Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles

Abstract

We find lower bounds on the minimum distance and characterize codewords of small weight in low-density parity check (LDPC) codes defined by (dual) classical generalized quadrangles. We analyze the geometry of the non-singular parabolic quadric in PG(4,q) to find information about the LDPC codes defined by Q (4,q), ${\mathcal{W}(q)}$ and ${\mathcal{H}(3,q^{2})}$ . For ${\mathcal{W}(q)}$ , and ${\mathcal{H}(3,q^{2})}$ , we are able to describe small weight codewords geometrically. For ${\mathcal{Q}(4,q)}$ , q odd, and for ${\mathcal{H}(4,q^{2})^{D}}$ , we improve the best known lower bounds on the minimum distance, again only using geometric arguments. Similar results are also presented for the LDPC codes LU(3,q) given in [Kim, (2004) IEEE Trans. Inform. Theory, Vol. 50: 2378–2388]