Abstract
Two-dimensional (2D) finite codes are defined as families of compact support sequences indexed in Z × Z and taking values in Fn, F a Galois field. Several properties of encoders, decoders and syndrome decoders are discussed under different hypotheses on the code structure, and related to the injectivity and primeness of the corresponding polynomial matrices in two variables. Dual codes are finally introduced as families of parity checks on a given modular code, and related to the standard theory of 2D behaviors.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G.D. Forney, “Convolutional Codes. I: Algebraic Structure,”IEEE Transactions on Information Theory, vol. 16, 1970, pp. 720–738.
G.D. Forney, “Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems,”SIAM Journal of Control, vol. 13, 1975, pp. 493–521.
J.C. Willems, “Models for Dynamics,” inDynamics Reported, vol. 2, U. Kirchgaber and H.O. Walther (eds.), New York: Wiley, 1989, pp. 171–269.
G.D. Forney and M.D. Trott, “The Dynamics of Group Codes: State Spaces, Trellis Diagrams and Canonical Encoders,” to appear.
H.A. Loeliger, G.D. Forney, T. Mittelholzer, and M.D. Trott, “Minimality and Observability of Group Systems,” to appear inLin. Alg. Appl., 1994.
E. Fornasini and M.E. Valcher, “Algebraic Aspects of 2D Convolutional Codes,” to appear inLin. Alg. Appl., 1994.
S. Lang,Algebra, Reading, MA: Addison-Wesley, 1967.
M. Morf, B.C. Lévy, and S.Y. Kung, “New Results in 2D Systems Theory: Part I,” inProceedings of the IEEE, vol. 65, 1977, pp. 861–872.
D.C. Youla and P.F. Pickel, “The Quillen-Suslin Theorem,”IEEE Transactions on Circuits and Systems, vol. 31, 1984, pp. 513–518.
P. Rocha, “Structure and Representation of 2D Systems,” Ph.D. Thesis, Rijksuniversiteit, Groningen, 1990.
W. Greub,Linear Algebra, New York: Springer-Verlag, 1975.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Valcher, M.E., Fornasini, E. On 2D finite support convolutional codes: An algebraic approach. Multidim Syst Sign Process 5, 231–243 (1994). https://doi.org/10.1007/BF00980707
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00980707