Design of nonbinary quasi-cyclic LDPC cycle codes

In this paper, we study the design of nonbinary low-density parity-check (LDPC) cycle codes over Galois field GF(q). First, we construct a special class of nonbinary LDPC cycle codes with low error floors. Our construction utilizes the cycle elimination algorithm to remove short cycles in the normal graph and to select nonzero elements in the parity-check matrix to reduce the number of low-weight codewords generated by short cycles. Furthermore, we show that simple modifications of such codes are parallel sparse encodable (PSE). The PSE code, consisting of a quasi-cyclic (QC) LDPC cycle code and a simple tree code, has the attractive feature that it is not only linearly encodable, but also allows parallel encoding which can reduce the encoding time significantly. We provide a systematic comparison between nonbinary coded systems and binary coded systems. For the MIMO channel considered, our results show that the proposed nonbinary system employing the PSE code outperforms not only the binary LDPC code specified in the 802.16e standard, but also the optimized binary LDPC code obtained using the EXIT chart methods.