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Designs, Codes and Cryptography

, Volume 42, Issue 1, pp 73–92 | Cite as

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

  • Jon-Lark Kim
  • Keith E. Mellinger
  • Leo Storme
Article

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]

Keywords

LDPC code Generalized quadrangle Minimum distance 

Classifications

51E12 94B05 

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Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  2. 2.Department of MathematicsUniversity of Mary WashingtonFredericksburgUSA
  3. 3.Department of Pure Mathematics and Computer AlgebraGhent UniversityGhentBelgium

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