On the coexistence of conference matrices and near resolvable 2-(2k+1,k,k-1) designs

We show that a near resolvable 2-(2k+1,k,k-1) design exists if and only if a conference matrix of order 2k+2 does. A known result on conference matrices then allows us to conclude that a near resolvable 2-(2k+1,k,k-1) design with even k can only exist if 2k+1 is the sum of two squares. In particular, neither a near resolvable 2-(21,10,9) design nor does a near resolvable 2-(33,16,15) design exist. For k⩽14, we also enumerate the near resolvable 2-(2k+1,k,k-1) designs and the corresponding conference matrices.