Graph-based codes for hierarchical recovery

In this paper, we consider approaches to designing Tanner codes to protect against symbol loss from multiple erasures. First, we note that Tanner codes inherit locality and availability from their inner codes, allowing one to design longer codes with specified locality and availability. Availability is desirable in that multiple disjoint repair groups increase the likelihood that symbols are available to repair erased ones. Even so, particular patterns of erasures well-distributed across the repair groups may prevent recovery. Hence, we consider an alternative using hierarchical locality which implements tiered recovery, where the tier utilized depends on the number of erasures. Finally, we define hierarchical stopping sets to characterize local message-passing decoder failure at the various repair levels.