Generalized quad, hex, and octagon LDPC codes

We use the theory of finite classical generalized polygons to derive and study low-density parity-check (LDPC) codes. The Tanner graph of a generalized polygon LDPC code is highly symmetric, inherits the diameter size of the parent generalized polygon, and has minimum (one half) diameter-to-girth ratio. We show formally that when the diameter is four or six or eight all codewords have even Hamming weight. When the generalized polygon has in addition equal number of points and lines, we see that the non-regular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length. A new minimum distance bound is presented for codes from non-regular polygons of even diameter and equal number of points and lines. Finally, we prove that all codes derived from finite classical generalized quadrangles are quasi-cyclic and we give the explicit size of the circulant blocks in the parity check matrix. Our simulation studies of several generalized polygon LDPC codes demonstrate powerful bit-error-rate performance when decoding is carried out via low complexity variants of belief propagation.