Designs, Codes and Cryptography

, Volume 72, Issue 2, pp 219–247 | Cite as

On the relative profiles of a linear code and a subcode

  • Zhuojun Zhuang
  • Bin Dai
  • Yuan Luo
  • A. J. Han Vinck


Relative dimension/length profile (RDLP), inverse relative dimension/length profile (IRDLP) and relative length/dimension profile (RLDP) are equivalent sequences of a linear code and a subcode. The concepts were applied to protect messages from an adversary in the wiretap channel of type II with illegitimate parties. The equivocation to the adversary is described by IRDLP and upper-bounded by the generalized Singleton bound on IRDLP. Recently, RLDP was also extended in wiretap network II for secrecy control of network coding. In this paper, we introduce new relations and bounds about the sequences. They not only reveal new connections among known results but also find applications in trellis complexities of linear codes. The state complexity profile of a linear code and that of a subcode can be bounded from each other, which is particularly useful when a tradeoff among coding rate, error-correcting capability and decoding complexity is considered. Furthermore, a unified framework is proposed to derive bounds on RDLP and IRDLP from an upper bound on RLDP. We introduce three new upper bounds on RLDP and use some of them to tighten the generalized Singleton bounds by applying the framework. The approach is useful to improve equivocation estimation in the wiretap channel of type II with illegitimate parties.


Generalized Hamming weight (GHW) Relative dimension/length profile (RDLP) Trellis complexity Wiretap channel Wiretap network 

Mathematics Subject Classification (2010)

94B05 94B65 


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Zhuojun Zhuang
    • 1
    • 2
  • Bin Dai
    • 1
    • 3
  • Yuan Luo
    • 1
  • A. J. Han Vinck
    • 4
  1. 1.Department of Computer Science and EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.The State Key Laboratory of Integrated Services NetworksXidian UniversityXi’anChina
  3. 3.National Mobile Communications Research LaboratorySoutheast UniversityNanjingChina
  4. 4.Institute for Experimental MathematicsDuisburg-Essen UniversityEssenGermany

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