On the structure of linear recurrent error-control codes
ESAIM: Control, Optimisation and Calculus of Variations (2002)
- Volume: 8, page 703-713
- ISSN: 1292-8119
Access Full Article
topAbstract
topHow to cite
topFliess, Michel. "On the structure of linear recurrent error-control codes." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 703-713. <http://eudml.org/doc/246026>.
@article{Fliess2002,
abstract = {We are extending to linear recurrent codes, i.e., to time-varying convolutional codes, most of the classic structural properties of fixed convolutional codes. We are also proposing a new connection between fixed convolutional codes and linear block codes. These results are obtained thanks to a module-theoretic framework which has been previously developed for linear control.},
author = {Fliess, Michel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {convolutional codes; linear recurrent codes; block codes; transducers; encoders; feedback decoding; linear systems; controllability; observability; input-output inversion; modules},
language = {eng},
pages = {703-713},
publisher = {EDP-Sciences},
title = {On the structure of linear recurrent error-control codes},
url = {http://eudml.org/doc/246026},
volume = {8},
year = {2002},
}
TY - JOUR
AU - Fliess, Michel
TI - On the structure of linear recurrent error-control codes
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 703
EP - 713
AB - We are extending to linear recurrent codes, i.e., to time-varying convolutional codes, most of the classic structural properties of fixed convolutional codes. We are also proposing a new connection between fixed convolutional codes and linear block codes. These results are obtained thanks to a module-theoretic framework which has been previously developed for linear control.
LA - eng
KW - convolutional codes; linear recurrent codes; block codes; transducers; encoders; feedback decoding; linear systems; controllability; observability; input-output inversion; modules
UR - http://eudml.org/doc/246026
ER -
References
top- [1] C. Berrou and A. Glavieux, Near-optimum error-correcting coding and decoding: Turbo-codes. IEEE Trans. Communicat. 44 (1996) 1261-1271.
- [2] R.E. Blahut, Theory and Practice of Error Control Codes. Addison-Wesley (1983). Zbl0569.94012MR698946
- [3] N. Bourbaki, Algèbre, Chap. 2. Hermann (1970). MR274237
- [4] G. Cohen, J.-L. Dornstetter and P. Godlewski, Codes correcteurs d’erreurs. Masson (1992).
- [5] R.M. Cohn, Difference Algebra. Interscience (1965). Zbl0127.26402MR205987
- [6] A. Dholakia, Introduction to Convolutional Codes with Applications. Kluwer (1994). Zbl0826.94001
- [7] F. Fagnani and S. Zampieri, System-theoretic properties of convolutional codes over rings. IEEE Trans. Inform. Theory 47 (2001) 2256-2274. Zbl1028.94033MR1873201
- [8] M. Fliess, Automatique en temps discret et algèbre aux différences. Forum Math. 2 (1990) 213-232. Zbl0706.93039MR1050406
- [9] M. Fliess, Some basic structural properties of generalized linear systems. Systems Control Lett. 15 (1990) 391-396. Zbl0727.93024MR1084580
- [10] M. Fliess, A remark on Willems’ trajectory characterization of linear controllability. Systems Control Lett. 19 (1992) 43-45. Zbl0765.93003
- [11] M. Fliess, Reversible linear and nonlinear discrete-time dynamics. IEEE Trans. Automat. Control 37 (1992) 1144-1153. Zbl0764.93058MR1178584
- [12] M. Fliess, Une interprétation algébrique de la transformation de Laplace et des matrices de transfert. Linear Algebra Appl. 203-204 (1994) 429-442. Zbl0802.93010MR1275520
- [13] M. Fliess, Variations sur la notion de contrôlabilité, in Journée Soc. Math. France. Paris (2000) 47-86. Zbl0992.93002MR1799559
- [14] M. Fliess and H. Bourlès, Discussing some examples of linear system interconnections. Systems Control Lett. 27 (1996) 1-7. Zbl0877.93064MR1375906
- [15] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of non-linear systems: Introductory theory and applications. Internat. J. Control 61 (1995) 1327-1361. Zbl0838.93022MR1613557
- [16] M. Fliess and R. Marquez, Continuous-time linear predictive control and flatness: A module-theoretic setting with examples. Internat. J. Control 73 (2000) 606-623. Zbl1006.93508MR1768024
- [17] M. Fliess and R. Marquez, Une approche intrinsèque de la commande prédictive linéaire discrète. APII J. Europ. Syst. Automat. 35 (2001) 127-147.
- [18] M. Fliess, R. Marquez, E. Delaleau and H. Sira–Ramírez, Correcteurs proportionnels-intégraux généralisés. ESAIM: COCV 7 (2002) 23-41. Zbl1037.93040
- [19] M. Fliess, R. Marquez and H. Mounier, An extension of predictive control, PID regulators and Smith predictors to some linear delay systems. Internat. J. Control (to appear). Zbl1021.93015MR1916231
- [20] M. Fliess and H. Mounier, Controllability and observability of linear delay systems: An algebraic approach. ESAIM: COCV 3 (1998) 301-314. Zbl0908.93013MR1644427
- [21] G.D. Forney Jr., Convolutional codes I: Algebraic structure. IEEE Trans. Inform. Theory 16 (1970) 720-738. Zbl0205.20702MR275988
- [22] G.D. Forney Jr., Minimal bases of rational vector spaces, with applications to multivariable linear systems. SIAM J. Control 13 (1975) 493-520. Zbl0269.93011MR378886
- [23] G.D. Forney Jr., Algebraic structure of convolutional codes and algebraic system theory, in Mathematical System Theory – The Influence of R.E. Kalman, edited by A.C. Antoulas. Springer (1991) 527-557. Zbl0751.94011
- [24] G.D. Forney Jr. and M.D. Trott, The dynamics of group codes: State-space, trellis diagrams and canonical encoders. IEEE Trans. Inform. Theory 39 (1993) 1491-1513. Zbl0801.94015MR1281706
- [25] G.D. Forney Jr., B. Marcus, N.T. Sindhushayana and M. Trott, A multilingual dictionary: System theory, coding theory, symbolic dynamics and automata theory, in Different Aspects of Coding Theory. Proc. Symp. Appl. Math. 50; Amer. Math. Soc. (1995) 109-138. Zbl0845.58024MR1368638
- [26] R. Johannesson and K. Sh. Zigangirov, Fundamentals of Convolutional Coding. IEEE Press (1999). Zbl0964.94024MR1768537
- [27] T. Kailath, Linear Systems. Prentice-Hall (1979). Zbl0454.93001MR569473
- [28] E.W. Kamen, P.P. Khargonekar and K.R. Poola, A transfer-function approach to linear time-varying discrete-time systems. SIAM J. Control Optim. 23 (1985) 550-565. Zbl0626.93039MR791887
- [29] T.Y. Lam, Lectures on Rings and Modules. Springer (1999). MR1653294
- [30] S. Lin and D.J. Costello Jr., Error Control Coding: Fundamentals and Applications. Prentice-Hall (1983). Zbl1310.94181
- [31] J.H. van Lint, Introduction to Coding Theory, Edition. Springer (1999). Zbl0936.94014MR1664228
- [32] H.-A. Loeliger, G.D. Forney Jr., T. Mittelholzer and M.D. Trott, Minimality and observability of group systems. Linear Algebra Appl. 205-206 (1994) 937-963. Zbl0813.93046MR1276847
- [33] J.L. Massey and M.K. Sain, Codes, automata and contnuous systems: Explicit interconnections. IEEE Trans. Automat. Control 12 (1967) 644-650.
- [34] R.J. McEliece, The algebraic theory of convolutional codes, in Handbook of Coding Theory, Vol. 1, edited by V. Pless and W.C. Huffman. Elsevier (1998) 1065-1138. Zbl0967.94020MR1667948
- [35] J.C. McConnel and J.C. Robson, Noncommutative Noetherian Rings. Wiley (1987). Zbl0644.16008MR934572
- [36] H. Mounier, P. Rouchon and J. Rudolph, Some examples of linear systems with delays. APII J. Europ. Syst. Automat. 31 (1997) 911-925.
- [37] P. Piret, Convolutional Codes, an Algebraic Approach. MIT Press (1988). Zbl0986.94510MR959540
- [38] J. Rosenthal, Connections between linear systems and convolutional codes, in Codes, Systems and Graphical Models, edited by B. Marcus and J. Rosenthal. Springer (2000) 39-66. Zbl0993.94559MR1861952
- [39] J. Rosenthal, J.M. Schumacher and E.V. York, On behaviors and convolutional codes. IEEE Trans. Informat. Theory 42 (1996) 1881-1891. Zbl0876.94042MR1465746
- [40] J. Rosenthal and E.V. York, BCH convolutional codes. IEEE Trans. Inform. Theory 45 (1999) 1833-1844. Zbl0958.94035MR1720637
- [41] J. Rotman, An Introduction to Homological Algebra. Academic Press (1979). Zbl0441.18018MR538169
- [42] A.J. Viterbi and J.K. Omura, Principles of Digital Communication and Coding. McGraw-Hill (1979). Zbl0495.94002
- [43] L. Weiss, Controllability, realization and stability of discrete-time systems. SIAM J. Control 10 (1972) 230-251. Zbl0238.93007MR392092
- [44] J.C. Willems, Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Automat. Control 36 (1991) 259-294. Zbl0737.93004MR1092818
- [45] G. Zémor, Cours de cryptographie. Cassini (2000). Zbl0978.94040
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.