Kernels and ranks of cyclic and negacyclic quaternary codes

Steven T. Dougherty, Cristina Fernández-Córdoba

Research output: Contribution to journalArticleResearchpeer-review

6 Citations (Scopus)


© 2015, Springer Science+Business Media New York. We study the rank and kernel of Z4 cyclic codes of odd length n and give bounds on the size of the kernel and the rank. Given that a cyclic code of odd length is of the form C= ⟨ fh, 2 fg⟩ , where fgh= xn- 1 , we show that ⟨ 2 f⟩ ⊆ K(C) ⊆ C and C⊆ R(C) ⊆ ⟨ fh, 2 g⟩ where K(C) is the preimage of the binary kernel and R(C) is the preimage of the space generated by the image of C. Additionally, we show that both K(C) and R(C) are cyclic codes and determine K(C) and R(C) in numerous cases. We conclude by using these results to determine the case for negacyclic codes as well.
Original languageEnglish
Pages (from-to)347-364
JournalDesigns, Codes, and Cryptography
Issue number2
Publication statusPublished - 1 Nov 2016


  • Cyclic codes
  • Kernel
  • Quaternary codes
  • Rank


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