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MDS codes are erasure-correcting codes that can correct the maximum number oferasures for a given number of redundancy or parity symbols. If an MDS code has$r$ parities and no more than $r$ erasures occur, then by transmitting all theremaining data in the code, the original information can be recovered. However,it was shown that in order to recover a single symbol erasure, only a fractionof $1/r$ of the information needs to be transmitted. This fraction is calledthe repair bandwidth (fraction). Explicit code constructions were given inprevious works. If we view each symbol in the code as a vector or a column oversome field, then the code forms a 2D array and such codes are especially widelyused in storage systems. In this paper, we address the following question:given the length of the column $l$, number of parities $r$, can we constructhigh-rate MDS array codes with optimal repair bandwidth of $1/r$, whose codelength is as long as possible? In this paper, we give code constructions suchthat the code length is $(r+1)\log_r l$.

Explicit MDS Codes for Optimal Repair Bandwidth