A New Construction for LDPC Codes using Permutation Polynomials over   Integer Rings
  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. 
Oscar Y. Takeshita