# LDPC for channel coding

I am working on OFDM over harsh channel, so the channel coding is an essential thing for achieving a reliable communication.

According to my reading, LDPC is almost the best channel coding we can use for channel coding, but I have a question regarding that kind of coding.

Assume we are using LDCP with rate $$1/2$$ which means that communication systems uses the half of data rate for coding and the other half as data.

If we have the OFDM symbol have $$1024$$ sub-carriers, with modulation order $$M=4$$ = which gives us a total bits of symbol $$2048$$. In that case, will the LPDC will take the whole $$2048$$ together? I mean it will take $$128$$ bits from the whole bits for coding and the rest for data OR it will take group by group, which means for example group of $$6$$ bits, and then code them $$3$$ for data and $$3$$ for parity or coding.

Which one is right ?

According to my reading, LDPC is almost the best channel coding we can use for channel coding, but I have a question regarding that kind of coding.

Mentally, something being "the best" should always instantly raise a mental flag for you, saying "under which conditions, according to which measure".

It is right that iterative LDPC decoders can achieve maximum likelihood performance (i.e. being the best possible decoder), but only when

1. The code is large and
2. there's an infinite number of iterations.

While 2. is never fulfilled, there's often a number of iterations after which gains are small enough for people to simply stop and limit complexity.

The first condition, however, is often a pretty complicated to fulfill one:

will the LPDC will take the whole 2048 (bits) together?

That is a small LDPC code (128 bits would really be tiny, and I don't think I've seen any sensible OFDM application do that; least-rate IoT modes might be interested in doing that on the uplink, but that doesn't match the OFDM approach).

At a small size like 2048 bits, less complex codes and decoders might be comparable or even better suited for your use case and error model. (PS: Have an error model before deciding on the code you use! There's an awesome website, http://pretty-good-codes.org/ , which is sadly offline now, which compares many codes with metrics.)

Try to put more bits into a single codeword if you want to harness the powers of LDPC codes. For example, in DVB-T2 (which is an OFDM system), the short codewords are 16000 bits long, the normal ones are 64800 bits long.

I mean it will take bits from the whole bits for coding and the rest for data OR it will take group by group, which means for example group of bits, and then code them for data and for parity or coding.

Neither. You take blocksize · rate (so, in your proposed system, 2048·1/2=1024, but really, use larger blocks and established LDPC codes) information bits and encode them as one. These are not systematic, usually, so you don't get separate redundancy bit. You get a code word that is blocksize bits long, and doesn't contain the original bits in any structured way, usually. (Systematic LDPCs are usually undesirable.)

You will need to use a decoder to get the original bits from the codeword.

• For one interested in the codes themselves, there is uni-kl.de/en/channel-codes/ml-simulation-results posting performance data of maximum likelihood decoders. Also, has it been proven that infinite-iteration iterative LDPC decoders achieve ML performance? If not, I would prefer saying "iterative LDPC decoders approach maximum likelihood performance". – AlexTP Jul 16 at 18:45
• I think it's proven, Burshtein 2004! (the wording I used is still borderline, they can achieve that) – Marcus Müller Jul 16 at 18:47
• thanks. I will read it. – AlexTP Jul 16 at 19:53
• 1- Do you mean taking 2048 bits at once is less-complexity for LDPC code ? .. 2- When saying LDPC $(15,16)$ does that mean the codeword is 16 bits? does that mean the code will take every 16bits and code them? – Fatima_Ali Jul 17 at 1:55
• @Fatima_Ali I addressed code sizes in my answer. codes of sizes like (16000,32000) or (32400, 64800) are appropriate if you want to harness the strengths of LDPC. (16,32): certainly not a sensible size for an LDPC. – Marcus Müller Jul 17 at 10:06