  A new construction is proposed for low density parity check (LDPC) codes
using quadratic permutation polynomials over finite integer rings. The
associated graphs for the new codes have both algebraic and pseudo-random
nature, and the new codes are quasi-cyclic. Graph isomorphisms and
automorphisms are identified and used in an efficient search for good codes.
Graphs with girth as large as 12 were found. Upper bounds on the minimum
Hamming distance are found both analytically and algorithmically. The bounds
indicate that the minimum distance grows with block length. Near-codewords are
one of the causes for error floors in LDPC codes; the new construction provides
a good framework for studying near-codewords in LDPC codes. Nine example codes
are given, and computer simulation results show the excellent error performance
of these codes. Finally, connections are made between this new LDPC
construction and turbo codes using interleavers generated by quadratic
permutation polynomials.
