An elementary derivation of the annihilator polynomial for extremal (2s+1)-designs

Abstract

Let D be a (2s+1)-design with parameters (v, k, λ_2s+1). It is known that D has at least s+1 block intersection numbers x₁, x₂, ..., x_s+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.