Construction of Binary Array Codes Based on Matrix Transformation

This paper proposes to construct binary array codes from non-binary codes by binary matrix dispersion. Thanks to their low-density and quasi-cyclic (QC) generator matrices, the proposed codes enjoy low encoding complexity, about $1/(m+1)$ of the XOR operations used in their counterparts over $\mathbf{G F}\left(2^m\right)$. Their maximum distance separable (MDS) property and repair can be efficiently analyzed via matrix transformation. The MDS property is ensured by introducing Vandermonde upper triangular form and pre-processing. Moreover, the efficient repair can be guaranteed by applying a matrix mapping called all-one superposition on the associated matrices. Analysis indicates that their code rate and repair ratio asymptotically approach those of their non-binary counterparts.