A Tight Rate Bound and a Matching Construction for Locally Recoverable Codes with Sequential Recovery From Any Number of Multiple Erasures

Abstract: An $[n,k]$ code $\mathcal{C}$ is said to be locally recoverable in the presence of a single erasure, and with locality parameter $r$, if each of the $n$ code symbols of $\mathcal{C}$ can be recovered by accessing at most $r$ other code symbols. An $[n,k]$ code is said to be a locally recoverable code with sequential recovery from $t$ erasures, if for any set of $s \leq t$ erasures, there is an $s$-step sequential recovery process, in which at each step, a single erased symbol is recovered by accessing at most $r$ other code symbols. This is equivalent to the requirement that for any set of $s \leq t$ erasures, the dual code contain a codeword whose support contains the coordinate of precisely one of the $s$ erased symbols. In this paper, a tight upper bound on the rate of such a code, for any value of number of erasures $t$ and any value $r \geq 3$, of the locality parameter is derived. This bound proves an earlier conjecture due to Song, Cai and Yuen. While the bound is valid irrespective of the field over which the code is defined, a matching construction of {\em binary} codes that are rate-optimal is also provided, again for any value of $t$ and any value $r\geq3$.