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Suppose that we have a LDPC code $C$ with a $(n -k )\times n $ parity check matrix $H$, and there exist approximately $ \sqrt n$ numbers of degree-2 columns. It means that there are approximately $\sqrt n$ numbers of degree-2 variable nodes($VN$) in the corresponding Tanner graph.

We suppose that the left $VN$s are with high degrees, and then $C$ is irregular. The girth of $C$ is $\geq 8$.

I want to know that whether these degree-2 $VN$s will affect the decoding performance seriously. What kind of effects will it have? Such as high error floors, degradation in the waterfall region?

Sincerely,
Bruce

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    $\begingroup$ The codes I tested (when our team was checking out and standardizing DVB-T2) with a very basic belief propagation algorithm had many of such nodes. They did contribute to some uncomfortably small trapping sets that showed as a persistent error floor at BER $10^{-11}$ or thereabouts. I was left with the impression that this was largely an oversight in the design of the code, as colleagues from Sony managed to get around the problem (my proposal did as well, but theirs was more extensively tested, and selected to the standard). $\endgroup$
    – Jyrki Lahtonen
    Nov 23, 2015 at 18:31
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    $\begingroup$ (cont'd) I don't recall witnessing any problems near the waterfalls. The large number of degree-2 nodes was then forced upon us as part of the decoding circuitry (dictating the use of a staircase block in the check matrix). I do recal having to warn colleagues NOT to puncture a large section of the staircase, because then erasure recovery would take hundreds of iterations. A quick simulation proved my point :-) $\endgroup$
    – Jyrki Lahtonen
    Nov 23, 2015 at 18:34
  • $\begingroup$ Thanks very much for your comments. I have made several simulations for my LDPC codes. Generally speaking, these degree-2 VNs can make the generator matrix more sparse. But the error-correction performance is not good enough. I am dealing with a response letter for a submitted paper at present. So the simulation work is interrupted. If I could improve the simulation performance, I will add a new comment. Regards, Bruce $\endgroup$
    – Bruce Fang
    Nov 25, 2015 at 1:10

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As maybe you know, the most good decoding algorithm for LDPC codes, for example iterative decoding, has not provable efficiency, except in special cases. But, it is believed that variable nodes with degree $2$, behave weaker in error protection against the higher degree variable nodes. Also, the important part of efficiency of the decoding algorithms depends to the first round executing. So, degree $2$ variable nodes shows their effects in the first stage for the failing of the algorithm.

For more details you can see the book:

"LDPC Coded Modulations" by "Michele Franceschini, Gianluigi Ferrari, Riccardo Raheli"

Also, there are some good references in the book which help you for finding good answer for your question.

Note that, the degree $2$ variable nodes effect on trapping sets and stopping sets of the LDPC codes which are good parameters for the efficiency of the decoding algorithm.

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  • $\begingroup$ Thanks very much for your suggestions. I will borrow the book from my library and have a look at it. I think the best way to find out the effects is to do a simulation, and I expect a good performance, at least some trade offs. LDPC coding is really interesting, I love it! $\endgroup$
    – Bruce Fang
    Aug 23, 2015 at 10:20

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