Hostname: page-component-7dd5485656-frp75 Total loading time: 0 Render date: 2025-10-29T06:59:15.718Z Has data issue: false hasContentIssue false
  • English
  • Français

On the structure of linear recurrent error-control codes

Published online by Cambridge University Press:  15 August 2002

Michel Fliess
Michel Fliess*
Affiliation:
Centre de Mathématiques et Leurs Applications, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, 94235 Cachan, France; fliess@cmla.ens-cachan.fr. Laboratoire GAGE, École polytechnique, 91128 Palaiseau, France.
Get access

Abstract

We are extending to linear recurrent codes, i.e., totime-varying convolutional codes, most of the classic structuralproperties of fixed convolutional codes. We are also proposing anew connection between fixed convolutional codes and linear blockcodes. These results are obtained thanks to a module-theoreticframework which has been previously developed for linear control.

Keywords

Convolutional codeslinear recurrent codesblockcodestransducersencodersfeedback decodinglinear systemscontrollabilityobservabilityinput-output inversionmodules.

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 703 - 713
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Berrou, C. and Glavieux, A., Near-optimum error-correcting coding and decoding: Turbo-codes. IEEE Trans. Communicat. 44 (1996) 1261-1271. CrossRef
R.E. Blahut, Theory and Practice of Error Control Codes. Addison-Wesley (1983).
N. Bourbaki, Algèbre, Chap. 2. Hermann (1970).
G. Cohen, J.-L. Dornstetter and P. Godlewski, Codes correcteurs d'erreurs. Masson (1992).
R.M. Cohn, Difference Algebra. Interscience (1965).
A. Dholakia, Introduction to Convolutional Codes with Applications. Kluwer (1994).
Fagnani, F. and Zampieri, S., System-theoretic properties of convolutional codes over rings. IEEE Trans. Inform. Theory 47 (2001) 2256-2274. CrossRef
Fliess, M., Automatique en temps discret et algèbre aux différences. Forum Math. 2 (1990) 213-232.
Fliess, M., Some basic structural properties of generalized linear systems. Systems Control Lett. 15 (1990) 391-396. CrossRef
Fliess, M., A remark on Willems' trajectory characterization of linear controllability. Systems Control Lett. 19 (1992) 43-45. CrossRef
Fliess, M., Reversible linear and nonlinear discrete-time dynamics. IEEE Trans. Automat. Control 37 (1992) 1144-1153. CrossRef
Fliess, M., Une interprétation algébrique de la transformation de Laplace et des matrices de transfert. Linear Algebra Appl. 203-204 (1994) 429-442. CrossRef
M. Fliess, Variations sur la notion de contrôlabilité, in Journée Soc. Math. France. Paris (2000) 47-86.
Fliess, M. and Bourlès, H., Discussing some examples of linear system interconnections. Systems Control Lett. 27 (1996) 1-7. CrossRef
Fliess, M., Lévine, J., Martin, P. and Rouchon, P., Flatness and defect of non-linear systems: Introductory theory and applications. Internat. J. Control 61 (1995) 1327-1361. CrossRef
Fliess, M. and Marquez, R., Continuous-time linear predictive control and flatness: A module-theoretic setting with examples. Internat. J. Control 73 (2000) 606-623. CrossRef
Fliess, M. and Marquez, R., Une approche intrinsèque de la commande prédictive linéaire discrète. APII J. Europ. Syst. Automat. 35 (2001) 127-147.
Fliess, M., Marquez, R., Delaleau, E. and Sira-Ramírez, H., Correcteurs proportionnels-intégraux généralisés. ESAIM: COCV 7 (2002) 23-41. CrossRef
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).
Fliess, M. and Mounier, H., Controllability and observability of linear delay systems: An algebraic approach. ESAIM: COCV 3 (1998) 301-314. CrossRef
Forney Jr, G.D.., Convolutional codes I: Algebraic structure. IEEE Trans. Inform. Theory 16 (1970) 720-738.
Forney Jr, G.D.., Minimal bases of rational vector spaces, with applications to multivariable linear systems. SIAM J. Control 13 (1975) 493-520.
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.
Forney Jr, G.D.. and M.D. Trott, The dynamics of group codes: State-space, trellis diagrams and canonical encoders. IEEE Trans. Inform. Theory 39 (1993) 1491-1513.
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.
R. Johannesson and K.Sh. Zigangirov, Fundamentals of Convolutional Coding. IEEE Press (1999).
T. Kailath, Linear Systems. Prentice-Hall (1979).
Kamen, E.W., Khargonekar, P.P. and Poola, K.R., A transfer-function approach to linear time-varying discrete-time systems. SIAM J. Control Optim. 23 (1985) 550-565. CrossRef
T.Y. Lam, Lectures on Rings and Modules. Springer (1999).
S. Lin and D.J. Costello Jr., Error Control Coding: Fundamentals and Applications. Prentice-Hall (1983).
J.H. van Lint, Introduction to Coding Theory, $3^{\rm rd}$ Edition. Springer (1999).
Loeliger, H.-A., Forney Jr, G.D.., T. Mittelholzer and M.D. Trott, Minimality and observability of group systems. Linear Algebra Appl. 205-206 (1994) 937-963. CrossRef
Massey, J.L. and Sain, M.K., Codes, automata and contnuous systems: Explicit interconnections. IEEE Trans. Automat. Control 12 (1967) 644-650. CrossRef
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.
J.C. McConnel and J.C. Robson, Noncommutative Noetherian Rings. Wiley (1987).
Mounier, H., Rouchon, P. and Rudolph, J., Some examples of linear systems with delays. APII J. Europ. Syst. Automat. 31 (1997) 911-925.
P. Piret, Convolutional Codes, an Algebraic Approach. MIT Press (1988).
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.
Rosenthal, J., Schumacher, J.M. and York, E.V., On behaviors and convolutional codes. IEEE Trans. Informat. Theory 42 (1996) 1881-1891. CrossRef
Rosenthal, J. and York, E.V., BCH convolutional codes. IEEE Trans. Inform. Theory 45 (1999) 1833-1844. CrossRef
J. Rotman, An Introduction to Homological Algebra. Academic Press (1979).
A.J. Viterbi and J.K. Omura, Principles of Digital Communication and Coding. McGraw-Hill (1979).
Weiss, L., Controllability, realization and stability of discrete-time systems. SIAM J. Control 10 (1972) 230-251. CrossRef
Willems, J.C., Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Automat. Control 36 (1991) 259-294. CrossRef
G. Zémor, Cours de cryptographie. Cassini (2000).