Efficient ADMM Decoder for Non-binary LDPC Codes based on Bit Embedding Technique

In this paper, we devise a new alternating direction method of multipliers (ADMM) decoder for non-binary low-density parity-check (LDPC) codes in Galois fields of characteristic two $\left( {{\mathbb{F}_{{2^q}}}} \right)$. Its main content are threefold: first, the procedure of formulating the maximum likelihood (ML) decoding problem in ${\mathbb{F}_{{2^q}}}$ to a linear integer problem in real space is presented; Second, after relaxing the integer problem to a continuous one, an efficient ADMM algorithm is customized to solve the latter, where all the entries of the variable vectors can be obtained in parallel; Third, we show that the proposed ADMM decoder satisfies the favorable codeword-independent property under some mild conditions and its computation complexity in each ADMM iteration is roughly $\mathcal{O}\left( {nq} \right)$, where n is code length of the considered non-binary LDPC code. Simulation results demonstrate that its performance, such as error-correction and decoding efficiency, is very competitive in comparison with state-of-the-art non-binary LDPC decoders.