In this paper, we propose two new classes of binary array codes, termed V-ETBR and V-ESIP codes, by reformulating and generalizing the variant technique of deriving the well-known generalized row-diagonal parity~(RDP) codes from shortened independent parity~(IP) codes. The V-ETBR and V-ESIP codes are both based on binary parity-check matrices and are essentially variants of two classes of codes over a special polynomial ring (termed ETBR and ESIP codes in this paper). To explore the conditions that make the variant codes binary Maximum Distance Separable~(MDS) array codes that achieve optimal storage efficiency, this paper derives the connections between V-ETBR/V-ESIP codes and ETBR/ESIP codes. These connections are beneficial for constructing various forms of the variant codes. By utilizing these connections, this paper also explicitly presents the constructions of V-ETBR and V-ESIP MDS array codes with any number of parity columns $r$, along with their fast syndrome computations. In terms of construction, all proposed MDS array codes have an exponentially growing total number of data columns with respect to the column size, while alternative codes have that only with linear order. In terms of computation, the proposed syndrome computations make the corresponding encoding/decoding asymptotically require $\lfloor \lg r \rfloor+1$ XOR~(exclusive OR) operations per data bit, when the total number of data columns approaches infinity. This is also the lowest known asymptotic complexity in MDS codes.

翻译：本文通过重新表述和推广从缩短独立奇偶校验(IP)码推导著名广义行对角奇偶校验(RDP)码的变体技术，提出了两类新的二进制阵列码，称为V-ETBR码和V-ESIP码。V-ETBR码和V-ESIP码均基于二进制奇偶校验矩阵，本质上是两类基于特殊多项式环的码（本文称为ETBR码和ESIP码）的变体。为探究使变体码成为达到最优存储效率的二进制最大距离可分(MDS)阵列码的条件，本文推导了V-ETBR/V-ESIP码与ETBR/ESIP码之间的联系。这些联系有助于构造各种形式的变体码。利用这些联系，本文还明确给出了具有任意奇偶列数$r$的V-ETBR和V-ESIP MDS阵列码的构造及其快速伴随式计算。在构造方面，所有提出的MDS阵列码的数据列总数随列大小呈指数增长，而其他码仅呈线性增长。在计算方面，当数据列总数趋近无穷大时，所提出的伴随式计算使得相应的编码/解码渐进地每数据位仅需$\lfloor \lg r \rfloor+1$次异或(XOR)运算。这也是MDS码中已知的最低渐近复杂度。