On the Generating Function and Minimum Free Distance for a Class of Convolutional Codes

Document Type : Original Article

Author

Assistant Professor, Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran.

Abstract

It is common knowledge that convolutional coding is one of the two main types of channel coding widely used to detect and correct errors. In convolutional coding, the encoder output is in the form of a code sequence generated from an input information sequence. In this paper, an important class of convolutional encoders, the convolutional encoders  were considered. The transfer function matrix for this class of encoders were computed and their state diagram drawn. Moreover, the general form of the generating function for this class was obtained using the modified state diagram. Degenerate states and catastrophic convolutional codes appeared for some multipliers. Then, the minimum free distance for all non-catastrophic convolutional codes which are obtained from non-degenerate states were computed. As it is shown, the maximum value of the minimum free distance for this class of encoders is equal to five and it is obtained only in two situations.

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References

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Karafan Quarterly Scientific Journal
Volume 19, Issue 3 - Serial Number 59
Technical and Engineering
December 2022
Pages 647-661

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History

  • Receive Date: 26 June 2021
  • Revise Date: 26 August 2021
  • Accept Date: 02 September 2021

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