On the coexistence of conference matrices and near resolvable designs

We show that a near resolvable 2 - ( 2 k + 1 , k , k - 1 ) design exists if and only if a conference matrix of order 2 k + 2 does. A known result on conference matrices then allows us to conclude that a near resolvable 2 - ( 2 k + 1 , k , k - 1 ) design with even k can only exist if 2 k + 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 - ( 2 k + 1 , k , k - 1 ) designs and the corresponding conference matrices.