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GF(4) Nonbinary SC-LDPC for DNA Data Storage

This repo implements nonbinary, terminated, spatially coupled LDPC codes over GF(2^m), with a DNA substitution channel model for GF(4).

1. Build

Compile the code. Re-run this command any time the source code changes.

clang++ -std=c++14 -O2 -IInclude LDPC/*.cpp -o build/ldpc

2. Concepts

  • Spec file (.ldspec): the serialized parity-check matrix H, plus code parameters (length, redundancy, GF extension). Produced by mode B, used by every other mode.
  • Nucleotide mapping: 0,1,2,3 ↔ A,C,G,T. As written in the testbed.cpp.
  • Channel matrix file: text file with convention,

$ \scriptsize Q = \begin{bmatrix} P(y=A\mid x=A) & P(y=A\mid x=C) & P(y=A\mid x=G) & P(y=A\mid x=T)\ P(y=C\mid x=A) & P(y=C\mid x=C) & P(y=C\mid x=G) & P(y=C\mid x=T)\ P(y=G\mid x=A) & P(y=G\mid x=C) & P(y=G\mid x=G) & P(y=G\mid x=T)\ P(y=T\mid x=A) & P(y=T\mid x=C) & P(y=T\mid x=G) & P(y=T\mid x=T)\ \end{bmatrix}^T $

  • Dimension = number of message symbols as input; Length = codeword length after encoding.
  • Substitution errors only, no insertions/deletions.

3. Step-by-step commands

Mode B: build the H matrix

Builds a terminated, spatially-coupled regular GF(2^Extension) parity-check matrix with random generation within each coupling layer.

./build/ldpc B <VNsPerPos> <CNsPerPos> <CouplingWidth> <CheckDegreePerOffset> <ChainLength> <Extension> <specFile>
  • VNsPerPos, CNsPerPos: number of variable/check nodes at each position along the chain
  • CouplingWidth (w): each VN connects to w consecutive check-node positions
  • CheckDegreePerOffset: edges each check node gets from each contributing position
  • ChainLength (L): number of VN positions (number of CN positions L+w-1)
  • Extension = 2 for GF(4)
  • specFile: location in the repo where H matrix is saved

Constraints on the parameters (IMPORTANT):

  • Length = VNsPerPos * ChainLength, Redundancy = CNsPerPos * (ChainLength + CouplingWidth - 1).
  • VNsPerPos must equal CNsPerPos * CheckDegreePerOffset Otherwise, error.
  • CheckDegreePerOffset should be ≥ 2. The decoder program assumes every check has degree ≥ 2. Otherwise, error.
  • Preferably, CouplingWidth should be far less than ChainLength .

Properties:

  • VN degrees: constantly equals to CouplingWidth
  • CN degrees: degree profile is a trapezoid, it ramps linearly from CheckDegreePerOffset up to the max CheckDegreePerOffset * CouplingWidth over the first w-1 positions, stays flat at the full value through the bulk of the chain, then ramps symmetrically back down to CheckDegreePerOffset over the last w-1 positions.

Mode H: View the H matrix

./build/ldpc H <specFile>

Prints Redundancy × Length matrix of GF(4) elements (0-3)

Mode E: encode a string

Message length must match H dimension exactly, otherwise error.

./build/ldpc E <specFile> <message>

Mode C: full pipeline: encode → simulate DNA channel → decode → recover

./build/ldpc C <specFile> <confusionMatrixFile> <message> <NumOfDecoderIterations>

confusionMatrixFile: channel model file (LDPC/acc-true_passQ-true_forward.txt)

Python code to check the errors:

def count_substitutions(clean, noisy):
    assert len(clean) == len(noisy)
    diffs = [(i, a, b) for i, (a, b) in enumerate(zip(clean, noisy)) if a != b]
    print(f"{len(diffs)} substitution(s) at position(s) {[i for i, _, _ in diffs]}")
    for i, a, b in diffs:
        print(f"  {i}: {a} -> {b}")
    return diffs

clean = 
noisy = 
count_substitutions(clean, noisy)

Mode D: simulation over random trials

For measuring error rate on randomly generated messages. Thus, this can be a quick way to test the LDPC code been designed.

./build/ldpc D <specFile> <NumOfDecoderIterations> <NumOfRemainingVals> <confMatrixFile> <NumOfErrors> <NumOfTrials>

NumOfErrors: stop early once this number errors are seen

NumOfTrials: max number of random trials to run

NumOfDecoderIterations: number of decoding iterations allowed at each trail

i.e. run random trials, stop if NumOfErrors of the trails have an error or NumOfTrials has been run, where each trial is an independent experiment: random message → encode → channel injects random substitution errors → decode → check errors.

NumOfRemainingVals must be ≥ the field size (4 for GF(4))

The final line in the output will have the format:

<trials run> <codeword error rate> <symbol error rate> <decode failure rate>

Mode T: encoding verification

Encodes random messages and confirms every codeword satisfies all parity checks (no channel involved).

./build/ldpc T <specFile> <NumOfIterations>

4. Example

LDPC initialization

./build/ldpc B 40 10 4 4 6 2 build/gf4_150.ldspec

i.e. 240 length code, 90 redundancy symbols, so 150 length message symbols per codeword. Rate = (150/240) * 2 = 1.25 bits per channel use.

The corresponding parity check matrix H:

./build/ldpc H build/gf4_150.ldspec 

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

It can be seen that the CNs degrees are lower at the ends, which means stronger checks.

Demo of the 150-nt message and its corresponding encoded codeword:

$ ./build/ldpc E build/gf4_150.ldspec AGTGCAACAAGTCAATCCGTTTCCCCAAGGAGGCCTCCTGGAACAATGAATTATGGCGCCAAGACATGGGGGATCCTAACTGGGGCGCCGACCTGGAGAAACGATCCGGAGGTGCCAGGATCGTCTCTGGAACGCTCCGAAAGTCTTGTT

Codeword: GTATAACGTTTTCCGTTCTAGAGCCACGTCGCCGTAATCAGCGGGGGGCTGGTGCCAGGAAGGCCTCCTGGGATCCTAACATCCACGGTAAGTGCAACAAGTCAATCCGTTTCCCCAAGGCGCCGAGGTCGAACAATGAATTATGGCGCCAAGACATGGGTTCACTTTGTTGGGGCGCCGACCTGGAGAAACGATCCGGACGCGTAGGGGTCGTCTCTGGAACGCTCCGAAAGTCTTGTT

Full pipeline implementation:

$ ./build/ldpc B 40 10 4 4 6 2 build/gf4_150.ldspec
$ ./build/ldpc C build/gf4_150.ldspec LDPC/acc-true_passQ-true_forward.txt \AGTGCAACAAGTCAATCCGTTTCCCCAAGGAGGCCTCCTGGAACAATGAATTATGGCGCCAAGACATGGGGGATCCTAACTGGGGCGCCGACCTGGAGAAACGATCCGGAGGTGCCAGGATCGTCTCTGGAACGCTCCGAAAGTCTTGTT 20

Message:        AGTGCAACAAGTCAATCCGTTTCCCCAAGGAGGCCTCCTGGAACAATGAATTATGGCGCCAAGACATGGGGGATCCTAACTGGGGCGCCGACCTGGAGAAACGATCCGGAGGTGCCAGGATCGTCTCTGGAACGCTCCGAAAGTCTTGTT
Clean codeword: GTATAACGTTTTCCGTTCTAGAGCCACGTCGCCGTAATCAGCGGGGGGCTGGTGCCAGGAAGGCCTCCTGGGATCCTAACATCCACGGTAAGTGCAACAAGTCAATCCGTTTCCCCAAGGCGCCGAGGTCGAACAATGAATTATGGCGCCAAGACATGGGTTCACTTTGTTGGGGCGCCGACCTGGAGAAACGATCCGGACGCGTAGGGGTCGTCTCTGGAACGCTCCGAAAGTCTTGTT
Received noisy: GTATAACGTTTTCCGTTCTAGAGCCACGTCGCCGTAATCAGCGGGGGGCTGGTGCGAGGAAGGCCTCCTGGGATCCTAACATCCACGGTAAGTGCAACAAGTCAATCCGTTTCCCCAAGGCGCCGAGGTCGAACAATGAATTCTGGCGCCAAGACATGGGTTCACTTTGTTGGGGCGCCGACCTGGAGAAACGATCCGGACGCGTAGGGGTCGTCTCTGGAACGCTCCGAAAGTCTTGTT
Decoded:        GTATAACGTTTTCCGTTCTAGAGCCACGTCGCCGTAATCAGCGGGGGGCTGGTGCCAGGAAGGCCTCCTGGGATCCTAACATCCACGGTAAGTGCAACAAGTCAATCCGTTTCCCCAAGGCGCCGAGGTCGAACAATGAATTATGGCGCCAAGACATGGGTTCACTTTGTTGGGGCGCCGACCTGGAGAAACGATCCGGACGCGTAGGGGTCGTCTCTGGAACGCTCCGAAAGTCTTGTT
Decode converged (all parity checks satisfied)
Recovered msg:  AGTGCAACAAGTCAATCCGTTTCCCCAAGGAGGCCTCCTGGAACAATGAATTATGGCGCCAAGACATGGGGGATCCTAACTGGGGCGCCGACCTGGAGAAACGATCCGGAGGTGCCAGGATCGTCTCTGGAACGCTCCGAAAGTCTTGTT  (matches original)

This run had just 2 real substitution error introduced by the channel (position 55, 142 in the codeword C→G, A→C) and the decoder corrected it — the recovered message matches the original exactly.

Simulation results for randomly generated messages:

./build/ldpc D build/gf4_150.ldspec 20 4 LDPC/acc-true_passQ-true_forward.txt 50 2000

1000 0.002
2000 0.002 0.001 0.002

Verification of the codewords:

./build/ldpc T build/gf4_150.ldspec 20

Verification successful

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Nonbinary Spatially-Coupled LDPC codes encoding and decoding implementation, and benchmark results for GF(4) DNA data storage applications.

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dnaldpcldpc-bp-decoder

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