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According to Google Scholar original Gallager's article Low-density parity-check codes is cited more than 10000 times. It looks scary for non-experts.

I suspect that the number of algorithms for constructing good sparse matrices for LDPC-codes is much less than 10000. Is there any survey which contains description of such algorithms?

Basically I'm interested in explicit constructions based on number theory methods.

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  • $\begingroup$ What's wrong with Margulis' construction which is based upon expanders? It provided some motivation to define expanders explicitly. $\endgroup$
    – Asaf
    Commented Dec 20, 2016 at 19:20
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    $\begingroup$ There is a new book coming out soon that might be what you're looking for: LDPC Code Designs, Constructions, and Unification by Li, Lin, Abdel-Ghaffar, Ryan, and Costello. $\endgroup$
    – Timothy Chow
    Commented Dec 20, 2016 at 22:15
  • $\begingroup$ @Asaf Nothing is wrong. I'm just looking for survey. $\endgroup$
    – Alexey Ustinov
    Commented Dec 21, 2016 at 6:50
  • $\begingroup$ @Timothy Chow I've found this book. It is well-written and contains a lot of useful information on LDPC codes. $\endgroup$
    – Alexey Ustinov
    Commented Dec 21, 2016 at 7:19

2 Answers 2

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Ryan and Lin's Channel Codes has several chapters (computer-based, finite geometries, finite fields, combinatorial designs) devoted to various constructions of LDPC codes, as well as a chapter devoted to nonbinary codes that also details constructions for that case.

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  • $\begingroup$ Thank you, very nice book. But for the first glance the book "LDPC Code Designs, Constructions, and Unification" from Timothy Chow's answer looks more fundamental. $\endgroup$
    – Alexey Ustinov
    Commented Dec 21, 2016 at 7:36
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An overview of LDPC codes constructed using methods from combinatorial mathematics and finite geometry is given in Low Density Parity Check Codes Based on Finite Geometries: A Rediscovery and New Results

In this paper, a geometric approach to the construction of LDPC codes is presented. Four classes of LDPC codes have been constructed based on the lines and points of the well known Euclidean and projective geometries over finite fields. Their performance is compared with that of randomly computer generated LDPC codes and irregular LDPC codes.

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