ADMM decoding of non-binary LDPC codes in F2m

In this paper, we develop an efficient algorithm for linear programming (LP) decoding of non-binary low-density parity-check (LDPC) codes. We build our algorithm on the decomposition method based on the alternating direction method of multipliers (ADMM). Although expressing the LP decoding problem using ADMM is not hard, a sub-routine of ADMM - projection onto a polytope formed from embeddings of the non-binary single parity-check code - is not straightforward. In this work, we focus on non-binary codes in fields of characteristic two. This allows us to use operations in F₂ to relax the polytope under consideration into a form that is computational friendly. We introduce a rotation step that normalizes the geometry under consideration. We then apply ADMM a second time to solve the projection problem, which is a quadratic program (QP). Our decoding algorithm scales linearly with block length, linearly with check degree, and quadratically with field size.