Fourier domain representation of non-binary LDPC codes

In this paper we propose a binary representation of non-binary codes, which consists in a q-fold covering of the non-binary graph, where q is the size of the non-binary alphabet. It is shown that this covering graph is actually the underlying graph of the Fourier-domain Belief-Propagation decoding, meaning that the exchanged messages carry information about the bit-nodes of the covering graph. Finally, a new coding scheme is proposed, in which the bits transmitted over the channel correspond to a set of bit-nodes of the Fourier domain representation. The proposed coding scheme allows using the same decoder, regardless of how many and which of the Fourier-domain bit-nodes have been transmitted. This considerably increases the flexibility of the system, without increasing its complexity.