Harmonic Index Designs in Binary Hamming Schemes


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).

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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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Correspondence to Kyoung-Tark Kim.

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Zhu, Y., Bannai, E., Bannai, E. et al. Harmonic Index Designs in Binary Hamming Schemes. Graphs and Combinatorics 33, 1405–1418 (2017). https://doi.org/10.1007/s00373-017-1784-5

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

Mathematics Subject Classification

  • 05B30
  • 05E30