Exact Cooperative Regenerating Codes with Minimum-Repair-Bandwidth for   Distributed Storage
  We give an explicit construction of exact cooperative regenerating codes at the MBCR (minimum bandwidth cooperative regeneration) point. Before the paper, the only known explicit MBCR code is given with parameters $n=d+r$ and $d=k$, while our construction applies to all possible values of $n,k,d,r$. The code has a brief expression in the polynomial form and the data reconstruction is accomplished by bivariate polynomial interpolation. It is a scalar code and operates over a finite field of size $q\geq n$. Besides, we establish several subspace properties for linear exact MBCR codes. Based on these properties we prove that linear exact MBCR codes cannot achieve repair-by-transfer. 
Anyu Wang and Zhifang Zhang
cs.IT math.IT