Graphs and Combinatorics

, Volume 33, Issue 6, pp 1405–1418 | Cite as

Harmonic Index Designs in Binary Hamming Schemes

  • Yan Zhu
  • Eiichi Bannai
  • Etsuko Bannai
  • Takuya Ikuta
  • Kyoung-Tark KimEmail author
Original Paper


We recall the “addition formula” for a commutative association scheme. From this formula we construct a linear programming for the size of harmonic index designs and define the notion of tight designs using Fisher type inequality. We examine the existence of tight harmonic index T-designs for some T in binary Hamming association schemes H(d, 2).


Harmonic index design Hamming association scheme Addition formula Fisher type lower bound Tight design 

Mathematics Subject Classification

05B30 05E30 



We thank the referees for helpful comments which lead to the improvement of the paper. In the original version, the cases of \(\{ 2 \}\)- and \(\{ 4,2 \}\)-designs were not explicitly mentioned. Eiichi Bannai was supported in part by NSFC Grant No. 11271257.


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Copyright information

© Springer Japan 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanghai Jiao Tong UniversityShanghaiChina
  2. 2.FukuokaJapan
  3. 3.Faculty of LawKobe Gakuin UniversityKobeJapan
  4. 4.Sogang research team for discrete and geometric structuresSogang UniversitySeoulKorea

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