Admissible parameters of symmetric designs satisfying v=4(k-\lambda )+2 and symmetric designs with inner balance

The admissible parameters of symmetric \((v,k,\lambda )\) designs satisfying \(v=4(k-\lambda )+2\) are shown to correspond with the solutions of a certain Pell equation. We then determine the feasible parameters of such designs that could have a quasi-symmetric residual design with respect to a block, and classify them into two possible families. Finally, we consider the feasible parameters of symmetric designs with inner balance as defined by Nilson and Heidtmann (Des. Codes Cryptogr. doi:10.1007/s10623-012-9730-2, (2012)), and show that (with one exception) they must all belong to one of these families.