Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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PRIMITIVE IDEMPOTENTS IN THE RING F4[x]/〈xpn-1〉 AND CYCLOTOMIC Q CODES

  • Batra, Sudhir (Department of Mathematics DCR University of Science and Technology) ;
  • Mathur, Rekha (Department of Mathematics DCR University of Science and Technology)
  • Received : 2017.05.18
  • Accepted : 2017.09.14
  • Published : 2018.05.31
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Abstract

The parity of cyclotomic numbers of order 2, 4 and 6 associated with 4-cyclotomic cosets modulo an odd prime p are obtained. Hence the explicit expressions of primitive idempotents of minimal cyclic codes of length $p^n$, $n{\geq}1$ over the quaternary field $F_4$ are obtained. These codes are observed to be subcodes of Q codes of length $p^n$. Some orthogonal properties of these subcodes are discussed. The minimal cyclic codes of length 17 and 43 are also discussed and it is observed that the minimal cyclic codes of length 17 are two weight codes. Further, it is shown that a Q code of prime length is always cyclotomic like a binary duadic code and it seems that there are infinitely many prime lengths for which cyclotomic Q codes of order 6 exist.

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References

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