ADMM decoding of error correction codes: From geometries to algorithms

Many code constraints can be represented using factor graphs. By relaxing these factorable coding constraints to linear constraints, it is straightforward to form a decoding optimization problem. Furthermore, by pairing these factor graphs with the alternating directions method of multipliers (ADMM) technique of large-scale optimization, one can develop distributed algorithms to solve the decoding optimization problems. However, the non-trivial part has always been developing an efficient algorithm for the subroutines of ADMM, which directly relates to the geometries of the relaxed coding constraints. In this paper, we focus on summarizing existing results and distilling insights to these problems. First, we review the ADMM formulation and geometries involved in the subroutines. Next, we present a linear time algorithm for projecting onto an ℓ1 ball with box constraints.