Explicit Construction of Families of LDPC Codes with Girth at Least Six

LDPC codes are serious contenders to Turbo codes in terms of decoding performance. One of the main problems is to give an explicit construction of such codes whose Tanner graphs have known girth. For a prime power q and m ≥ 2, Lazebnik and Ustimenko construct a q-regular bipartite graph D(m, q) on 2qm vertices, which has girth at least 2dm/2e+ 4. We regard these graphs as Tanner graphs of binary codes LU(m, q). We can determine all the parameters of LU(2, q). We know that their girth is 6 and their diameter is 4. We know that LU(3, q) has girth 8 and diameter 6 and we conjecture its dimension. We find some interesting LDPC codes by our partial row construction.