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Relative t-Designs in Nonbinary Hamming Association Schemes

Abstract

Relative t-designs in binary Hamming association schemes are equivalent to weighted regular t-wise balanced designs which are studied to some extent. In the paper, we extend the investigation to relative t-designs in nonbinary Hamming association schemes. Each nontrivial shell of a nonbinary Hamming association scheme is a symmetric association scheme which is called q-ary or nonbinary Johnson association scheme. Using the addition formula for Krawtchouk polynomials, we prove that the subset on each shell of a relative t-design in nonbinary Hamming association schemes supported by p shells is a weighted \(\mathcal T\)-design in nonbinary Johnson association scheme for \(\mathcal T=\{(k,h)\mid 0\le h\le k\le t+1-p\}\). We also give a combinatorial characterization of the subset of a relative t-design on each shell making use of regular semilattice. In addition, we obtain some lower bounds on the size as well as the degree of \(\mathcal T\)-designs in nonbinary Johnson association schemes.

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Acknowledgements

The author is supported by NSFC Grant No. 11801353.

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Correspondence to Yan Zhu.

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Dedicated to Professor Eiichi Bannai on the occasion of his 75-th birthday.

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Zhu, Y. Relative t-Designs in Nonbinary Hamming Association Schemes. Graphs and Combinatorics 37, 1775–1791 (2021). https://doi.org/10.1007/s00373-021-02389-4

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  • DOI: https://doi.org/10.1007/s00373-021-02389-4

Keywords

  • Relative t-design
  • nonbinary Hamming association scheme
  • nonbinary Johnson association scheme
  • addition formula
  • regular semilattice