Distance-Regular Graphs withbt=1 and Antipodal Double-Covers

LetΛbe a distance-regular graph of diameterdand valencyk>2. Ifbt=1 and 2t⩽d, thenΛis an antipodal double-cover. Consequently, iff>2 is the multiplicity of an eigenvalue of the adjacency matrix ofΛand ifΛis not an antipodal double-cover thend⩽2f−3. This result is an improvement of Godsilʹs bound.