New Constructions of q-Ary MDS Array Codes With Multiple Parities and Their Effective Decoding

From the perspective of parity-check matrices, we present new constructions of
$q$
-ary maximum distance separable (MDS) array codes with multiple parities. Applying these constructions, some new types of MDS array codes with array numbers
$m-\tau $
can be derived, where
${\mathrm{ gcd}}(m,q)=1$
. Moreover, an explicit construction of binary MDS array codes is also presented. Compared to the existing MDS array codes, one important characteristic of these codes is that their available code lengths are much longer, which is suitable for large-scale storage systems. In some particular cases, the maximum code lengths of these codes and their extension can be up to
$2^{m-\tau }$
and
$2^{m-\tau }+1$
(or
$2^{m-\tau }+2$
), respectively. Moreover, to demonstrate the applicability of our constructed MDS array codes, we present an effective generic decoding method for the erased errors. In particular, when there are no more than three erasures occurring, a scheduled algorithm for the syndrome computation of our explicit construction is further proposed, whose computational complexity is asymptotically optimal. Furthermore, this algorithm can be directly applied to the encoding procedure of their extended form. The simulation shows that our new MDS array codes have better encoding and decoding performances than the corresponding extended RS codes coupled with different algorithms.