New Construction of MDS Array Codes and Explicit Characterization of Decoding Matrices

Row-Diagonal-Parity (RDP) codes and EVENODD codes are classical systematic array codes and most attention in the literature has been on the generalization of RDP codes. In this work, as generalization of not only RDP codes but also EVENODD codes, we present new construction of $\phi (L)$ -dimensional $(k+r, k)$ systematic array codes with $r \leq 4$ , where L is an odd integer and $\phi (L)$ represents the Euler’s totient function of L. We explicitly characterize sufficient conditions on the selection of L to make the codes maximum distance separable (MDS). Compared with EVENODD codes and RDP codes, the largest k that can be supported by the new codes is nearly doubled, and the asymptotic encoding complexity of the new codes is same, that is, asymptotically approaches r XORs per original data bit with increasing L and k. Moreover, for prime L, $r = 2$ and $k = 2L-3$ , the new code exactly achieves the optimal encoding complexity. For the case $r = 4$ , the largest k that can be supported by the new codes is larger than the recently proposed so-called Variants of Extended Shortened Independent-Parity (V-ESIP) systematic array code in a number of code dimension selections, and meanwhile, the obtained explicit conditions on L to guarantee the MDS property of the new codes also apply to classical EVENODD codes and RDP codes, but are more general than well known explicit ones in the literature. The decoding process of the new array codes is also discussed. In particular, the $r\times r$ block inverse matrix involved in decoding is explicitly characterized, which applies to all MDS array codes generalized from RDP or EVENODD codes in the literature.