On Constructing Low-Density Parity-Check Codes

This thesis focuses on designing Low-Density Parity-Check (LDPC) codes for forward-error-correction. The target application is real-time multimedia communications over packet networks. We investigate two code design issues, which are important in the target application scenarios, designing LDPC codes with low decoding latency, and constructing capacity-approaching LDPC codes with very low error probabilities. On designing LDPC codes with low decoding latency, we present a framework for optimizing the code parameters so that the decoding can be fulfilled after only a small number of iterative decoding iterations. The brute force approach for such optimization is numerical intractable, because it involves a difficult discrete optimization programming. In this thesis, we show an asymptotic approximation to the number of decoding iterations. Based on this asymptotic approximation, we propose an approximate optimization framework for finding near-optimal code parameters, so that the number of decoding iterations is minimized. The approximate optimization approach is numerically tractable. Numerical results confirm that the proposed optimization approach has excellent numerical properties, and codes with excellent performance in terms of number of decoding iterations can be obtained. Our results show that the numbers of decoding iterations of the codes by the proposed design approach can be as small as one-fifth of the numbers of decoding iterations of some previously well-known codes. The numerical results also show that the proposed asymptotic approximation is generally tight for even non-extremely limiting cases. On constructing capacity-approaching LDPC codes with very low error probabilities, we propose a new LDPC code construction scheme based on 2-lifts. Based on stopping set distribution analysis, we propose design criteria for the resulting codes to have very low error floors. High error floors are the main problems of previously constructed capacity-approaching codes, which prevent them from achieving very low error probabilities. Numerical results confirm that codes with very low error floors can be obtained by the proposed code construction scheme and the design criteria. Compared with the codes by the previous standard construction schemes, which have error floors at the levels of 10−3 to 10−4, the codes by the proposed approach do not have observable error floors at the levels higher than 10−7. The error floors of the codes by the proposed approach are also significantly lower compared with the codes by the previous approaches to constructing codes with low error floors.