The Bounds of Rigid Sphere Design

Abstract We give some necessary conditions for the existence of rigid sphere designs. In particular, we settle the conjecture proposed by Bannai (J Fac Sci Univ Tokyo Sect IA Math 34(3), 485-489, 1987). Given fixed positive integers t and d , we show that there exist only finitely many rigid t -designs on $$S^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , up to orthogonal transformations.