There are two gradient descent decoding procedures for binary codes proposed
independently by Liebler and by Ashikhmin and Barg. Liebler in his paper
mentions that both algorithms have the same philosophy but in fact they are
rather different. The purpose of this communication is to show that both
algorithms can be seen as two ways of understanding the reduction process
algebraic monoid structure related to the code. The main tool used for showing
this is the Gr\"obner representation of the monoid associated to the linear
code.