Characterizations of regularity for certain $Q$-polynomial association schemes

Understand

It is shown that linked systems of symmetric designs with $a_1^*=0$ and mutually unbiased bases (MUB) are triply regular association schemes.

• In this paper, we characterize triple regularity of linked systems of symmetric designs by its Krein number.
• And we prove that maximal MUB carries a quadruply regular association scheme and characterize the quadruple regularity of MUB by its parameter.