  Code annealing, a new method of designing good codes of short block length,
is proposed, which is then concatenated with cyclic lifting to create finite
codes of low frame error rate (FER) error floors without performance outliers.
The stopping set analysis is performed on the cyclically lifted code ensemble
assuming uniformly random lifting sequences, and the suppressing effect/weight
of the cyclic lifting is identified for the first time, based on which the
ensemble FER error floor can be analytically determined and a scaling law is
derived. Both the first-order and high-order suppressing effects are discussed
and quantified by different methods including the explicit expression, an
algorithmic upper bound, and an algebraic lower bound.
  The mismatch between the suppressing weight and the stopping distances
explains the dramatic performance discrepancy among different cyclically lifted
codes when the underlying base codes have degree 2 variable nodes or not. For
the former case, a degree augmentation method is further introduced to mitigate
this metric mismatch, and a systematic method of constructing irregular codes
of low FER error floors is presented. Both regular and irregular codes of very
low FER error floors are reported, for which the improvement factor ranges from
10^6-10^4 when compared to the classic graph-based code ensembles.
