Feedback equivalence of convolutional codes over finite rings

Noemí DeCastro-García 1
  • 1 Departamento de Matematicas, 24071, León, Spain

Abstract

The approach to convolutional codes from the linear systems point of view provides us with effective tools in order to construct convolutional codes with adequate properties that let us use them in many applications. In this work, we have generalized feedback equivalence between families of convolutional codes and linear systems over certain rings, and we show that every locally Brunovsky linear system may be considered as a representation of a code under feedback convolutional equivalence.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Climent J.J., Herranz V., Perea C., Tomás V., Un criptosistema de clave pública basado en códigos convolucionales, In Proceedings of XXI Congreso de Ecuaciones Diferenciales y Aplicaciones & XI Congreso de Matemática aplicada (21-25 Septiembre, Ciudad Real, Spain), 2009, 1-8.

  • [2]

    Petsche, T., Dickinson B. W., A Trellis - structured Neural Network, Neural Infor. Proc. Systems, 1988, 592 - 601.

  • [3]

    Forney Jr. G.D., Concatenated Codes, MIT Press, Cambridge, MA, 1966.

  • [4]

    Rosenthal J., Codes, systems and graphical models, IMA, vol. 123, ch. Connections between linear systems and convolutional codes, Springer - Verlag, 2001, 39 - 66,

  • [5]

    García-Planas M.I., Souidi El.M., Um L.E., Convolutional codes under linear systems point of view. Analysis of output-controllability. Wseas Transactions on Mathematics, 2012, 11 (4), 324-333.

  • [6]

    Climent J-J., Herranz V., Perea C., A first approximation of concatenated convolutional codes from linear systems theory viewpoint, Linear Algebra Appl., 2007, 425 ( 2-3), 673-699.

  • [7]

    Rosenthal J., Smarandache R., Maximum distance separable convolutional codes, Appl. Algebr. Eng. Comm., 1999, 10 (1), 15-32.

  • [8]

    Zerz E., On multidimensional convolutional codes and controllability properties of multidimensional systems over finite rings, Asian J. Control, 2010, 12 (2), 119-126.

  • [9]

    Mahapakulchai S., Van Dyck R., Design of ring convolutional trellis codes for MAP decoding of MPEG-4 images, IEEE Trans.Commun., 2004, 52 (7), 1033 -1037.

  • [10]

    Jouhari H., Souidi E. M., Improving Embedding Capacity by using the Z4-linearity of Preparata Codes. Int J Comput Appl., 2012, 53 (18), 1-6.

  • [11]

    Massey J.L., Mittelholzer T., Convolutional codes over rings, In Proceedings Joint Swedish-Soviet Int. Workshop on Inform. Theory, (Gotland, Sweeden), 1989, 14-18.

  • [12]

    Massey J.L., Mittelholzer T., Systematicity and rotational invariance of convolutional codes over rings. In Proceedings Int. Workshop on Alg. and Combinatorial Coding Theory (Leningrad, Russia),1990, 154-158.

  • [13]

    Fagnani, F., Zampieri S., System-theoretic properties of convolutional codes over rings, IEEE T. Infor. Theory, 2001, 47 (6), 2256-2274.

  • [14]

    Johannesson R., Wan Z., Wittenmark E., Some structural properties of convolutional codes over rings, IEEE T. Infor. Theory, 1998, 44 (2), 839-845.

  • [15]

    Kuijper M., Pinto R., On minimality of convolutional ring encoders, IEEE T. Inform. Theory, 2009, 55 (11), 4890-4897.

  • [16]

    El Oued M., Napp D., Pinto R., Toste M., The dual of convolutional codes over Zpr. volume entitled Applied and Computational Matrix Analysis. Springer Verlag, 2017, 192, 79–91.

  • [17]

    El Oued M., Sole P., MDS convolutional codes over a finite ring. IEEE T. Inform. Theory.,2013, 59 (11), 7305 –7313.

  • [18]

    Napp D., Pinto R., Toste M., On MDS convolutional codes over Zpr, Des. Codes Cryptogr., 2017, 83, 101–114.

  • [19]

    Carriegos M., DeCastro-García N., Muñoz Castañeda A., Kernel representations of convolutional codes over rings, Preprint available in https://arxiv.org/pdf/1609.05043v1.pdf, unpublished data.

  • [20]

    DeCastro-García N., Feedback equivalence in linear systems and convolutional codes. Applications to Cybernetics, Coding and Cryptography, Ph. D. DThesis, Universidad de León, Spain, 2016.

  • [21]

    Rosenthal J., Schumacher, J. M., York E. V., On behaviors and convolutional codes, IEEE T. Inform. Theory, 1996, 42 (6), 1881-1891.

  • [22]

    Rosenthal J., York E. V., BCH Convolutional Codes, IEEE T. Inform. Theory, 1999, 45 (6), 1833-1844,

  • [23]

    York E. V., Algebraic description and construction of error correcting codes, a systems theory point of view, Ph.D. thesis, Univ. Notre Dame, France, 1997.

  • [24]

    Brewer J. W., Klingler L., On feedback invariants for linear dynamical systems, Linear Algebra Appl., 2001, 325 (1-3), 209-220.

  • [25]

    Brewer J. W., Bunce J. W., Van Vleck F. S., Linear Systems over Commutative Rings, Marcel Dekker, New York, 1986.

  • [26]

    Hautus M.L.J., Sontag E.D., New results on pole-shifting for parametrized families of systems, J. Pure Appl. Algeb, 1986, 40, 229-244.

  • [27]

    MacWilliams F. J., A theorem on the distribution of weights in a systematic code. Bell Syst. Tech. J., 1963, 42, 79–94.

  • [28]

    Gluesing-Luerssen H., Schneider G., State space realizations and monomial equivalence for convolutional codes, Linear Algebra Appl., 2007, 425 (2-3), 518-533.

  • [29]

    Quang Dinh H., Łódź-Permouth S. R., On the equivalence of codes over rings and modules, Finite Fields Th. App., 2004, 10, 615–625.

  • [30]

    Carriegos M., Hermida-Alonso J.A., Sánchez-Giralda T., The pointwise feedback relation for linear dynamical systems, Linear Algebra Appl., 1998, 279 (1-3), 119-134.

  • [31]

    Hermida-Alonso J.A., Perez M.P., Sanchez-Giralda T., Brunovsky's canonical form for linear dynamical Systems over commutative rings, Linear Algebra Appl., 1996, 233, 131-147.

  • [32]

    Carriegos M. V., Enumeration of classes of linear systems via equations and via partitions in a ordered abelian monoid, Linear Algebra App., 2013, 438 (3), 1132-1148.

  • [33]

    Hautus M.L.J., Controllability and observability condition for linear autonomous system. Ned. Akad. Wetenschappen, Proc. Ser. A, 1969, 72, 443-448.

  • [34]

    Brunovsky P. A., A classification of linear controllable systems, Kibernetika, 1970, 6 (3), 173-187.

  • [35]

    Kalman R. E., Kronecker invariants and Feedback, in Ordinary Differential Equations, Academic Press (New York), 1972, 459-471.

  • [36]

    DeCastro-García N., Carriegos M.V., Muñoz Castañeda A., A characterization of von Neumann rings in terms of linear systems, Linear Algebra App., 2016, 494, 236-244.

  • [37]

    McEliece R. J., The algebraic theory of convolutional codes. In V. Pless and W.C. Huffman, editors, Handbook of Coding Theory, 1998, 1, 1065-1138.

  • [38]

    Massey J.L., Sain M.K., Codes, automata and continuous systems: explicit interconnections, IEEE Trans. Automat. Contr., 1967, 12 (6), 644–650.

  • [39]

    Kitchens B. Symbolic Dynamics and Convolutional Codes. In: Marcus B., Rosenthal J. (eds) Codes, Systems, and Graphical Models. The IMA Volumes in Mathematics and its Applications, Springer, New York, NY, 2001, 123.

  • [40]

    Willems J. C., From time series to linear system-part I. Finite dimensional linear time invariant systems, Automatica (Journal of IFAC), 1986, 22 (5), 561-580.

  • [41]

    Kuijper M., First Order Representations of Linear Systems, Ph.D. Thesis, Boston, MA: Birkhäuser, 1994

  • [42]

    Kuijper M., Schumacher J. M., Realization of Autoregressive Equations in Pencil and Descriptor Form, SIAM J. Control Optim., 1990, 28 (5), 1162-1189.

  • [43]

    Fornasini, Valcher M.E., Multidimensional systems with finite support behaviors, Signal structure generation and detection, SIAM J. Control Op., 1998, 36(2), 760 779.

  • [44]

    Popov V. M., Invariant description of linear time-invariant controllable systems, SIAM J. Control Opl, 1972, 10 (2), 252-264.

OPEN ACCESS

Journal + Issues

Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant and original works in all areas of mathematics. The journal publishes both research papers and comprehensive and timely survey articles. Open Math aims at presenting high-impact and relevant research on topics across the full span of mathematics.

Search