  The $(n,k,d)$ regenerating code is a class of $(n,k)$ erasure codes with the
capability to recover a lost code fragment from other $d$ existing code
fragments. This paper concentrates on the design of exact regenerating codes at
Minimum Bandwidth Regenerating (MBR) points. For $d=n-1$, a class of
$(n,k,d=n-1)$ Exact-MBR codes, termed as repair-by-transfer codes, have been
developed in prior work to avoid arithmetic operations in node repairing
process. The first result of this paper presents a new class of
repair-by-transfer codes via congruent transformations. As compared with the
prior works, the advantages of the proposed codes include: i) The minimum of
the finite field size is significantly reduced from $n \choose 2$ to $n$. ii)
The encoding complexity is decreased from $n^4$ to $n^3$. As shown in
simulations, the proposed repair-by-transfer codes have lower computational
overhead when $n$ is greater than a specific constant. The second result of
this paper presents a new form of coding matrix for product-matrix Exact-MBR
codes. The proposed coding matrix includes a number of advantages: i). The
minimum of the finite field size is reduced from $n-k+d$ to $n$. ii). The fast
Reed-Solomon erasure coding algorithms can be applied on the Exact-MBR codes to
reduce the time complexities.
