Abstract
In this paper we study periodically time-varying convolutional codes by means of input-state-output representations. Using these representations we investigate under which conditions a given time-invariant convolutional code can be transformed into an equivalent periodic time-varying one. The relation between these two classes of convolutional codes is studied for period 2. We illustrate the ideas presented in this paper by constructing a periodic time-varying convolutional code from a time-invariant one. The resulting periodic code has larger free distance than any time-invariant convolutional code with equivalent parameters.
This work was supported in part by the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013 and also by Project POCI-01-0145-FEDER-006933 - SYSTEC - Research Center for Systems and Technologies - funded by FEDER funds through COMPETE2020 - Programa Operacional Competitividade e Internacionalização (POCI) - and by national funds through FCT - Fundação para a Ciência e a Tecnologia.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aleixo, J.C., Rocha, P., Willems, J.C.: State space representation of SISO periodic behaviors. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA, pp. 1545–1550, 12–15 December 2011
Almeida, P., Napp, D., Pinto, R.: A new class of superregular matrices and MDP convolutional codes. Linear Algebra Appl. 439(7), 2145–2147 (2013)
Bocharova, I., Kudryashov, B.: Rational rate punctured convolutional codes for soft-decision Viterbi decoding. IEEE Trans. Inf. Theory 43(4), 1305–1313 (1997)
Climent, J.-J., Herranz, V., Perea, C., Tomás, V.: A systems theory approach to periodically time-varying convolutional codes by means of their invariant equivalent. In: Bras-Amorós, M., Høholdt, T. (eds.) AAECC 2009. LNCS, vol. 5527, pp. 73–82. Springer, Heidelberg (2009). doi:10.1007/978-3-642-02181-7_8
Costello, D.: Free distance bounds for convolutional codes. IEEE Trans. Inf. Theory 20(3), 356–365 (1974)
Fekri, F., Sartipi, M., Mersereau, R.M., Schafer, R.W.: Convolutional codes using finite-field wavelets: time-varying codes and more. IEEE Trans. Signal Process. 53(5), 1881–1896 (2005)
Gluesing-Luerssen, H., Schmale, W.: Distance bounds for convolutional codes and some optimal codes, arXiv:math/0305135v1 (2003)
Gluesing-Luerssen, H., Schneider, G.: State space realizations and monomial equivalence for convolutional codes. Linear Algebra Appl. 425, 518–533 (2007)
Gluesing-Luerssen, H., Rosenthal, J., Smarandache, R.: Strongly-MDS convolutional codes. IEEE Trans. Inf. Theory 52(2), 584–598 (2006)
Johannesson, R., Zigangirov, K.S.: Fundamentals of Convolutional Coding. IEEE press, New York (1999)
Justesen, J.: New convolutional code constructions and a class of asymptotically good time-varying codes. IEEE Trans. Inf. Theory 19(2), 220–225 (1973)
Kuijper, M., Polderman, J.W.: Reed-Solomon list decoding from a system-theoretic perspective. IEEE Trans. Inf. Theory 50(2), 259–571 (2004)
Kuijper, M., Willems, J.C.: A behavioral framework for periodically time-varying systems. In: Proceedings of the 36th IEEE Conference on Decision & Control - CDC 1997, vol. 3, San Diego, California USA, 10–12 December 1997, pp. 2013–2016 (1997)
La Guardia, G.: Convolutional codes: techniques of construction. Comput. Appl. Math. 35(2), 501–517 (2016)
Lee, P.J.: There are many good periodically time-varying convolutional codes. IEEE Trans. Inf. Theory 35(2), 460–463 (1989)
Mooser, M.: Some periodic convolutional codes better than any fixed code. IEEE Trans. Inf. Theory 29(5), 750–751 (1983)
Napp, D., Perea, C., Pinto, R.: Input-state-output representations and constructions of finite support 2D convolutional codes. Adv. Math. Commun. 4(4), 533–545 (2010)
Palazzo, R.: A time-varying convolutional encoder better than the best time-invariant encoder. IEEE Trans. Inf. Theory 39(3), 1109–1110 (1993)
Porras, J.M., Curto, J.I.: Classification of convolutional codes. Linear Algebra Appl. 432(10), 2701–2725 (2010)
Rosenthal, J.: Connections between linear systems and convolutional codes. In: Marcus, B., Rosenthal, J. (eds.) Codes, Systems, and Graphical Models, vol. 123, pp. 39–66. Springer, New York (2001). doi:10.1007/978-1-4613-0165-3_2
Rosenthal, J., Smarandache, R.: Maximum distance separable convolutional codes. Appl. Algebra Eng. Commun. Comput. 10, 15–32 (1999)
Truhachev, D., Zigangirov, K., Costello, D.: Distance bounds for periodically time-varying and tail-biting LDPC convolutional codes. IEEE Trans. Inf. Theory 56(9), 4301–4308 (2010)
Viterbi, A.J.: Convolutional codes and their performance in communication systems. IEEE Trans. Commun. Technol. 19(5), 751–772 (1971)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Napp, D., Pereira, R., Rocha, P. (2017). A State Space Approach to Periodic Convolutional Codes. In: Barbero, Á., Skachek, V., Ytrehus, Ø. (eds) Coding Theory and Applications. ICMCTA 2017. Lecture Notes in Computer Science(), vol 10495. Springer, Cham. https://doi.org/10.1007/978-3-319-66278-7_20
Download citation
DOI: https://doi.org/10.1007/978-3-319-66278-7_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66277-0
Online ISBN: 978-3-319-66278-7
eBook Packages: Computer ScienceComputer Science (R0)