Layered Superposition Alphabet Enabling Binary Soft-Decision Decoding for Composite DNA Storage

Composite deoxyribonucleic acid storage represents information by nucleotide mixture ratios and can exceed 2 bits per synthesis cycle. Readout requires inferring the transmitted composite letter from finite sequencing counts subject to sampling randomness. For this purpose, we propose a layered one-hot superposition alphabet. Each layer maps a pair of coded bits from a binary component code to a one-hot nucleotide indicator, and a weighted sum of these indicators forms a composite letter. The layered structure supports multi-level coding and multi-stage decoding, allowing count-based likelihoods that account for finite sampling to be converted into soft inputs for binary component decoders. We use alphabet-constrained capacity to guide layer-weight selection and derive layer-wise capacities as references for component-code rate allocation. We also introduce a hard-decision-aided update that reduces marginalization over candidate layer combinations by fixing the nucleotide choices of previously decoded layers. Finite-length simulations with standardized low-density parity-check component codes from the Digital Video Broadcasting–Satellite–Second Generation standard demonstrate binary soft-decision decoding under finite sampling and substitution errors, and characterize the sequencing-depth requirements and sensitivity to mixture-ratio mismatch and oligo-wise coverage variation. Under symmetric substitution errors with probability $0.05$, a four-layer design achieves $5.000$ bits per synthesis cycle with a bit error rate below ${10}^{-4}$ at 57 reads per position.