Abstract: We propose a new and unified construction method, general supplementary difference sets (GSDS)s, for near-Hadamard designs when the run sizes are n ≡ 2 (mod 4). These designs possess high D-efficiencies. Ehlich (1964) derived an upper bound for the determinant of matrices of order n ≡ 2 (mod 4) achievable only if 2n - 2 is a sum of two squares. Between 1 to 100, there are 6 parameters, 22,34,58,70,78, and 94, that do not fulfill this condition. We formulate a class of near-Hadamard designs whose determinants are very close to Ehlich’s upper bound, and construct these designs for many values of n.

Key words and phrases: Difference sets, general supplementary difference sets, Hadamard design, near-Hadamard design, optimal designs.