  Binary array codes are widely used in storage systems to prevent data loss,
such as the Redundant Array of Independent Disks~(RAID). Most designs for such
codes, such as Blaum-Roth~(BR) codes and Independent-Parity~(IP) codes, are
carried out on the polynomial ring F_2[x]/<\sum_{i=0}^{p-1}x^i >, where F_2 is
a binary field, and p is a prime number. In this paper, we consider the
polynomial ring F_2[x]/<\sum_{i=0}^{p-1}x^{i\tau}>, where p>1 is an odd number
and \tau \geq 1 is any power of two, and explore variant codes from codes over
this polynomial ring. Particularly, the variant codes are derived by mapping
parity-check matrices over the polynomial ring to binary parity-check matrices.
  Specifically, we first propose two classes of variant codes, termed V-ETBR
and V-ESIP codes. To make these variant codes binary maximum distance
separable~(MDS) array codes that achieve optimal storage efficiency, this paper
then derives the connections between them and their counterparts over
polynomial rings. These connections are general, making it easy to construct
variant MDS array codes from various forms of matrices over polynomial rings.
Subsequently, some instances are explicitly constructed based on Cauchy and
Vandermonde matrices. In the proposed constructions, both V-ETBR and V-ESIP MDS
array codes can have any number of parity columns and have the total number of
data columns of exponential order with respect to $p$. In terms of computation,
two fast syndrome computations are proposed for the Vandermonde-based V-ETBR
and V-ESIP MDS array codes, both meeting the lowest known asymptotic complexity
among MDS codes. Due to the fact that all variant codes are constructed from
parity-check matrices over simple binary fields instead of polynomial rings,
they are attractive in practice.
