Conventionally, an interweave cognitive radio (CR) system can be developed based on the assumption of spectrum sharing between a wide-band CR system and some narrow-band primary systems. In this paper, we explode this assumption by constructing an interweave CR system using a hierarchical wide-band two-dimensional (2D) spread multi-carrier directed-sequenced code division multiple access (MC-DS-CDMA). In this hierarchical system, a small CR cell is embedded in a large primary cell such that the limited serving area provides one geographical protection for the primary users (PUs). To facilitate the spreading code management, we define the white, gray and black spreading codes for the CR system. Properly allocating the white and gray spreading codes but the black ones can achieve a higher spectrum efficiency without harming the PUs, which provides another dimension for the PU protection in the proposed dynamic spectrum access (DSA) algorithm. For the purpose of evaluating the candidacy of the gray spreading codes, a sophisticated interference analysis is conducted so as to define a new multiple access interference (MAI) coefficient to quantize the amount of interference imposed on a secondary user (SU). With the help of the new MAI coefficient, a less interfered white or gray spreading code can be assigned to an SU, which can bring about an extra 55% of call admissions in one of our considered cases.