Fissioned Triangular Schemes via the Cross-ratio

AbstractA construction of association schemes is presented; these are fission schemes of the triangular schemesT (n) where n=q+ 1 with q any prime power. The key observation is quite elementary, being that the natural action of PGL(2, q) on the 2-element subsets of the projective line PG(1, q) is generously transitive. In addition, some observations on the intersection parameters and fusion schemes of these association schemes are made

Fissioned Triangular Schemes Via the Cross-Ratio

A construction of association schemes is presented; these are fission schemes of the triangular schemes T(n) where n=q + 1with q any prime power. The key observation is quite elementary, being that the natural action of PGL(2,q) on the 2-element subsets of the projective line PG(1,q) is generously transitive. Also some observations on the intersection parameters and fusion schemes of these association schemes are made.

On the subschemes of the Johnson scheme J(v,d)

For two association schemes $chi$ and ${chi}'$, defined on the same set, we call ${chi}'$ a subscheme of $chi$ if each relation of ${chi}'$ is a union of some relations of $chi$. In this paper we prove that the Johnson scheme $J(v, d)$ has no non-trivial subscheme if $v > 2d + \frac{3}{2} + sqrt{( d - \frac{7}{2} )^2 + 6}$. This slightly improves the earlier result of Muzichuk that the conclusion holds if $v geqq 3d + 4$