The factor graph of an LDPC barcoding system built from a 4-ary LDPC code of size <i>n</i> = 12 able to carry <i>k</i> = 3 informative quads and thus, to induce 64 candidate barcode sequences.

<p>Each codeword symbol <i>c</i>_<i>i</i>, <i>i</i> = 1, …, 12, is constrained by exactly <i>j</i> = 3 parity subcodes. The LDPC code is built from <i>m</i> = 9 parity subcodes, e.g., <i>c</i>₁ + <i>c</i>₈ + <i>c</i>₁₁ + 2 <i>c</i>₁₂ = 0 holds. A <i>QSC</i> generates mismatch sequencing errors with probabilities <i>p</i>_<i>i</i> = <i>p</i>_<i>s</i> and thus, corrupted barcode bases <i>r</i>_<i>i</i> are observed after sequencing, <i>i</i> = 1, …, 12. At <i>p</i>_<i>s</i> = 0.01 this system can multiplex up to <i>M</i> = 15 samples with <i>p</i>_<i>e</i> = 10⁻⁴ and <i>p</i>_<i>u</i> ≈ 0.