verlag, 1982, doi: 10.1007/978-1-4757-0163-0.
[11]	P. Piret, Structure and constructions of cyclic convolutional codes, IEEE Trans. Inf. Theory, 22 (1976), 147-155. doi: 10.1109/TIT.1976.1055531.

show all references

References:

[1]	S. Estrada, J. R. García-Rozas, J. Peralta and E. Sánchez-García, Group convolutional codes, Adv. Math. Commun., 2 (2008), 83-94. doi: 10.3934/amc.2008.2.83.
[2]	G. D. Forney Jr., Convolutional codes I: Algebraic structure, IEEE Trans. Inf. Theory, 16 (1970), 720-738, doi: 10.1109/TIT.1970.1054541.
[3]	H. Gluesing-Luerssen and W. Schmale, On cyclic convolutional codes, Acta Appl. Math., 82 (2004), 183-237. doi: 10.1023/B:ACAP.0000027534.61242.09.
[4]	J. Gómez-Torrecillas, F. J. Lobillo and G. Navarro, Ideal codes over separable ring extensions, preprint, arXiv:1408.1546
[5]	J. Gómez-Torrecillas, F. J. Lobillo and G. Navarro, Cyclic convolutional codes over separable extensions, in Coding Theory and Applications (eds. R. Pinto, P. Rocha Malonek and P. Vettori), Springer, 2015, 209-215. doi: 10.1007/978-3-319-17296-5_22.
[6]	K. Hirata and K. Sugano, On semisimple extensions and separable extensions over non commutative rings, J. Math. Soc. Japan, 18 (1966), 360-373. doi: 10.2969/jmsj/01840360.
[7]	R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, 1994. doi: 10.1017/CBO9780511840371.
[8]	N. Jacobson, Basic Algebra: II, W. H. Freeman Company, 1980.
[9]	S. R. López-Permouth and S. Szabo, Convolutional codes with additional algebraic structure, J. Pure Appl. Algebra, 217 (2013), 958-972. doi: 10.1016/j.jpaa.2012.09.017.
[10]	R. Pierce, Associative Algebras, Springer-Verlag, 1982, doi: 10.1007/978-1-4757-0163-0.
[11]	P. Piret, Structure and constructions of cyclic convolutional codes, IEEE Trans. Inf. Theory, 22 (1976), 147-155. doi: 10.1109/TIT.1976.1055531.

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