Universal and Dynamic Locally Repairable Codes with Maximal Recoverability via Sum-Rank Codes

Locally repairable codes (LRCs) are considered with equal or unequal localities, local distances and local field sizes. An explicit two-layer architecture with a sum-rank outer code are obtained, having disjoint local groups and achieving maximal recoverability (MR) for all families of local linear codes (MDS or not) simultaneously, up to a prescribed maximum locality r. Furthermore, the local linear codes (thus the localities, local distances and local fields) can be efficiently and dynamically modified without global recoding or changes in architecture or outer code, while preserving MR, easily adapting to new hot and cold data. In addition, local groups and file components can be added, removed or updated without global recoding. The construction requires global fields of size roughly $g^{r}$, for g local groups and maximum locality r. For equal localities, these global fields are smaller than those of previous MR-LRCs when $ r\leq h$ (global parities). For unequal localities, they provide an exponential field size reduction on all previous best known MR-LRCs. For bounded localities and a large number of local groups, the global erasure-correction complexity of the given construction is comparable to that of Tamo-Barg codes or Reed-Solomon codes with local replication, while local repair is as efficient as for the Cartesian product of the local codes. Reed-Solomon codes with local replication and Cartesian products are recovered from the given construction when $r = 1$ and $h = 0$, respectively. Finally, subextension subcodes and sum-rank alternant codes are introduced to obtain further exponential field size reductions, at the expense of lower information rates.