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


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]

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Correspondence to Jon-Lark Kim.

Additional information

Communicated by D. Jungnickel.

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Kim, JL., Mellinger, K.E. & Storme, L. Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles. Des Codes Crypt 42, 73–92 (2007).

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  • LDPC code
  • Generalized quadrangle
  • Minimum distance


  • 51E12
  • 94B05