An Upper Bound for the Cardinality of an s-Distance Set in Euclidean Space

In this paper we show that if X is an s-distance set in ℝm and X is on p concentric spheres then \( {\left| X \right|} \leqslant {\sum\nolimits_{i = 0}^{2p - 1} {{\left( {\begin{array}{*{20}c} {{m + s - i - 1}} \\ {{s - i}} \\ \end{array} } \right)}} } \) Moreover if X is antipodal, then \( {\left| X \right|} \leqslant 2{\sum\nolimits_{i = 0}^{p - 1} {{\left( {\begin{array}{*{20}c} {{m + s - 2i - 2}} \\ {{m - 1}} \\ \end{array} } \right)}} } \).

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Correspondence to Etsuko Bannai.

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Bannai, E., Kawasaki, K., Nitamizu, Y. et al. An Upper Bound for the Cardinality of an s-Distance Set in Euclidean Space. Combinatorica 23, 535–557 (2003). https://doi.org/10.1007/s00493-003-0032-1

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Mathematics Subject Classification (2000):

  • 05E99
  • 05B99
  • 51M99
  • 62K99