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On the State Approach Representations of Convolutional Codes over Rings of Modular Integers

Author

Listed:
  • Ángel Luis Muñoz Castañeda

    (Department of Mathematics, Universidad de León, 24007 León, Spain)

  • Noemí DeCastro-García

    (Department of Mathematics, Universidad de León, 24007 León, Spain)

  • Miguel V. Carriegos

    (Department of Mathematics, Universidad de León, 24007 León, Spain)

Abstract

In this study, we prove the existence of minimal first-order representations for convolutional codes with the predictable degree property over principal ideal artinian rings. Further, we prove that any such first-order representation leads to an input/state/output representation of the code provided the base ring is local. When the base ring is a finite field, we recover the classical construction, studied in depth by J. Rosenthal and E. V. York. This allows us to construct observable convolutional codes over such rings in the same way as is carried out in classical convolutional coding theory. Furthermore, we prove the minimality of the obtained representations. This completes the study of the existence of input/state/output representations of convolutional codes over rings of modular integers.

Suggested Citation

  • Ángel Luis Muñoz Castañeda & Noemí DeCastro-García & Miguel V. Carriegos, 2021. "On the State Approach Representations of Convolutional Codes over Rings of Modular Integers," Mathematics, MDPI, vol. 9(22), pages 1-19, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2962-:d:683763
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    References listed on IDEAS

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    1. Rosenthal, J. & Schumacher, J.M. & York, E.V., 1996. "On behaviors and convolutional codes," Other publications TiSEM 1c23f564-7bce-463e-898b-6, Tilburg University, School of Economics and Management.
    2. Kuijper, M. & Schumacher, J.M., 1990. "Realization of autoregressive equations in pencil and descriptor form," Other publications TiSEM 3c164c8f-c092-448d-8ae7-9, Tilburg University, School of Economics and Management.
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