  In distributed storage systems, erasure codes with locality $r$ is preferred
because a coordinate can be recovered by accessing at most $r$ other
coordinates which in turn greatly reduces the disk I/O complexity for small
$r$. However, the local repair may be ineffective when some of the $r$
coordinates accessed for recovery are also erased.
  To overcome this problem, we propose the $(r,\delta)_c$-locality providing
$\delta -1$ local repair options for a coordinate. Consequently, the repair
locality $r$ can tolerate $\delta-1$ erasures in total. We derive an upper
bound on the minimum distance $d$ for any linear $[n,k]$ code with information
$(r,\delta)_c$-locality. For general parameters, we prove existence of the
codes that attain this bound when $n\geq k(r(\delta-1)+1)$, implying tightness
of this bound. Although the locality $(r,\delta)$ defined by Prakash et al
provides the same level of locality and local repair tolerance as our
definition, codes with $(r,\delta)_c$-locality are proved to have more
advantage in the minimum distance. In particular, we construct a class of codes
with all symbol $(r,\delta)_c$-locality where the gain in minimum distance is
$\Omega(\sqrt{r})$ and the information rate is close to 1.
