Unequal locality and recovery for Locally Recoverable Codes with availability

A code is said to be a Locally Recoverable Code (LRC) with availability if every coordinate can be recovered from multiple disjoint sets of other coordinates called recovering sets. The size of a recovering set is called locality. In this work, we consider LRCs with availability under two different settings-unequal recovery set sizes fixed for all coordinates or unequal locality over multiple coordinates. For each setting, we derive bounds for the minimum distance that generalize previously known bounds to the cases of unequal recovery and unequal locality, and are proved using similar techniques. For the case of equal recovery and locality with availability, we show that a known polynomial-evaluation construction extends to optimal codes with all-symbol locality in a specific case.