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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

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.  Google Scholar

[2]

G. D. Forney Jr., Convolutional codes I: Algebraic structure, IEEE Trans. Inf. Theory, 16 (1970), 720-738, doi: 10.1109/TIT.1970.1054541.  Google Scholar

[3]

H. Gluesing-Luerssen and W. Schmale, On cyclic convolutional codes, Acta Appl. Math., 82 (2004), 183-237. 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, P. Rocha Malonek and P. Vettori), Springer, 2015, 209-215. doi: 10.1007/978-3-319-17296-5_22.  Google Scholar

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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.  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), 958-972. 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), 147-155. 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., 2 (2008), 83-94. 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), 720-738, doi: 10.1109/TIT.1970.1054541.  Google Scholar

[3]

H. Gluesing-Luerssen and W. Schmale, On cyclic convolutional codes, Acta Appl. Math., 82 (2004), 183-237. 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, P. Rocha Malonek and P. Vettori), Springer, 2015, 209-215. 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), 360-373. 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), 958-972. 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), 147-155. doi: 10.1109/TIT.1976.1055531.  Google Scholar

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