An Explicit Construction of $q\text{-ary}$ MDS Array Codes and Their Efficient Decoding

In this short work, a new explicit construction of
$q-\mathbf{ary}$
MDS array codes with multiple parities will be provided, whose code lengths can be up to
$q^{m-1}$
, where
$m-1$
is the size of subpackage. As far as we know, this may be the first explicit construction of practical MDS array codes with such long code lengths for general
$q$
. In addition, to demonstrate the applicability of our MDS array codes, by the LU factorization of Vandermonde matrices, we present an efficient decoding method aimed at the erased errors, whose computational complexity is
$O(m^{2})$
in total. Furthermore, if one stores a small number of polynomials in advance or computes the syndrome in a scheduled algorithm, the decoding efficiency of these MDS array codes can be further improved.