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In this paper, we develop efficient decoders for non-binary low-densityparity-check (LDPC) codes using the alternating direction method of multipliers(ADMM). We apply ADMM to two decoding problems. The first problem is linearprogramming (LP) decoding. In order to develop an efficient algorithm, we focuson non-binary codes in fields of characteristic two. This allows us totransform each constraint in $\mathbb{F}_{2^m}$ to a set of constraints in$\mathbb{F}_{2}$ that has a factor graph representation. Applying ADMM to theLP decoding problem results in two types of non-trivial sub-routines. The firsttype requires us to solve an unconstrained quadratic program. We solve thisproblem efficiently by leveraging new results obtained from studying the abovefactor graphs. The second type requires Euclidean projection onto polytopesthat are studied in the literature, a projection that can be solved efficientlyusing off-the-shelf techniques, which scale linearly in the dimension of thevector to project. ADMM LP decoding scales linearly with block length, linearlywith check degree, and quadratically with field size. The second problem weconsider is a penalized LP decoding problem. This problem is obtained byincorporating a penalty term into the LP decoding objective. The purpose of thepenalty term is to make non-integer solutions (pseudocodewords) more expensiveand hence to improve decoding performance. The ADMM algorithm for the penalizedLP problem requires Euclidean projection onto a polytope formed by embeddingthe constraints specified by the non-binary single parity-check code, which canbe solved by applying the ADMM technique to the resulting quadratic program.Empirically, this decoder achieves a much reduced error rate than LP decodingat low signal-to-noise ratios.

ADMM LP decoding of non-binary LDPC codes in $\mathbb{F}_{2^m}$