Abstract
We construct various classes of low-density parity-check codes using point-line incidence structures in the classical projective plane PG(2,q). Each incidence structure is based on the various classes of points and lines created by the geometry of a conic in the plane. For each class, we prove various properties about dimension and minimum distance. Some arguments involve the geometry of two conics in the plane. As a result, we prove, under mild conditions, the existence of two conics, one entirely internal or external to the other. We conclude with some simulation data to exhibit the effectiveness of our codes.
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References
J Cannon C Playoust (1994) An introduction to magma University of Sydney Sydney, Australia
RG Gallager (1962) ArticleTitleLow density parity-check codes IRE Trans Infom Theory IT-8 21–28 Occurrence Handle0107.11802 Occurrence Handle136009 Occurrence Handle10.1109/TIT.1962.1057683
Hirschfeld JWP (1998) Projective geometries over finite fields, 2nd edn. Oxford University Press
Huffman WC, Pless V (2003) Fundamentals of error-correcting codes. Cambridge University Press
Kim J-L, Peled UN, Perepelitsa I, Pless V (2002) Explicit construction of families of LDPC codes of girth at least six. In: Proc. 40th Allerton conf. on communication, control and computing, In: Voulgaris PG, Srikant R (eds) Oct. 2–4, 2002, pp 1024–1031
Y Kuo S Lin MPC Fossorier (2001) ArticleTitleLow-density parity-check codes based on finite geometries: a rediscovery and new results IEEE Trans Inform Theory 47 IssueID7 2711–2736 Occurrence Handle1872835 Occurrence Handle10.1109/18.959255
DJC MacKay RM Neal (1996) ArticleTitleNear Shannon limit performance of low density parity-check codes Electron Lett 32 IssueID18 1645–1646 Occurrence Handle10.1049/el:19961141
KE Mellinger (2004) ArticleTitleLDPC codes and triangle-free line sets Designs Codes Cryptog 32 IssueID1-3 341–350 Occurrence Handle1052.51008 Occurrence Handle2072337 Occurrence Handle10.1023/B:DESI.0000029233.20866.41
Mellinger KE Classes of codes from quadratic surfaces of PG(3,q). Submitted
A Passmore J Stovall (2004) ArticleTitleOn codes generated by quadratic surfaces of PG(3,q) Rose-Hulman Inst Technol Undergrad Res J 5 1
Rosenthal J, Vontobel PO (2000) Constructions of LDPC codes using Ramanujan graphs and ideas from Margulis. In: Proc. of the 38th annual Allerton conference on communication, control, and computing, pp 248–257
B Segre (1955) ArticleTitleOvals in a finite projective plane Can J Math 7 414–416 Occurrence Handle0065.13402 Occurrence Handle71034
M Sipser DA Spielman (1996) ArticleTitleExpander codes IEEE Trans Inform Theory 42 IssueID6 1710–1722 Occurrence Handle0943.94543 Occurrence Handle1465731 Occurrence Handle10.1109/18.556667
Storer T (1967) Cyclotomy and difference sets. Markham Publishing Company
RM Tanner (1981) ArticleTitleA recursive approach to low complexity codes IEEE Trans Inform Theory IT-27 533–547 Occurrence Handle650686 Occurrence Handle10.1109/TIT.1981.1056404
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Communicated by R. Hill
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Droms, S.V., Mellinger, K.E. & Meyer, C. LDPC codes generated by conics in the classical projective plane. Des Codes Crypt 40, 343–356 (2006). https://doi.org/10.1007/s10623-006-0022-6
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DOI: https://doi.org/10.1007/s10623-006-0022-6