Multiple Repair Localities with Distinct Erasure Tolerance

In distributed storage systems, codes with lower repair locality for each coordinate are much more desirable since they can reduce the disk I/O complexity for repairing a failed node. The ith coordinate of a linear code 𝒞 is said to have $(r_i,{\delta }_i)$ locality if there exist δ_i non-overlapping local repair sets of size no more than r_i, where a local repair set of one coordinate is defined as the set of some other coordinates by which one can recover the value at this coordinate. In this paper, we consider linear codes with information $(r_{\min },{\delta }_{\min };r_{\max },{\delta }_{\max })$ locality, where there exists an information set I such that for each $i\in I$, the ith coordinate has $(r_i,{\delta }_i)$ locality and $\min \{r_i:i\in I\}=r_{\min ,}\max \{r_i:i\in I\}=r_{\max ,}\min \{{\delta }_i:i\in I\}={\delta }_{\min }$ and $max\{{\delta }_i:i\in I\}={\delta }_{\max }$. We derive a lower bound on the codeword length n for any linear [n, k, d] code with information $(r_{\min },{\delta }_{\min };r_{\max },{\delta }_{\max })$ locality. Particularly, we indicate that some existing bounds can be deduced from our result by restrictions on parameters.