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

Variant Codes Based on A Special Polynomial Ring and Their Fast Computations