AN ELEMENTARY DERIVATION OF THE ANNIHILATOR POLYNOMIAL FOR EXTREMAL (2S+1)-DESIGNS

Let D be a (2s + 1)-design with parameters (v, k, λ2s + 1). It is known that D has at least s + 1 block intersection numbers x1, x2, ..., xs + 1. Suppose now D is an extremal (2s + 1)-design with exactly s + 1 intersection numbers. In this case we give a short proof of the following known result of Delsarte:. The s + 1 intersection numbers are roots of a polynomial whose coefficients depend only on the design parameters. Delsarte's result, proved more generally, for designs in Q-polynomial association schemes, uses the notion of the annihilator polynomial. Our proof relies on elementary ideas and part of an algorithm used for decoding BCH codes. © 1990.