Designs, Codes and Cryptography

, Volume 86, Issue 5, pp 1063–1083 | Cite as

Constructions of cyclic quaternary constant-weight codes of weight three and distance four



A cyclic \((n,d,w)_q\) code is a cyclic q-ary code of length n, constant-weight w and minimum distance d. A cyclic \((n,d,w)_q\) code with the largest possible number of codewords is said to be optimal. Optimal nonbinary cyclic \((n,d,w)_q\) codes were first studied in our recent paper (Lan et al. in IEEE Trans Inf Theory 62(11):6328–6341, 2016). In this paper, we continue to discuss the constructions of optimal cyclic \((n,4,3)_q\) codes. We establish the connection between cyclic \((n,4,3)_{q}\) codes and \(q-1\) mutually orbit-disjoint cyclic (n, 3, 1) difference packings (briefly (n, 3, 1)-CDPs). For the case of \(q=4\), we construct three mutually orbit-disjoint (n, 3, 1)-CDPs by constructing a pair of strongly orbit-disjoint (n, 3, 1)-CDPs, which are obtained from Skolem-type sequences. As a consequence, we completely determine the number of codewords of an optimal cyclic \((n,4,3)_{4}\) code.


Constant-weight code Cyclic Optimal Cyclic difference packing Orbit-disjoint 

Mathematics Subject Classification




Supported by the NSFC under Grant 11431003 (Y. Chang), and the NSFC under Grant 11401582 and the NSFHB under Grant A2015507019 (L. Wang).


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute of MathematicsBeijing Jiaotong UniversityBeijingPeople’s Republic of China
  2. 2.Basic Subject Application and Development Research CenterChinese People’s Armed Police Force AcademyLangfangPeople’s Republic of China

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