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Maximum Distance Separable (MDS) codes with a sparse and balanced generatormatrix are appealing in distributed storage systems for balancing andminimizing the computational load. Such codes have been constructed viaReed-Solomon codes over large fields. In this paper, we focus on small fields.We prove that there exists an $[n,k]_q$ MDS code that has a sparse and balancedgenerator matrix for any $q\geq n$ provided that $n\leq 2k$, by designingseveral algorithms with complexity running in polynomial time in $k$ and $n$.

Sparse and Balanced MDS Codes over Small Fields