The low-density parity check codes whose performance is closest to the Shannon limit are "Gallager codes" based on irregular graphs. We compare alternative methods for constructing these graphs and present two results. First, we find a "super-Poisson" construction which gives a small improvement in empirical performance over a random construction. Second, whereas Gallager codes normally take N/sup 2/ time to encode, we investigate constructions of regular and irregular Gallager codes that allow more rapid encoding and have smaller memory requirements in the encoder. We find that these "fast encoding" Gallager codes have equally good performance.