How to Read and Update Coded Distributed Storage Robustly and Optimally?

We consider the problem of robust dynamic coded distributed storage (RDCDS) that is associated with the coded distributed storage of a message with $N$ servers where 1) it suffices to recover the message from the storage at any $R_r$ servers; and 2) each of the servers stores a coded portion of the message that is at most $\frac{1}{K_c}$ the size of the message. The goal is to enable two main functionalities: the read operation and the update operation of the message. Specifically, at time slot $t$, the user may execute either the read operation or the update operation, where the read operation allows the user to recover the message from the servers, and the update operation allows the user to update the message to the servers in the form of an additive increment so that any up to $X^{(t)}$ colluding servers reveal nothing about the increment. The two functionalities are robust if at any time slot $t$ 1) they tolerate temporarily dropout servers up to certain thresholds (the read threshold is $R_r$ and the update threshold is denoted as $R_u^{(t)}$); and 2) the user may remain oblivious to prior server states. The communication efficiency is measured by the download cost $C_r^{(t)}$ of the read operation and the upload cost $C_u^{(t)}$ of the update operation. Given $K_c$ and $R_r$, we are curious about the optimal $(R_u^{(t)},C_r^{(t)},C_u^{(t)})$ tuple. In this work, we settle the fundamental limits of RDCDS. In particular, denoting the number of dropout servers at time slot $t$ as $|\mathcal{D}^{(t)}|$, we first show that 1) $R_u^{(t)}\geq N-R_r+\lceil K_c\rceil+X^{(t)}$; and 2) $C_r^{(t)}\geq \frac{N-|\mathcal{D}^{(t)}|}{N-R_r+\lceil K_c\rceil-|\mathcal{D}^{(t)}|}, C_u^{(t)}\geq \frac{N-|\mathcal{D}^{(t)}|}{R_r-X^{(t)}-|\mathcal{D}^{(t)}|}$. Then, inspired by the idea of staircase codes, we construct an RDCDS scheme that simultaneously achieves the above lower bounds.