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