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Recent developments in decoding of Tanner codes with maximum-likelihoodcertificates are based on a sufficient condition called local-optimality. Wedefine hierarchies of locally-optimal codewords with respect to two parameters.One parameter is related to the minimum distance of the local codes in Tannercodes. The second parameter is related to the finite number of iterations usedin iterative decoding. We show that these hierarchies satisfy inclusionproperties as these parameters are increased. In particular, this implies thata codeword that is decoded with a certificate using an iterative decoder after$h$ iterations is decoded with a certificate after $k\cdot h$ iterations, forevery integer $k$.

Hierarchies of Local-Optimality Characterizations in Decoding of Tanner Codes