Decoding Algorithms for Nonbinary LDPC Codes Over GF(q)

In this paper, we address the problem of decoding nonbinary LDPC codes over finite fields GF(q), with reasonnable complexity and good performance. In a first part of the paper, we recall the original Belief Propagation (BP) decoding algorithms and its Fourier domain implementation. We show that the use of tensor notations for the messages is very convenient for the algorihtm description and understanding. In the second part of the paper, we introduce a simplified decoder which is inspired from the min-sum decoder for binary LDPC codes. We called this decoder Extended Min-Sum (EMS). We show that it is possible to greatly reduce the computationnal complexity of the check node processing by computing approximate reliability measures with a limited number of values in a message. By choosing appropriate correction factors or offsets, we show that the EMS decoder performance are quite good, and in some cases better than the regular BP decoder. The optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution. Our simulations on ultra-sparse codes over very high order fields show that non-binary LDPC codes are promising for applications which require low FER for small or moderate codeword lengths. The EMS decoder is a good candidate for practical hardware implementations of such codes.