Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ON THE IDEMPOTENTS OF CYCLIC CODES OVER 𝔽2t

  • Received : 2022.05.30
  • Accepted : 2022.12.01
  • Published : 2022.12.30
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Abstract

We study cyclic codes of length n over 𝔽2t. Cyclic codes can be viewed as ideals in 𝓡n = 𝔽2t [x]/(xn − 1). It is known that there is a unique generating idempotent for each ideal. Let e(x) ∈ 𝓡n. If t = 1 or t = 2, then there is a necessary and sufficient condition that e(x) is an idempotent. But there is no known similar result for t ≥ 3. In this paper we give an answer for this problem.

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Acknowledgement

The author would like to thank the referees for their valuable comments which improved the clarity of this paper.

References

  1. W. C. Huffman, V. S. Pless, Fundamentals of Error-correcting Codes, Cambridge: Cambridge University Press, 2003.
  2. W. W. Peterson, Error-Correcting Codes, Cambridge, MA: MIT Press, 1961.
  3. W. W. Peterson, E. J. Weldon, Jr., Error-Correcting Codes, 2nd ed. Cambridge, MA.: MIT Press, 1972.