This paper considers the parity-check matrix of a maximum distance separable (MDS) array code as a superposition of two matrices, block diagonal matrix A and side matrix S. By matching their entries, the syndrome calculation of the matrix A can share computations with that of the side matrix S. Then, a low-complexity encoding, referred to as side encoding, is proposed to encode matched MDS array codes efficiently. Moreover, it can combine with the Reed-Muller transform-based (RMTB) Reed-Solomon encoding algorithm to reduce the encoding complexity further. The analysis indicates that for the MDS array code [1] with four parity nodes, the number of multiplications is reduced by 85.4% and 52.6% compared to the traditional encoding and only RMTB encoding, respectively.

To achieve various tradeoffs between storage and repair bandwidth, this article proposes to construct exact repair codes by grafting two codesC1andC2. By replacing certain nonzero entries in the generator matrix ofC1by zero, the repair bandwidth of the resulting grafting part decreases. However, it may no longer keep the maximum-distance-separable (MDS) property. As a result, the grafted codeC2takes these nonzero entries into account such that the entire graftage code can keep the MDS property. The relationship between the bandwidth reduction ofC1and the file size ofC2is derived to optimize graftage codes. Our analysis indicates that these graftage codes may provide better tradeoffs than space-sharing.

Recently, several remarkable works [1]-[5] constructed regenerating codes to offer intermediate tradeoffs between storage and bandwidth. Unlike regenerating codes, this paper proposes to graft codes together to provide various such intermediate tradeoffs. It shows that the linear relations in the generator matrices of grafting codes can be transferred to those of grafted codes without any loss of reconstruction capability. A construction based on minimum storage regenerating codes shows that the resulted graftage codes may provide better tradeoffs than space-sharing and approach cut-set bounds, with the cost of fixed access of helper nodes.