In this thesis, design details of binary irregular Low-Density Parity-Check (LDPC) codes are investigated. We especially focus on the trade-off between the average variable node degree, wa, and the number of length-6 cycles of an irregular code. We observe that the performance of the irregular code improves with increasing wa up to a critical value, but deteriorates for larger wa because of the exponential increase in the number of length-6 cycles. We have designed an irregular code of length 16,000 bits with average variable node degree wa=3.8, that we call &lsquo / 2/3/13&rsquo / since it has some variable nodes of degree 2 and 13 in addition to the majority of degree-3 nodes. The observed performance is found to be very close to that of the capacity approaching commercial codes. Time spent for decoding 50,000 codewords of length 1800 at Eb/No=1.6 dB for an irregular 2/3/13 code is measured to be 19% less than that of the regular (3, 6) code, mainly because of the smaller number of decoding failures.