  In this paper, we propose two new constructions of exact-repair minimum
storage regenerating (exact-MBR) codes. Both constructions obtain the encoded
symbols by first treating the message vector over GF(q) as a linearized
polynomial and then evaluating it over an extension field GF(q^m). The
evaluation points are chosen so that the encoded symbols at any node are
conjugates of each other, while corresponding symbols of different nodes are
linearly dependent with respect to GF(q). These properties ensure that data
repair can be carried out over the base field GF(q), instead of matrix
inversion over the extension field required by some existing exact-MBR codes.
To the best of our knowledge, this approach is novel in the construction of
exact-MBR codes. One of our constructions leads to exact-MBR codes with
arbitrary parameters. These exact-MBR codes have higher data reconstruction
complexities but lower data repair complexities than their counterparts based
on the product-matrix approach; hence they may be suitable for applications
that need a small number of data reconstructions but a large number of data
repairs.
