Codes of lengths 120 and 136 meeting the Grey-Rankin bound and quasi-symmetric designs

In this correspondence, we give a characterization of certain quasi-cyclic self-complementary codes with parameters [120,9,56] and [136,9,64], some quasi-cyclic self-complementary codes are also constructed with parameters [496,11,240] and [528,11,256]. These codes are optimal in the sense that they meet the Grey-Rankin bound, new quasi-symmetric SDP (symmetric difference property) designs are constructed from these codes.