Regenerating codes provide an efficient way to recover data at failed nodes
in distributed storage systems. It has been shown that regenerating codes can
be designed to minimize the per-node storage (called MSR) or minimize the
communication overhead for regeneration (called MBR). In this work, we propose
new encoding schemes for $[n,d]$ error-correcting MSR and MBR codes that
generalize our earlier work on error-correcting regenerating codes. We show
that by choosing a suitable diagonal matrix, any generator matrix of the
$[n,\alpha]$ Reed-Solomon (RS) code can be integrated into the encoding matrix.
Hence, MSR codes with the least update complexity can be found. By using the
coefficients of generator polynomials of $[n,k]$ and $[n,d]$ RS codes, we
present a least-update-complexity encoding scheme for MBR codes. A decoding
scheme is proposed that utilizes the $[n,\alpha]$ RS code to perform data
reconstruction for MSR codes. The proposed decoding scheme has better error
correction capability and incurs the least number of node accesses when errors
are present. A new decoding scheme is also proposed for MBR codes that can
correct more error-patterns.