Weight-2 input sequences of $ 1/n $ convolutional codes from linear systems point of view

Victoria Herranz; Diego Napp; Carmen Perea; Victoria Herranz; Diego Napp; Carmen Perea

doi:10.3934/math.2023034

AIMS Mathematics

2023, Volume 8, Issue 1: 713-732. doi: 10.3934/math.2023034

Previous Article Next Article

Research article

Weight-2 input sequences of $ 1/n $ convolutional codes from linear systems point of view

1.
Institute Center of Operations Research, Miguel Hernández University, Spain
2.
Departament of Mathematics, University of Alicante, Spain

Received: 15 June 2022 Revised: 19 August 2022 Accepted: 05 September 2022 Published: 11 October 2022
MSC : 94B10, 93C05, 11T71

Convolutional codes form an important class of codes that have memory. One natural way to study these codes is by means of input state output representations. In this paper we study the minimum (Hamming) weight among codewords produced by input sequences of weight two. In this paper, we consider rate $ 1/n $ and use the linear system setting called $ (A, B, C, D) $ input-state-space representations of convolutional codes for our analysis. Previous results on this area were recently derived assuming that the matrix $ A $, in the input-state-output representation, is nonsingular. This work completes this thread of research by treating the nontrivial case in which $ A $ is singular. Codewords generated by weight-2 inputs are relevant to determine the effective free distance of Turbo codes.
- convolutional codes,
- input-state-output representations,
- linear time-invariant systems,
- effective free distance
Citation: Victoria Herranz, Diego Napp, Carmen Perea. Weight-2 input sequences of $ 1/n $ convolutional codes from linear systems point of view[J]. AIMS Mathematics, 2023, 8(1): 713-732. doi: 10.3934/math.2023034

Related Papers:

Abstract

Convolutional codes form an important class of codes that have memory. One natural way to study these codes is by means of input state output representations. In this paper we study the minimum (Hamming) weight among codewords produced by input sequences of weight two. In this paper, we consider rate $ 1/n $ and use the linear system setting called $ (A, B, C, D) $ input-state-space representations of convolutional codes for our analysis. Previous results on this area were recently derived assuming that the matrix $ A $, in the input-state-output representation, is nonsingular. This work completes this thread of research by treating the nontrivial case in which $ A $ is singular. Codewords generated by weight-2 inputs are relevant to determine the effective free distance of Turbo codes.

References

[1]	S. Benedetto, G. Montorsi, Design of parallel concatenated convolutional codes, IEEE T. Commun., 44 (1996), 591–600. https://doi.org/10.1109/26.494303 doi: 10.1109/26.494303
[2]	C. Berrou, A. Glavieux, P. Thitimajshima, Near Shannon limit error-correcting coding and decoding: Turbo Codes (1), Proc. of IEEE ICC 93– IEEE International Conference on Communications, 2 (1993), 1064–1070. https://doi.org/10.1109/ICC.1993.397441 doi: 10.1109/ICC.1993.397441
[3]	R. Bru, R. Cantó, B. Ricarte, V. Rumchev, A Basic Canonical Form of Discrete-time Compartmental Systems, Int. J. Contemp. Math. Sciences, 2 (2007), 261–273. http://dx.doi.org/10.12988/ijcms.2007.07020 doi: 10.12988/ijcms.2007.07020
[4]	P. Campillo, A. Devesa, V. Herranz, C. Perea, Modelization of turbo encoder from linear system point of view, Proceedings of the 10th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE 2010), (2010), 314–317.
[5]	B. Cantó, C. Coll, E. Sánchez, On positive behaviour of periodic control systems, Appl. Math. Comput., 161 (2005), 779–786. https://doi.org/10.1016/j.amc.2003.03.001 doi: 10.1016/j.amc.2003.03.001
[6]	M. V. Carriegos, N. De Castro-García, Partitions of elements in a monoid and its applications to systems theory, Linear Algebra Appl., 491 (2016), 161–170. https://doi.org/10.1016/j.laa.2015.05.034 doi: 10.1016/j.laa.2015.05.034
[7]	N. De Castro-García, M. V. Carriegos, A. L. Muñoz Castañeda, A characterization of von Neumann rings in terms of linear systems, Linear Algebra Appl., 494 (2016), 236–244. https://doi.org/10.1016/j.laa.2016.01.019 doi: 10.1016/j.laa.2016.01.019
[8]	J.-J. Climent, V. Herranz, C. Perea, A first approximation of concatenated convolutional codes from linear systems theory viewpoint, Linear Algebra Appl., 425 (2007), 673–699. https://doi.org/10.1016/j.laa.2007.03.017 doi: 10.1016/j.laa.2007.03.017
[9]	J.-J. Climent, V. Herranz, C. Perea, Linear system modelization of concatenated block and convolutional codes, Linear Algebra Appl., 429 (2008), 1191–1212. https://doi.org/10.1016/j.laa.2007.09.006 doi: 10.1016/j.laa.2007.09.006
[10]	J.-J. Climent, D. Napp, C. Perea, R. Pinto, Maximum distance separable 2D convolutional codes, IEEE T. Inform. Theory, 62 (2016), 669–680. https://doi.org/10.1109/TIT.2015.2509075 doi: 10.1109/TIT.2015.2509075
[11]	J. J. Climent, V. Herranz, C. Perea, Parallel concatenated convolutional codes from linear systems theory viewpoint, Systems and Control Letters, 96 (2016), 15–22. https://doi.org/10.1016/j.sysconle.2016.06.016 doi: 10.1016/j.sysconle.2016.06.016
[12]	D. Divsalar, F. Pollara, Low Rate Turbo Codes for Deep Space Communications, Proceedings of 1995 IEEE Int. Symp. Info. Theory, (1995). https://doi.org/10.1109/ISIT.1995.531137
[13]	D. Divsalar, F. Pollara, Multiple turbo codes for deep-space communications, The Telecommunications and Data Acquisition Progress Report, (1995).
[14]	D. Divsalar, R. J. McEliece, The effective free distance of turbo codes, Electron. Lett., 32 (1996), 445–446. https://doi.org/10.1049/el:19960321 doi: 10.1049/el:19960321
[15]	D. Divsalar, R. J. McEliece, On the design of generalized concatenated coding systems with interleavers, TMO Progress Report 42-134, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA, (1998).
[16]	M. I. García-Planas, E. M. Soudit, L. E. Um, Convolutional codes under linear systems point of view. Analysis of output-controllability, WSEAS Press. World Scientific and Engineering Academy and Society, 11 (2012), 2224–2880.
[17]	M. I. García-Planas, N. deCastro, Concatenated linear systems over rings and their application to construction of concatenated families of convolutional codes, Linear algebra appl., 542 (2017), 624–647. https://doi.org/10.1016/j.laa.2017.12.009 doi: 10.1016/j.laa.2017.12.009
[18]	V. Herranz, D. Napp, C. Perea, Serial concatenation of a block code and a 2D convolutional code, Multidim. syst. sign. p., 30 (2019), 1113–1127. https://doi.org/10.1007/s11045-018-0591-3 doi: 10.1007/s11045-018-0591-3
[19]	V. Herranz, D. Napp, C. Perea, $1/n$ Turbo Codes from linear system point of view, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 114 (2020). https://doi.org/10.1007/s13398-020-00850-2
[20]	S. Hong, R. Wu, On deep holes of generalized Reed-Solomon codes, AIMS Mathematics, 1 (2016), 96–101. https://doi.org/10.3934/Math.2016.2.96 doi: 10.3934/Math.2016.2.96
[21]	J. Lieb, J. Rosenthal, Erasure decoding of convolutional codes using first order representations, Math. Control Signal., 33 (2021), 499–513. https://doi.org/10.1007/s00498-021-00289-9 doi: 10.1007/s00498-021-00289-9
[22]	T. Kailath, Linear Systems, Prentice Hall information and system sciences series, Prentice-Hall, 1980.
[23]	J. L. Massey, M. K. Sain, Codes, automata, and continuous systems: explicit interconnections, IEEE T. Automat. Contr., 12 (1967), 644–650.} https://doi.org/10.1109/TAC.1967.1098736 doi: 10.1109/TAC.1967.1098736
[24]	R. J. McEliece, The algebraic theory of convolutional codes, Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds. North-Holland: Elsevier (1998), 1065–1138.
[25]	A. L. M. Castañeda, J. M. Muñoz-Porras, F. J. Plaza-Martín, Rosenthal's Decoding Algorithm for Certain 1-Dimensional Convolutional Codes, IEEE T. Inform. Theory, 65 (2019), 7736–7741. https://doi.org/10.1109/TIT.2019.2921370 doi: 10.1109/TIT.2019.2921370
[26]	D. Napp, R. Pereira, R. Pinto, P. Rocha, Periodic state-space representations of periodic convolutional codes, Cryptography and Communications, 11 (2019), 585–595. https://doi.org/10.1007/s12095-018-0313-6 doi: 10.1007/s12095-018-0313-6
[27]	B. Ricarte, S. Romero-Vivó, An algebraic approach to the structural properties of positive state control systems, Math. Method. Appl. Sci., 41 (2018), 2370–2378. https://doi.org/10.1002/mma.4351 doi: 10.1002/mma.4351
[28]	J. Rosenthal, J. M. Schumacher, E. V. York, On behaviors and convolutional codes, IEEE T. Inform. Theory, 42 (1996), 1881–1891. https://doi.org/10.1109/18.556682 doi: 10.1109/18.556682
[29]	J. Rosenthal, E. V. York, BCH convolutional codes, IEEE T. Inform. Theory, 45 (1999), 1833–1844. https://doi.org/10.1109/18.782104 doi: 10.1109/18.782104
[30]	J. Rosenthal, Connections between linear systems and convolutional codes, Codes, Systems and Graphical Models, ser. The IMA Volumes in Mathematics and its Applications, B. Marcus and J. Rosenthal, Eds. New York: Springer-Verlag, 123 (2001), 39–66. https://doi.org/10.1007/978-1-4613-0165-3_2
[31]	R. M. Roth, A. Lempel, On MDS codes via Cauchy matrices, IEEE T. Inform. Theory, 35 (1989), 1314–1319. https://doi.org/10.1109/18.45291 doi: 10.1109/18.45291
[32]	R. Smarandache, J. Rosenthal, Construction of Convolutional Codes using Methods from Linear Systems Theory, Proc. of the 35-th Annual Allerton Conf. on Commun., Control, and Computing, (1997), 953–960.
[33]	R. Smarandache, J. Rosenthal, A state space approach for constructing MDS rate $1/n$ convolutional codes, Proc. of the 1998 IEEE Inform. Theory Workshop (ITW 1998), (1998), 116–117. https://doi.org/10.1109/ITW.1998.706461
[34]	X. F. Xu, Y. C. Xu, S. F. Hong, Some results on ordinary words of standard Reed-Solomon codes, AIMS Mathematics, 4 (2019), 1336–1347. https://doi.org/10.3934/math.2019.5.1336 doi: 10.3934/math.2019.5.1336

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)