Load-Balanced Fractional Repetition Codes

We introduce load-balanced fractional repetition (LBFR) codes, which are a strengthening of fractional repetition (FR) codes. LBFR codes have the additional property that multiple node failures can be sequentially repaired by downloading no more than one block from any other node. This allows for better use of the network, and can additionally reduce the number of disk reads necessary to repair multiple nodes. We characterize LBFR codes in terms of their adjacency graphs, and use this characterization to present explicit constructions LBFR codes with storage capacity comparable existing FR codes. Surprisingly, in some parameter regimes, our constructions of LBFR codes match the parameters of the best constructions of FR codes

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

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

This paper considers the natural extension of locally recoverable codes (LRC) to the case of t > 1 erased symbols. While several approaches have been proposed for the handling of multiple erasures, in the approach considered here, the t erased symbols are recovered in succession, each time contacting at most r other symbols for assistance. Under the local-recovery constraint, this sequential approach is the most general and hence offers the maximum possible code rate. We characterize the rate of an LRC with sequential recovery for any r \geq 3 and any t, by first deriving an upper bound on the code rate and then constructing a binary code achieving this optimal rate. The upper bound derived here proves an earlier conjecture. Our approach permits us to deduce the structure of the parity-check matrix of a rate-optimal LRC with sequential recovery. The derived structure of parity-check matrix leads to a graphical description of the code used in code construction. A subclass of binary codes that are both rate and block-length optimal, are shown to correspond to certain regular graphs known as Moore graphs, that have the smallest number of vertices for a given girth. A connection with Tornado codes is also made