Extending Symmetric Designs

Abstract If a symmetric 2-design with parameters ( v , k , λ ) is extendable, then one of the following holds: v = 4 λ + 3, k = 2 λ + 1; or v = ( λ + 2)( λ 2 + 4 λ + 2), k = λ 2 + 3 λ + 1; or v = 111, k = 11, λ = 1; or v = 495, k = 39, λ = 3. In particular, there are at most three sets of extendable symmetric design parameters with any given value of λ. As a consequence, the only twice-extendable symmetric design is the 21-point projective plane.