Various types of recovery algorithms for batch codes have been considered previously, including asynchronous recovery and recovery algorithms arising from Almost Affinely Disjoint (AAD) families. We initiate a systematic investigation of linear batch codes equipped with recovery algorithms. We introduce online and strongly online batch codes, investigate their relations with asynchronous batch codes, and introduce (m,L)-strongly online batch codes, which provide particularly simple recovery algorithms. We then study simple batch codes associated with arbitrary bipartite graphs, obtaining graph-theoretic conditions for such codes and generalizing several known results.

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