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The primitive idempotents of irreducible constacyclic codes and LCD cyclic codes

Abstract

In this paper, we investigate all irreducible factors of \(x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a\) over \(\mathbb {F}_{q}\) and obtain all primitive idempotents in \(\mathbb {F}_{q}[x]/\langle x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a \rangle \), where \(a \in \mathbb {F}_{q}^{*}\), l1, l2 are two distinct odd prime divisors of qt − 1 with \(\gcd (l_{1}l_{2},q(q-1))= 1\) for prime t. Furthermore, the weight distributions of all irreducible constacyclic codes of length \(l_{1}^{m_{1}}l_{2}^{m_{2}}\) are presented for t = 2. As an application, we determine all linear complementary dual cyclic codes of length \(l_{1}^{m_{1}}l_{2}^{m_{2}}\) over \(\mathbb {F}_{q}\).

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Acknowledgments

The authors are very grateful to the reviewers and the Editor for their comments and suggestions that improved the quality and presentation of this paper. This research is supported by the National Natural Science Foundation of China under Grant No. 61571243.

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Correspondence to Zexia Shi.

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Shi, Z., Fu, F. The primitive idempotents of irreducible constacyclic codes and LCD cyclic codes. Cryptogr. Commun. 12, 29–52 (2020). https://doi.org/10.1007/s12095-019-00362-w

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Keywords

  • Primitive idempotent
  • Irreducible constacyclic code
  • Weight distribution
  • LCD code

Mathematics Subject Classification (2010)

  • 11T71
  • 94B05
  • 94B15