On the MDS Condition of Generalized Expanded-Blaum-Roth Codes

Generalized Expanded-Blaum-Roth (GEBR) codes [You et al. 2020] 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 [Hou et al. 2023] for α = (p −1)τ. However, the (n, k) recoverable condition of GEBR codes is unknown when α < (p − 1)τ. Note that the (n, k) recoverable condition is reduced to be MDS condition if there is no local parity symbol in each column. In this paper, we present the (n, k) recoverable condition for GEBR codes for α < (p − 1)τ. Moreover, we characterize the necessary and sufficient (n, k) recoverable condition for GEBR codes when p is a special prime.