The exact-repair regenerating codes for distributed storage system are studied in this work. A novel coding scheme is proposed for code construction for any $(n,k,d=k)$ system. It is shown that the proposed codes are optimum, in the sense that any operating point satisfying the lower bound for the storage-bandwidth trade-off can be achieved with the proposed construction. As a consequence, the optimum linear trade-off of exact-regenerating system is fully characterized for any systems with $d=k$ . The proposed codes are linear, and can be generated by multiplication of an encoder matrix with a so-called data matrix. The exact regenerating property is provided based on fundamental properties of matrix determinants, and in particular, generalized Laplace expansion for determinant. Thus the code is called determinant code . Importantly, the field size required for this code construction is $\Theta (n)$ , the total number of nodes in the system.