A Generalization of Array Codes With Local Properties and Efficient Encoding/Decoding

An
$(n,k)$
recoverable property array code is composed of
$m\times n$
arrays such that any
$k$
out of
$n$
columns suffice to retrieve all the information symbols, where
$n > k$
. Note that maximum distance separable (MDS) array code is a special
$(n,k)$
recoverable property array code of size
$m\times n$
with the number of information symbols being
$km$
. Expanded-Blaum-Roth (EBR) codes and Expanded-Independent-Parity (EIP) codes are two classes of
$(n,k)$
recoverable property array codes that can repair any one symbol in a column by locally accessing some other symbols within the column, where the number of symbols
$m$
in a column is a prime number. By generalizing the constructions of EBR and EIP codes, we propose new
$(n,k)$
recoverable property array codes, such that any one symbol can be locally recovered and the number of symbols in a column can be not only a prime number but also a power of an odd prime number. Also, we present an efficient encoding/decoding method for the proposed generalized EBR (GEBR) and generalized EIP (GEIP) codes based on the LU factorization of a Vandermonde matrix. We show that the proposed decoding method has less computational complexity than existing methods. Furthermore, we show that the proposed GEBR codes have both a larger minimum symbol distance and a larger recovery ability of erased lines for some parameters when compared to EBR codes. We also present a necessary and sufficient condition of enabling EBR codes to recover any
$r$
erased lines of a slope for any parameter
$r$
, which was an open problem. Moreover, we show that EBR codes can recover any
$r$
consecutive erased lines of any slope for any parameter
$r$
.