  Generalized Expanded-Blaum-Roth (GEBR) codes [1] are designed for large-scale
distributed storage systems that have larger recoverability for single-symbol
failures, multi-column failures and multi-row failures, compared with locally
recoverable codes (LRC). GEBR codes encode an $\alpha\times k$ information
array into a $p\tau\times (k+r)$ array such that lines of slope $i$ with $0\leq
i\leq r-1$ have even parity and each column contains $p\tau-\alpha$ local
parity symbols, where $p$ is an odd prime and $k+r\leq p\tau$. Necessary and
sufficient conditions for GEBR codes to be $(n,k)$ recoverable (i.e., any $k$
out of $n=k+r$ columns can retrieve all information symbols) are given in [2]
for $\alpha=(p-1)\tau$. However, the $(n,k)$ recoverable condition of GEBR
codes is unknown when $\alpha<(p-1)\tau$. In this paper, we present the $(n,k)$
recoverable condition for GEBR codes for $\alpha< (p-1)\tau$. In addition, we
present a sufficient condition for enabling GEBR codes to recover some erased
lines of any slope $i$ ($0\leq i\leq p\tau-1$) for any parameter $r$ when
$\tau$ is a power of $p$. Moreover, we present the construction of Generalized
Blaum-Roth (GBR) codes that encode an $\alpha\times k$ information array into
an $\alpha\times (k+r)$ array. We show that GBR codes share the same MDS
condition as the $(n,k)$ recoverable condition of GEBR codes, and we also
present a sufficient condition for GBR codes to recover some erased lines of
any slope $i$ ($0\leq i\leq \alpha-1$).
