Regenerating codes are a class of distributed storage codes that allow for efficient repair of failed nodes, as compared to traditional erasure codes. An $[n, k, d]$ regenerating code permits the data to be recovered by connecting to any $k$ of the $n$ nodes in the network, while requiring that a failed node be repaired by connecting to any $d$ nodes. The amount of data downloaded for repair is typically much smaller than the size of the source data. Previous constructions of exact-regenerating codes have been confined to the case $n=d+1$ . In this paper, we present optimal, explicit constructions of (a) Minimum Bandwidth Regenerating (MBR) codes for all values of $[n, k, d]$ and (b) Minimum Storage Regenerating (MSR) codes for all $[n, k, d\geq 2k-2]$ , using a new product-matrix framework. The product-matrix framework is also shown to significantly simplify system operation. To the best of our knowledge, these are the first constructions of exact-regenerating codes that allow the number $n$ of nodes in the network, to be chosen independent of the other parameters. The paper also contains a simpler description, in the product-matrix framework, of a previously constructed MSR code with $[n=d+1, k, d\geq 2k-1]$ .

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