  Distributed storage systems suffer from significant repair traffic generated
due to frequent storage node failures. This paper shows that properly designed
low-density parity-check (LDPC) codes can substantially reduce the amount of
required block downloads for repair thanks to the sparse nature of their factor
graph representation. In particular, with a careful construction of the factor
graph, both low repair-bandwidth and high reliability can be achieved for a
given code rate. First, a formula for the average repair bandwidth of LDPC
codes is developed. This formula is then used to establish that the minimum
repair bandwidth can be achieved by forcing a regular check node degree in the
factor graph. Moreover, it is shown that given a fixed code rate, the variable
node degree should also be regular to yield minimum repair bandwidth, under
some reasonable minimum variable node degree constraint. It is also shown that
for a given repair-bandwidth requirement, LDPC codes can yield substantially
higher reliability than currently utilized Reed-Solomon (RS) codes. Our
reliability analysis is based on a formulation of the general equation for the
mean-time-to-data-loss (MTTDL) associated with LDPC codes. The formulation
reveals that the stopping number is closely related to the MTTDL. It is further
shown that LDPC codes can be designed such that a small loss of
repair-bandwidth optimality may be traded for a large improvement in
erasure-correction capability and thus the MTTDL.
