Symmetric Designs Attached to Four-Weight Spin Models

H. Guo and T. Huang studied the four-weight spin models (X, W ₁, W ₂, W ₃, W ₄;D) with the property that the entries of the matrix W ₂ (or equivalently W ₄) consist of exactly two distinct values. They found that such spin models are always related to symmetric designs whose derived design with respect to any block is a quasi symmetric design. In this paper we show that such a symmetric design admits a four-weight spin model with exactly two values on W ₂ if and only if it has some kind of duality between the set of points and the set of blocks. We also give some examples of parameters of symmetric designs which possibly admit four-weight spin models with exactly two values on W ₂.