On MDS Condition and Erased Lines Recovery of Generalized Expanded-Blaum-Roth Codes and Generalized Blaum-Roth Codes
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Generalized Expanded-Blaum-Roth (GEBR) codes [1] are designed for large-scale distributed storage systems that have larger recoverability for single-symbol failures, multi-column failures and multi-row failures, compared with locally recoverable codes (LRC). GEBR codes encode an α× k information array into a pτ× (k+r) array such that lines of slope i with 0≤ i≤ r-1 have even parity and each column contains pτ-α local parity symbols, where p is an odd prime and k+r≤ pτ. Necessary and sufficient conditions for GEBR codes to be (n,k) recoverable (i.e., any k out of n=k+r columns can retrieve all information symbols) are given in [2] for α=(p-1)τ. However, the (n,k) recoverable condition of GEBR codes is unknown when α<(p-1)τ. In this paper, we present the (n,k) recoverable condition for GEBR codes for α< (p-1)τ. In addition, we present a sufficient condition for enabling GEBR codes to recover some erased lines of any slope i (0≤ i≤ pτ-1) for any parameter r when τ is a power of p. Moreover, we present the construction of Generalized Blaum-Roth (GBR) codes that encode an α× k information array into an α× (k+r) array. We show that GBR codes share the same MDS condition as the (n,k) recoverable condition of GEBR codes, and we also present a sufficient condition for GBR codes to recover some erased lines of any slope i (0≤ i≤α-1).
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