DNA diamond formulates a decomposable composite letter constellation model for DNA data storage

Domain: chemistry
Published: 2026-01-31
Editorial outcome: major_revision

Abstract
Oligonucleotide multiplicity is an inherent property of current DNA synthesis technology. Composite letter DNA storage exploits this property to improve logical density and reduce costs. However, letter indistinguishability and high molecular diversity pose challenges for reliable recovery. Here, we formulate a composite letter constellation model, named DNA diamond, consisting of 15 decomposable points. Inspired by set partitioning in telecommunications, we propose a two-stage letter detection framework that partitions these letters into four distinguishable subsets based on their discrete entropy. Furthermore, we incorporate encoded double-end indices to eliminate crosstalk between synthesis sites and simultaneously apply length filtering to suppress error propagation during readout. We validate the eight-letter and 15-letter composite letter DNA storage under DNA diamond model, each with 10,000 composite strands. The eight-letter system achieves a payload density of 2.5 bits per letter and enables error-free recovery at 14× coverage, surpassing the storage density of prior six-letter systems while requiring lower coverage. The full 15-letter constellation enables 3.125 bits per letter for payload with error-free recovery at 33× coverage, corresponding to a density of 2.23 bits per letter for payload plus indices. The proposed decomposable DNA diamond model advances a practical and scalable framework for high-density composite DNA data storage.

Introduction
With the explosive growth of global data, traditional storage media may not satisfy the demands of future long-term data storage 1–4. DNA data storage, which encodes digital information into synthetic DNA, has emerged as a promising alternative 5–8. The theoretical upper limit of 2 bits per synthesis cycle (per nucleotide) is established by using four natural DNA bases (e.g., A:00, T:01, G:10, C:11). Recent advancements have demonstrated an achievable information density of up to 1.57 bits per nucleotide using only natural bases 9. However, due to the limited number of natural bases, it remains challenging to further increase information density in a cost-efficient manner 10–13. To address this limitation and reduce overall costs, two approaches have been explored: (i) incorporating non-natural bases to expand the base variety 14–17 and (ii) employing composite letters 18,19. Composite letters, which are mixtures of the four natural bases in controllable ratios, effectively enhance logical density (Fig. 1a). Given that multiple copies of an individual DNA sequence are synthesized at a unique synthesis site (Fig. 1b), composite letters exploit the inherent redundancy of DNA synthesis 20–23.

Despite these advantages, composite DNA storage faces challenges during readout, particularly elevated error rates and the requirement of high sequencing coverage. These issues stem from two fundamental and interacting sources. First, synthesis and sequencing errors perturb the intended base mixture proportions. As the mixture complexity increases, this perturbation leads to higher inference error rates for composite letters at a given sequencing depth (Fig. 1c). Second, statistical sampling bias at low sequencing coverage leads to fluctuations in the observed base-frequency distributions, making it difficult to accurately infer the original composition. Together, these factors reduce the discriminability of similar base mixtures, thereby increasing error rates and necessitating higher coverage to ensure reliable readout (Fig. 1d). For example, in a DNA storage system with a 15-letter composite alphabet 18, error-free data recovery required a sequencing coverage of at least 250×, whereas conventional natural DNA data storage only required about 5× coverage 24,25. The molecular diversity increases sharply as the length of the composite strand increases and thus renders the traditional clustering scheme ineffective 19. An effective scheme is needed to identify molecules originating from the same synthesis site, as these molecules collectively represent a single composite letter. However, this aspect has not been thoroughly explored in existing studies. In prior work with a six-letter composite alphabet (Σ6), Anavy et al. 19 demonstrated data recovery at 29× sequencing coverage using Kullback–Leibler (KL) divergence inference, Fountain coding, and Reed–Solomon (RS) coding 26–28. More recently, Xu et al. 29 applied a soft-decision decoding algorithm (Derrick-cp) to the same alphabet, reducing the practical coverage threshold to 17×. For larger composite alphabets, such as a 256-letter scheme targeting ultra-high logical density (theoretically up to 8 bits/letter), simulations by Anavy et al. showed that approximately 2000× coverage would be required. The recent Derrick-cp algorithm reduced this requirement to 490×, highlighting the potential of advanced decoding methods 29. Importantly, while increasing the composite alphabet size enhances storage density, it also inherently raises the required sequencing depth. Offsetting this trade-off requires robust detection and decoding strategies, such as those incorporating soft information, which can substantially lower coverage requirements and reduce overall sequencing costs. Nowadays, there are some theoretical investigations on the design of composite letter DNA storage 30–32. In this study, we adopt set partitioning (SP), an information-theoretical paradigm widely used in communication modems, to rationally devise the composite alphabet for DNA storage.

Here, we develop an information-theoretic framework to achieve low-coverage and reliable data readout for composite letter DNA data storage (Fig. 1e). First, we formulate a decomposable constellation model, termed DNA diamond. It comprises 15 points and increases the theoretical logical density to log2(15) ≈ 3.9 bits per synthesis cycle (per letter). Second, leveraging the discrete property of the information entropy of these 15 letters, we propose a two-stage composite letter detection method (Supplementary Fig. 1). These 15 letters are partitioned into four subsets based on their discrete entropy values. Maximum likelihood estimation (MLE) is then applied within each subset to infer specific composite letters. Third, given that reliable index sequences are critical for accurate data readout 33, we propose the encoded double-end indices to eliminate crosstalk. Simultaneously, we introduce a length filtering step to discard corrupted reads with indel errors. To verify the proposed scheme, we designed error correction codes tailored to different alphabets and conducted experiments on both column-based and array-based synthesis platforms, along with simulation tests (Supplementary Figs. 2–4). Specifically, we implemented the eight-letter alphabet using column-based synthesis. Error-free data recovery was achieved at 14× sequencing coverage. This scheme increased the information density from the canonical 2 bits per letter to 2.5 bits per letter (excluding primers and indices). The logical density of payload and indices was 1.97 bits per letter. In comparison with state-of-the-art schemes, our system achieved comparable performance in terms of density and sequencing coverage (Fig. 1f and Supplementary Table 1). We further designed large-scale oligonucleotide pools containing 10,000 composite strands. Both the eight-letter and 15-letter alphabets were experimentally validated using array-based synthesis. For the 15-letter system, our composite letter detection method enabled error-free recovery at 33× sequencing coverage. The system achieved a logical density of 3.125 bits per letter for payload, corresponding to a density of 2.23 bits per letter for payload and indices. Our work validated the feasibility of using a 15-letter alphabet for DNA data storage (10,000 strands synthesized in a single pool) and demonstrated that error-free recovery can be achieved at relatively low sequencing coverage.

Results
Decomposable DNA diamond constellation model and signal set partitioning
The composite DNA storage faces two previously unencountered challenges. First, the letter error rate increases with the number of constituent natural bases per composite letter 19. Second, the number of unique DNA strands is determined by the number of composite letters and increases explosively as the alphabet extends (Supplementary Figs. 5 and 6). High error rates and sequence diversity impose greater demands on sequencing coverage to ensure reliable data recovery. To overcome these limitations, we developed a model for composite letter DNA storage and adopted a well-established concept from information theory to improve composite letter detection.

First, we proposed the DNA diamond, a 15-letter decomposable alphabet constellation model. The DNA diamond can be geometrically represented as a tetrahedral simplex whose four vertices represent the natural bases {A, T, G, C}. Composite letters are formed by combining these bases in defined ratios and can be represented as four-dimensional vectors. Fifteen representative points are selected on the tetrahedron to form a finite set of decomposable composite letters, including its vertices, edge centers, face centers, and the centroid. Unlike previous resolution-based composite alphabets 19, the finite set of 15 letters exhibits only four discrete information entropy values (Supplementary Fig. 7, Supplementary Table 2). This property enables entropy-guided partitioning of 15 letters into four distinguishable subsets.

Then, we developed an innovative two-stage composite letter detection framework (Supplementary Fig. 8). In the first stage, entropy-based SP separates the 15 letters into four subsets. In the second stage, MLE is employed to infer the optimal result. In practice, synthesis and sequencing noise broaden the originally discrete entropy values into distinguishable distributions, as observed in experimental data. This strategy, adapted from coded modulation techniques in telecommunications 34,35, reduces the error rate in composite letter detection. A similar entropy-based method has been applied in the base calling of Solexa sequencing 36. Building on this composite letter detection approach, we established a complete data recovery pipeline. In brief, it begins with paired-end sequencing reads and involves primer identification, length filtering, index identification, and composite letter detection within each group (Supplementary Fig. 9).

Robust coding for non-clustering composite letter grouping
In conventional DNA storage systems using only natural bases, sequencing reads exhibit high homogeneity. This enables the use of sequence-level redundancy and clustering with consensus calling to reduce errors 37,38. However, this strategy fails for composite DNA strands due to their intrinsic heterogeneity (Fig. 2a). To quantify this issue, we measured the number of ordinary strands corresponding to each composite strand. Analysis results revealed an exceptionally high sequence diversity, estimated between 10^14 and 10^20 (Supplementary Table 3). At this scale, the traditional clustering-based error correction scheme becomes computationally infeasible. To enable non-clustering composite letter grouping and correct letter-level errors, we designed reliable double-end indices and applied efficient error correction codes to the payload (Supplementary Note 1).

First, we designed double-end indices to improve the reliability of read identification. The 7-bit address was encoded into two 8-nt natural DNA sequences using Bose-Chaudhuri-Hocquenghem (BCH) and repetition encoding, forming forward and reverse indices (Fig. 2b). Specifically, the forward index was encoded using BCH(15, 7) with an additional parity check bit for error detection. The reverse index employed a parity checksum and a cyclic left-shift mechanism. These encoded double-end indices enable the elimination of crosstalk between synthesis sites.

Then, we verified the applicability of discrete-entropy composite letters by testing multiple storage schemes across different coding schemes and alphabets (Fig. 2c, Supplementary Figs. 10 and 11, Supplementary Table 4). The encoding and writing process of the composite DNA storage system included three stages: data encoding, composite letter mapping, and DNA synthesis.

For data encoding, in the eight-letter system, 945 bytes of text were first encoded using NB-LDPC(3780, 1260) code (R=0.33) over Galois Field (GF(64)), with interleaving to mitigate burst erasures 39 (Supplementary Fig. 12). To validate high logical density, 2,355 bytes of text data were encoded with RS(1890, 1575) code (R=0.83) over GF(2^12), achieving 2.5 bits/letter (Supplementary Fig. 13). We further extended the eight-letter scheme to large-scale pools, where a 187,500-byte image file was encoded using RS(3750, 3125) code (Supplementary Fig. 14). In the 15-letter system, both experiments and simulations were performed. A 234,375-byte image file was encoded with RS(30000, 25000) code (R=0.83, 3.125 bits/letter) for experimental verification (Supplementary Fig. 15). A 780,713-byte image file was encoded with RS(32767, 27767) code (R=0.85, 3.18 bits/letter) for simulations (Supplementary Note 2, Supplementary Fig. 16).

For composite letter mapping, we chose four composite letter alphabets (Fig. 2d) and analyzed the impact of different bit group sizes on mapping efficiency (Supplementary Figs. 17 and 18). The eight-letter alphabet was implemented by converting every three bits into a single composite letter. The mapping efficiency is 100%. For the 15-letter alphabet, a group of 15 bits was converted into four composite letters, corresponding to an optimized efficiency of 96%. We chose the group size of 15 for satisfactory efficiency and error propagation within the group. Details can be found in Supplementary Note 3. Finally, encoded letters were partitioned into strands of 60 letters, padded with indices and primers, generating pools of either 126 or 10,000 strands for synthesis validation (Fig. 2e, Supplementary Tables 5 and 6).

For practical writing into DNA, two synthesis platforms were used to test different composite alphabet implementations (Supplementary Fig. 4). First, 126-strand pools were constructed with eight-letter alphabets using column-based phosphoramidite synthesis. Each strand was synthesized in an individual well, with composite letters implemented by volumetric mixing of A, T, C, and G phosphoramidite monomers. Second, large-scale pools containing 10,000 composite strands were synthesized by array-based inkjet phosphoramidite chemistry. In all experiments, payloads were uniformly 60 letters, flanked with double-end indices and PCR primer sequences (Fig. 2e). For the column-based pools of 126 strands, each carried two indices (8 nt) and two primers (20 nt) at the 5′ and 3′ ends. For the array-based pools of 10,000 strands, the indices were 6 nt or 12 nt at the two ends, respectively.

A two-stage composite letter detection via discrete information entropy
Accurate data readout in composite DNA storage requires inferring the intended composite letter at each synthesis site based on observed frequencies of constituent natural bases. However, this task is complicated by sampling bias and letter degradation. Inspired by SP in communication modems, we developed a two-stage detection method that exploits the fact that the 15 composite letters exhibit only four distinct information entropy values (Supplementary Notes 4 and 5).

The proposed framework integrates entropy-based letter partitioning with likelihood-based estimation, which reduces the size of the candidate letter set and thereby improves detection accuracy (Fig. 3a, b). In the first stage, we calculated the information entropy of observed base frequencies at each synthesis site and assigned the signals to one of four subsets. Raw sequencing data at various coverages revealed that sampling depth significantly influences the accuracy of observed base frequencies (Supplementary Figs. 19–21). For large-scale pools (10,000 composite strands) with eight-letter or 15-letter composite alphabets, the base frequencies were still guaranteed (Supplementary Figs. 22–25). Nevertheless, the transformation from raw signal distributions to entropy values enabled clear discrimination between composite letter types (Fig. 3c and Supplementary Fig. 26). In the second stage, MLE was applied within each subset to identify the most probable letter. The inferred probabilities were normalized and visualized using radar plots, where the radial distance from the origin quantified divergence from the candidate letters (Supplementary Fig. 27). Optimal letter detection was achieved by comparing with predetermined letters in each subset.

To assess the performance, we first developed a simulation model capable of generating base-frequency distributions, sequencing coverage, and user-defined random errors (Supplementary Fig. 28). The coverage parameter in the model was obtained by fitting experimental copy number distributions at different average sequencing depths. For column-based synthesis pools, the observed read counts per strand followed a negative binomial distribution (Supplementary Figs. 29 and 30). Consistently, in array-based synthesis pools containing up to 10,000 oligonucleotides, the read count distributions also conformed to the negative binomial model (Supplementary Figs. 31–33). This reflects stochastic sampling and strand-specific differences in PCR amplification efficiency.

Then, we evaluated composite letter detection performance. Simulated sequencing datasets under varying coverage were then processed through our standard readout pipeline. Error rate analysis at both detection stages revealed that most inference errors arose during the SP phase and were more prominent at low coverage (Supplementary Fig. 34). Detection accuracy within each subset improved as coverage increased. To further investigate misclassification behavior, we analyzed spatial distances among the 15 letters in the DNA diamond space and observed that misidentification was strongly correlated with spatial distance (Supplementary Fig. 35).

Finally, we compared the end-to-end recovery performance of our SP method with the KL method 19 and the maximum a posteriori probability (MAP) method 29 with computer simulations (Fig. 3d, e). MAP performs global inference and shows the highest detection accuracy but the longest runtime. SP reduces the candidate space through partitioning, resulting in a slight degradation in accuracy, as well as a reduction in computational complexity. Both SP and MAP methods achieved lower error rates than the KL method. With 1000 independent trials using 16 threads on the same server (Intel Xeon Gold 5220R CPU @ 2.20 GHz and 256 GB RAM), SP consumed only 25% of the time compared to the MAP method. In the simulation settings, MAP achieved error-free recovery at an average coverage of 25×, SP at 30×, whereas KL required 45× coverage. The performance in vitro experiments will be further verified in the following sections.

Performance evaluation on constituent base-frequency variation at low coverage
The accuracy of composite letter detection is sensitive to sequencing coverage, as lower coverage leads to increased noise in the observed base frequencies. In particular, insufficient sampling may lead to signal degradation of composite letters. We evaluated the robustness of our detection method at low coverage using synthetic pool datasets, focusing on entropy distribution, error rate, and recovery performance.

First, lower sequencing coverage significantly impairs composite letter detection accuracy due to increased noise in observed base frequencies. Downsampling experiments were performed at coverages ranging from 300× to 10× (Fig. 4a), revealing that lower coverage amplifies sampling noise and broadens the entropy distributions of composite letters. This overlap between distributions decreases their separability (Fig. 4b). To quantify this signal degradation, we evaluated letter detection error rates using three different eight-letter composite alphabets. As expected, lower coverage resulted in increased errors across both stages of the two-stage detection process (Supplementary Fig. 36). Experimental results showed substitution error patterns that closely matched simulation predictions (Supplementary Figs. 37–39). Notably, errors predominantly arose from substitutions between composite letters and their constituent natural bases, while within-subset substitutions were rare. Across all tested alphabets, composite letters were more prone to substitution errors than natural bases (Fig. 4c). Moreover, letters composed of more complex mixtures exhibit higher error rates at the same sequencing coverage (Supplementary Figs. 40 and 41). Although all three eight-letter alphabets had the same size and comparable logical density, Alphabet 1, which contains a higher proportion of natural bases, exhibited significantly lower letter error rates.

Then, the minimum sequencing coverage for data recovery was evaluated with 1000 independent trials on two sequencing datasets. Both the erasure error rate and the letter error rate progressively decreased as the average coverage increased (Fig. 4d, e). At low sequencing coverage, erasures dominated, while with increasing average coverage, erasures were suppressed, enabling reliable recovery. Using the NB-LDPC code with a code rate of 0.33, error-free recovery was achieved at an average coverage of 9×, even with an erasure error rate as high as 7% (Fig. 4f). Using the RS code with a code rate of 0.83, the coverage of 14× was required for error-free recovery.

Third, we benchmarked the SP method against the KL and MAP approaches at low sequencing coverages. In the column-based eight-letter experiments and the array-based experiments of 10,000 strands (including two eight-letter and two 15-letter systems), MAP consistently achieved the highest detection accuracy (Supplementary Tables 7 and 8). SP, as the second-best in accuracy, showed clear advantages in runtime over both MAP and KL (Supplementary Figs. 42 and 43). For the eight-letter pool of 10,000 strands, MAP enabled error-free recovery at 15× coverage, compared with 16× for SP. Executed with 10 threads, SP completed 300 validation runs in ~10 s (Supplementary Figs. 44 and 45). For the 15-letter system, MAP also provided the highest accuracy (Supplementary Figs. 46 and 47), while SP showed boundary-related precision loss yet still achieved error-free recovery at 33× coverage, consistent with simulations.

Finally, we applied symbol-level interleaving to improve the error correction capability. Low sequencing coverage not only introduces fluctuations in observed base frequencies but also causes molecule loss due to random sampling bias. The resulting consecutive erasure errors within codewords increase the difficulty of data recovery. Simulations confirmed that random interleaving disperses burst errors and enhances decoding robustness (Supplementary Figs. 48–51). Recovery tests showed that NB-LDPC codes with interleaving achieved error-free decoding at 9× average coverage, whereas RS codes with a higher code rate (0.83) required 17× average coverage (Supplementary Fig. 52). In contrast, a non-interleaved control based on Alphabet 3 showed poor resilience to molecule loss, requiring much higher coverage for reliable recovery (Supplementary Fig. 53). Additionally, recovery in the experimental data from Alphabet 2 was further impaired by the synthesis failure of a single composite strand.

Performance evaluation on strand crosstalk, error propagation, and DNA degradation
The reliability of composite letter detection is crucial for accurate data recovery in DNA storage. However, three factors compromise performance: crosstalk caused by interference reads, error propagation introduced by indels, and strand degradation. Crosstalk stems from the potential assignment errors between the indices, leading to misidentification of strands and altered entropy values (Fig. 5a, b). Indel-induced propagation errors distort the length and structure of reads, further interfering with the letter detection process (Fig. 5c). Notably, homopolymers are unavoidable in sequencing reads, and long homopolymers may introduce more indels (Supplementary Fig. 54). Fortunately, long homopolymers occurred rarely. Homopolymer length distributions in composite systems are similar to four-letter DNA storage without homopolymer optimization or random sequences (Supplementary Fig. 55). Several DNA storage systems guaranteed homopolymer length through appropriate encoding, e.g., rotation coding 4 or homopolymer filtering 9. We did not apply homopolymer optimization, balancing the efficiency and performance 40. To address these challenges, we incorporated two tailored strategies into the readout pipeline: robust double-end indices to mitigate strand crosstalk and DNA degradation; length filtering to reduce the impact of indel errors during synthesis and sequencing.

First, based on paired-end sequencing data, we established a pipeline that stitches read pairs, aligns primers, and extracts the central region containing the index and payload (Supplementary Figs. 56–58). The retained payload regions, which may contain errors, were grouped based on matching index sequences, each representing multiple copies of a unique payload. Analysis revealed that 99% of indices in our sequencing reads were error-free, and the remaining errors involved fewer than two substitutions (Fig. 5d and Supplementary Fig. 59). We retained reads in which both ends contained fewer than two substitution errors and were mapped to the same reference composite strand.

Then, the double-end strand structure enables robust data retrieval even when partial degradation occurs, for example, a strand break. Any fragment retaining at least one intact primer–index pair can still be correctly clustered and aligned according to its index (Supplementary Fig. 60). To evaluate the performance on strand break, we performed accelerated thermal aging experiments. The composite pool containing 126 strands of 116 letters was incubated at 80°C (2.5 h), 85°C (2 h), 90°C (2 h), and 95°C (2 h), with an unheated sample as the control. Under higher temperatures, the read lengths showed a pronounced shift toward shorter fragments. Consistently, the proportion of correctly assembled 116 nt reads decreased, while truncated fragments accumulated (Supplementary Figs. 61–63). At 95°C, nearly 14% of reads contained only a single-end primer (Supplementary Fig. 64). Despite this degradation, data recovery remained highly robust. Letter error rates increased moderately with degradation temperature. Error-free recovery was still achieved at sufficient sequencing coverage (Supplementary Fig. 65). Stress tests using short fragments revealed that the proposed scheme could recover the original data error-free, though the required sequencing coverage was increased (Supplementary Fig. 66).

Finally, length filtering was introduced to suppress error propagation caused by indels. Under the same average coverage, we evaluated the effectiveness of length filtering in reducing the symbol error rate by comparing two schemes. Across both coding schemes (R=0.83 and R=0.33), filtering consistently improved letter detection accuracy (Fig. 5e, f and Supplementary Fig. 67). At the coverage of 14×, filtering reduced the average composite-letter error rate from 2.8% to 1.0% (accuracy > 99%). With the RS code (R=0.83), length filtering enabled error-free recovery in all trials. Without filtering, only ~30% of trials achieved error-free recovery under the same coverage. Although filtering decreased the number of usable reads, 94.39% were retained after primer identification, length filtering, and index identification, ensuring high-quality input for downstream detection (Fig. 5g). Supplementary Table 9 presents the results of all pools, including small-scale pools and large-scale pools.

Cost analysis on composite letter DNA storage at low coverage
We performed a cost analysis in two application modes, consistent with prior studies, quantifying both synthesis and sequencing costs for our eight-letter and 15-letter schemes (Supplementary Note 6). Synthesis cost was modeled from logical density and sequencing cost from average coverage 19,25.

First, we estimated the total cost of storing 1-MB user data in DNA, including synthesis and sequencing. To compute the overall storage cost of 1-MB user data, we applied unit prices of $0.05/100 nt for synthesis 9 and $0.0000012/100 nt for sequencing 41 (Supplementary Fig. 68). We estimated the nucleotides needed to store 1-MB user data. Under these assumptions, the synthesis-to-sequencing cost ratio can be as high as 40,000:1, with the synthesis remaining the dominant cost. On the one hand, only considering the payload part, our eight-letter and 15-letter systems reduced the synthesis cost by 47% and 57%, respectively, compared with the conventional four-base DNA storage system 25. On the other hand, considering the index overhead (excluding primers), for a similar alphabet size of six or eight, we obtained similar cost reductions. For the 15-letter alphabet, we achieved large-scale validation using 10,000 strands, also with a slight increase in synthesis cost (Supplementary Fig. 68). This was primarily due to the adoption of a more robust double-end index scheme that introduced additional redundancy to improve reliability.

Then, as synthesis technology continues to improve and costs decline, we further normalized total cost under representative synthesis-to-sequencing ratios (e.g., 1000:1 and 500:1) relative to a baseline natural-base system with 5× coverage 25 (Supplementary Table 10). Only considering the region excluding primers at a 1000:1 ratio, the eight-letter and 15-letter systems reduced the total storage cost by 47% and 56%, respectively. When the ratio was reduced to 500:1, the total cost reductions remained at 46% and 55%, respectively.

Methods
A decomposable diamond for composite letter DNA storage
A decomposable diamond model was proposed to represent the composition of composite letters and partition the 15 decomposable constellations into four different subsets.
• Subset 1 (Vertices): this subset consists of the four natural bases, {A, T, G, C}, positioned at the vertices of the diamond constellation model.
• Subset 2 (Edges): defined by the six central points on the edges of the diamond, this subset includes {R, M, Y, K, S, W}. Each composite letter in this group is synthesized as an equimolar mixture of any two natural bases.
• Subset 3 (Faces): corresponding to the four central points on the faces of the diamond, this subset comprises {H, B, V, D}, where each composite letter is formed as an equimolar mixture of any three natural bases.
• Subset 4 (Centroid): representing the center point of the diamond model, this subset consists of a single composite letter {N}, synthesized as an equimolar mixture of all four natural bases.

Each letter in the diamond model is characterized by a probability vector (σA, σT, σG, σC) that represents the projection relationship between natural bases and composite letters (Supplementary Table 11). Zero-probability events were replaced with a smoothing term (ϵ=10−6) to ensure numerical stability, where σA + σT + σG + σC = 1.

Encoding scheme for experimental proofs with eight-letter and 15-letter alphabets
Two sets of experiments on the composite-letter DNA storage were conducted based on the constructed composite-letter alphabets (Supplementary Data 1). We devised four alphabets as follows.
• Alphabet 1: {A, T, G, C, R, Y, M, K};
• Alphabet 2: {A, T, R, Y, M, K, S, W};
• Alphabet 3: {A, T, G, C, H, B, V, D};
• Alphabet 4: {A, T, G, C, R, M, Y, K, S, W, H, B, V, D, N}.

In the small-scale composite strand storage, three eight-letter alphabets (Alphabets 1–3) were employed. Correspondingly, three different coding schemes were used. They were the NB-LDPC(3780, 1260) over GF(64), the NB-LDPC(3780, 1260) over GF(64) with random symbol interleaving, and the RS(1890, 1575) over GF(2^12). Each encoding process produced 126 composite strands of 60 letters, which were synthesized into four oligonucleotide pools. In the eight-letter system, every 3 bits were converted into one composite letter, and when encoded with RS(1890, 1575) at a code rate of 0.83, a logical density of 2.5 bits per letter was achieved (excluding indices and primers).

In the large-scale composite strand storage, an eight-letter alphabet (Alphabet 1) and a 15-letter system (Alphabet 4) were tested. For the eight-letter system, a 187,500-byte digital file was segmented into 40 data blocks and encoded using RS(3750, 3125) over GF(2^12), yielding 10,000 composite strands. For the 15-letter system, a 234,375-byte image file was segmented into 5 data blocks and encoded using RS(30000, 25000) over GF(2^15), yielding 10,000 composite strands. Every 15 bits was mapped to four composite letters, corresponding to a logical density of 3.125 bits/letter (excluding indices and primers). In both systems, each payload strand consisted of 60 composite letters flanked by indices and primers.

Double-end indices for homology identification of sequencing reads
To minimize the interference of crosstalk during data readout, double-end indices were designed to label the order of composite strands used in experimental validation. These indices consisted exclusively of natural bases and were generated through the following steps:
1) A 7-bit address in the range [1, 126] was assigned to each of the 126 DNA sequences.
2) The 5′ end index was generated using a BCH(15, 7) code with an additional parity check bit. The encoded sequence was mapped to DNA bases using the rule: 00 → A, 01 → T, 10 → G, 11 → C.
3) For the 3′ end index, a parity check bit was first generated based on the original 7-bit address. This address was then circularly left-shifted by one bit to generate a second 7-bit sequence. The 16-bit index was structured as follows: parity bit–original bits–parity bit–left-shifted address bits.
4) Following the same mapping rule as in Step 2, the encoded address bit sequence obtained in Step 3 was converted into an 8-nt index DNA sequence and appended to the 3′ end of the composite strand.
5) Each payload was flanked with an 8-nt index and a 20-nt amplification primer at both ends, resulting in a total sequence length of 116 letters.

Synthesis of small-scale pools (column-based synthesis)
A total of four small-scale composite-letter oligonucleotide pools were prepared. For each pool, 126 composite DNA sequences were synthesized using an Oligo-192 DNA Synthesizer (Hippobio, Huzhou, China). Each strand was synthesized independently in a dedicated column, with composite letters implemented through volumetric mixing of A, T, G, and C phosphoramidite monomers. Oligonucleotides were purified by high-performance liquid chromatography (HPLC). All 126 samples were then manually combined in equimolar amounts to generate one master pool. Four such master pools were obtained, with final DNA concentrations of 2482, 2515, 301, and 459 ng/μL, respectively.

Synthesis of large-scale pools (array-based inkjet printing synthesis)
Four large-scale oligonucleotide pools were designed, each comprising 10,000 composite strands: two based on the eight-letter alphabet and two based on the 15-letter alphabet. These pools were synthesized by Dynegene Technologies (Shanghai, China) using an array-based inkjet printing synthesizer. During synthesis, four inkjet nozzles dispensed monomer droplets onto array chips according to predefined base-mixing ratios. The eight-letter system used the alphabet {A, T, G, C, R, Y, M, K}, and the 15-letter system used the alphabet {A, T, G, C, R, M, Y, K, S, W, H, B, V, D, N}. Each large-scale pool was delivered as a lyophilized master pool containing 3 µg of DNA.

Library preparation of small-scale pools
All four small-scale pools were diluted to 0.1 ng and amplified in 25 μL PCR reactions using Phusion High-Fidelity DNA Polymerase (HotStart Flex 2X Master Mix, New England Biolabs, Cat# M0536S) (Supplementary Table 12). The PCR program included an initial denaturation at 98°C for 45 s, followed by 30 cycles of 98°C for 10 s, 60°C for 30 s, and 72°C for 30 s, and a final extension at 72°C for 1 min. Amplification products were purified using AMPure XP Beads (Beckman Coulter, Cat# A63880) at a 3:1 bead-to-DNA ratio and eluted in 20 μL of double-distilled water (ddH2O).

Furthermore, two small-scale pools designed with Alphabet 1 {A, T, G, C, R, Y, M, K} were subjected to low-cycle PCR with the KAPA HiFi HotStart PCR Kit (Roche, Cat# KK2502). The PCR program consisted of an initial denaturation at 98°C for 45 s, followed by 10 cycles of 98°C for 10 s, 60°C for 30 s, and 72°C for 30 s, with a final extension at 72°C for 1 min (Supplementary Table 13). Each 25 μL reaction contained 0.8 ng of DNA sample, 0.75 μL of KAPA dNTP Mix (10 mM each), 1.5 μL of 10 μM forward primer, 1.5 μL of 10 μM reverse primer, 5 μL of 5× KAPA HiFi Fidelity Buffer, and 0.75 μL of KAPA HiFi HotStart DNA Polymerase (1 U/μL). The final volume was adjusted to 25 μL using ddH2O. PCR products were purified using AMPure XP Beads (Beckman Coulter, Cat# A63880) at a 1.8× volume ratio and quantified using the Qubit 1X dsDNA HS Assay Kit (Thermo Fisher Scientific, Cat# Q33231).

Library preparation of large-scale pools
Libraries from four large-scale pools were PCR amplified for 10 cycles using the KAPA HiFi HotStart PCR Kit (Roche, Cat# KK2502) following the same reaction composition and thermal cycling parameters (Supplementary Table 13). PCR products were purified using AMPure XP Beads (Beckman Coulter, Cat# A63880) at a 1.8× volume ratio. Purified DNA was eluted in ddH2O and quantified using the Qubit 1X dsDNA HS Assay Kit (Thermo Fisher Scientific, Cat# Q33231).

Sequencing and data generation
After PCR amplification, purified PCR products were sequenced by Novogene (Beijing, China) and LC-Bio Technology Co., Ltd. (Hangzhou, China). The two small-scale pools (based on Alphabet 1) and the four large-scale pools (10,000 strands each) were sequenced by Novogene on the Illumina NovaSeq X Plus platform, generating 150-nt paired-end (PE150) raw reads. In parallel, for cross-platform validation, all four small-scale pools were also sequenced by LC-Bio Technology using Illumina sequencing platforms.

Accelerated aging test of oligo pools
Accelerated aging experiments were performed on a small-scale pool consisting of 126 composite strands encoded with the RS code. Four identical samples, each 25 µL, were incubated at 95°C for 2 h, 90°C for 2 h, 85°C for 2 h, and 80°C for 2.5 h in the absence of light and then stored at −20°C until sequencing.

DNA concentrations before and after aging were measured using the Qubit 1X dsDNA Assay Kit (Thermo Fisher Scientific, Cat# Q33231). Quantification revealed initial DNA masses of 57, 66, 59, and 82 ng, which decreased to 34, 35, 41, and 49 ng, respectively, after accelerated aging. The final four degraded DNA libraries and the non-degraded control DNA libraries were sequenced by Novogene on the Illumina NovaSeq X Plus, generating 150-nt paired-end (PE150) raw reads.

Simulation verification of the information density
To evaluate the information density of composite letter DNA storage, we conducted simulations using the RS(32767, 27767) code over GF(2^15) with a code rate of 0.85 to 15-bit symbols, achieving a mapping efficiency of 96%. Using this approach, a 780,713-byte image was encoded into 15 RS codewords, generating 32,767 composite strands (60 letters per strand), resulting in a logical density of 3.18 bits per letter.

To assess system performance, a simulation model was constructed. The model involves fixed parameters: (i) sequencing coverage M, (ii) the error rates (insertions, deletions, and substitutions), and (iii) random instantiation as a permutation over A, T, G, and C. The specific simulation process follows these steps:
(1) Based on real sequencing reads, a Poisson-Gamma distribution (equivalent to a negative binomial distribution) was fitted to model the copy number of each sequence under varying sequencing coverage M. This modeling resulted in a set of shape parameters r and success probabilities p (Supplementary Table 14). Simulations were performed to determine the number of strand copies under varying coverage conditions, using 32,767 composite strands with a 15-letter alphabet.
(2) Each position containing a composite letter was instantiated as random combinations of A, T, G, and C according to a predefined probability vector (σA, σT, σG, σC).
(3) The error model assumed that each position of instantiated DNA sequences was subject to a total error rate Perror, defined as the sum of the substitution error rate Psub, the insertion error rate Pins, and the deletion error rate Pdel. For each nucleotide i (where i ∈ {A, T, G, C}), the error probability Perrori was defined as: Perrori = Pins + Pdel + Psub. Given the total number of positions k and the count of each nucleotide αi, the total expected number of errors across all positions was calculated as: Σαi × Perrori = k × (Pins + Pdel + Psub).

In practice, we selected three types of errors (Pins = 0.001, Pdel = 0.001, Psub = 0.002) and performed 1000 repetitions at each sequencing coverage. The simulated data were processed using the same readout pipeline as real sequencing reads.

A fast data recovery pipeline from paired-end sequencing reads
The data recovery of the composite DNA storage consists of the following steps:
(1) Read assembly: paired-end reads were assembled using the paired-end read merger program PEAR 46, and the target region (including index and payload) was extracted based on the alignment with paired-end primers.
(2) Length filtering: reads that did not match the ideal length for the target region after primer identification were discarded.
(3) Index identification: payload data were identified and grouped using double-end indices. All copies of reads containing the same index sequence correspond to a unique composite strand.
(4) Composite letter detection: a two-stage detection method was applied, incorporating SP and MLE.
(5) Error correction and decoding: the inferred composite letters were decoded and corrected using the LDPC decoding or RS decoding, achieving error-free data recovery.

Two-stage letter detection method
To accurately recover data, we proposed a two-stage composite letter detection method (Supplementary Notes 4 and 5). In the first stage, the observed base frequencies were mapped to information entropy and partitioned into several subsets. At each synthesis position i, the observation probability for each natural base Pi,α was defined as Pi,α = ri,α / Σα∈{A,T,G,C} ri,α, where ri,A, ri,T, ri,G, ri,C are the counts of natural bases at position i, and Σα∈{A,T,G,C} Pi,α = 1. Information entropy for each synthesis position i was defined as hi = − Σα∈{A,T,G,C} Pi,α log2(Pi,α), where hi quantifies the uncertainty associated with the observed base frequencies. Each of the 15 composite letters was associated with a theoretical Shannon entropy of 0, 1, 1.5, or 2 bits. The measured entropy of each letter was compared to these four discrete values, and the letter was assigned to the subset corresponding to the closest entropy level:
• S1 = {A, T, G, C}, natural bases.
• S2 = {Y, M, K, R, S, W}, consisting of two types of natural bases.
• S3 = {H, B, V, D}, consisting of three types of natural bases.
• S4 = {N}, consisting of four types of natural bases.

In the second stage, a more precise comparison of each candidate letter was performed using a MLE method within each subset, as detailed in the following:
First, the divergence between the observed frequency and the candidate letter was quantified. A likelihood function l(σ, ri) was formulated based on the observed natural base counts at each synthesis position ri and the original probability vector (σA, σT, σG, σC) of each candidate composite letter within the subset. The likelihood function was defined as l(σ, ri) = Σα∈{A,T,G,C} ri,α log(σα).

Then, the probabilities obtained from the MLE were normalized to determine the composite letter with the highest likelihood as the detection result for each synthesis position.
Finally, the detected composite letter was mapped to its corresponding original bit sequence, followed by an error correction scheme to resolve residual substitution or erasure errors. This process enables accurate reconstruction of the original digital files.

Statistics and reproducibility
No statistical method was used to predetermine sample size. No data were excluded from the analyses. The experiments were not randomized. The investigators were not blinded to allocation during experiments and outcome assessment.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This work discussed an interesting approach, named DNA diamond, which a composite letter constellation model for encoding in DNA data storage. This model expands the letters available for DNA data storage from 4 (simply natural bases A, T, C, G) to 15, considering the probability distribution of natural bases. Compared with other composite letter approaches, this method achieved the highest bit-to-letter coding efficiency with a relatively low sequencing coverage required to fully recover the data. The authors also developed methods to enable accurate and lossless recovery of data, including set partitioning, maximum likelihood estimation (MLE), appending double-end indices and perform length filtering.\n\nThe concept of DNA diamond is great. However, in practice, one of the key challenges of DNA data storage is the synthesis cost, which is measured by how many bits are encoded in one nucleotide. Although this work did a great job in improving the letter efficiency, I am not able to find any elaborations on the actual coding potential of this coding strategy. How many nucleotides, in total, are synthesized to encode each experimental (7,560-bit and 18,900-bit data) and simulation (780,713-byte image)? In my understanding, as shown in Supplementary Figure 2, N strands of 60-nt DNA only yielded 60 letters. If 1 letter can encode 3 bits in their wet experiments, then these strands just encoded180 bits of data. I suggest the authors to calculate and clearly state the coding potential (not considering the indices, PCR primers and error correction codes) and the net information density (all synthesized nucleotides considered) using DNA diamond, with a unit of “bits/nt”.\n\nExcept the major concern above, I also have other concerns in terms of the feasibility of this coding strategy in real settings:\n1. In practice, the existence of homopolymers (3 or more) of natural DNA bases is detrimental to sequencing-based data recovery. However, by mapping 1 letter to 3-bit of data, it is impossible to avoid the appearance of homopolymers in synthesized DNA strands. How would the authors overcome this issue?\n2. In their simulation experiment, the mapping between 15 binary bits (2^15 = 32,768) and 4 letters (15^4 = 50625) left so many combinations of letters not used in encoding and therefore is not efficient. This origins from the fact that this encoding strategy is very sensitive to the total bits of information to be coded, and the coding efficiency is not always high for different data size. In addition, I noticed that for different sizes of data, a different size of block should be carefully selected (6-bit in wet experiment and 15-bit in simulation). Did the authors investigate the impact of the size of block on coding potential?\n3. I am not clear on how the authors can accurately control the read count of each DNA strand through sequencing. I assume this was achieved through adjustment of copy numbers of each strand. However, this significantly complicates experimental procedure and is apparently not practical in commercial DNA data storage settings. In addition, the efficiency of PCR is different for different amplicon sequences (the payload in this study). For example, the amplification may be difficult to proceed when there are homopolymers in the amplicon. When this encoding strategy scale up to 10,000 or more strands in one oligo pool (which is very typical in real DNA data storage applications), can the probability distribution of each natural DNA base still be guaranteed?\n4. The degradation of DNA is also a concern in practice, and the degradation rate of DNA strands with different sequences may also be non-homogeneous, which cause distortion of the assumed probability distribution of natural bases. I suggest the authors to perform accelerated aging experiments on their synthesized DNA pools (elevated temperatures, acidic and basic conditions, repeated freeze-thaw cycles) to investigate if this encoding strategy is less resistant to DNA degradations compared with conventional encoding strategies.", "reviewer_2": "This manuscript introduces a “diamond” composite-letter alphabet for DNA storage and an entropy-guided, two-stage detection pipeline: composite letters are first partitioned into entropy-defined subsets and subsequently identified within subsets by maximum-likelihood inference. Wet-lab experiments on an eight-letter alphabet report achieving ~2.5 bits per letter and error‑free recovery at 18× Illumina coverage. Although these proof-of-concept results are promising, the study currently exhibits several empirical, methodological, and contextual shortcomings.\n\nMajor:\n1. The manuscript contains several instances of imprecise descriptions and inaccurate data. In particular, the authors should more accurately clarify the challenges of composite-letter storage and carefully contextualize prior studies.\n(1) Lines 42-47: While the authors identify key manifestations of the composite-DNA readout challenge—namely, difficulty in discriminating similar base mixtures and obscuration of constituent components by sequencing noise—the discussion remains somewhat surface-level and does not sufficiently articulate the fundamental underlying causes. In fact, the elevated error rates in composite DNA storage arise from two interacting sources: (i) synthesis and sequencing errors, which perturb the intended base mixture proportions; and (ii) statistical sampling bias, especially under low coverage, which leads to fluctuations in observed base‑frequency distributions.\n(2) Lines 54–61: The current summary of prior work on composite-letter DNA storage is incomplete and may inadvertently mislead readers about the actual performance trade-offs.\n(i) Six-letter experiments (Σ₆). Anavy et al. (2019) demonstrated data recovery at 29× coverage using KL-divergence inference followed by Fountain and Reed–Solomon coding. More recently, Xu et al. (2024) showed that a soft-decision decoder (Derrick-cp) applied to the same six-letter alphabet reduced the practical coverage threshold to 17×. Including this update will provide a realistic comparison point.\n(ii) Large alphabets (256 letters). The early estimates of 2000× coverage from Anavy et al. (2019)’s simulations with a 256-letter composite alphabet stemmed from high-density aims (~8 bits/letter). Xu et al. (2024)’s Derrick-cp soft-decision decoding algorithm reduced requirements to 490×. Importantly, these elevated coverage needs are not signs of irrational design, but rather inherent consequences of pursuing high‑density information encoding. The key question is whether advanced strategies (KL-divergence, soft-decision, or the present set-partitioning) can offset this cost effectively.\n(3) Lines 80-82: The current Fig. 1f and Extended Data Table 1 inaccurately report Xu et al.’s metrics: Their six-letter performance derived from in vitro experiments following Anavy et al., not simulations; The value of 2.06 bits/cycle includes primer and index regions, while Extended Data Table 1 consistently omits these overheads. Excluding primers/indexes, the correct density is 2.39 bits/cycle. The revised methods and corresponding numerical values are listed in bellowed table. Therefore, in direct comparison with Xu et al.—considering both logical density and coverage—the performance of the current eight-letter system is comparable, but does not demonstrate notable advantages. I recommend updating Fig. 1f and Extended Data Table 1 to correct these values and revising the accompanying text to reflect that actual performance.\n2. The main motivation for composite alphabets is to reduce synthesis cycles, which dominate the costs in DNA Storage system. However, the current manuscript does not quantify this critical trade-off. A comparative cost analysis aligned with prior studies would clarify the practical benefits of the proposed scheme.\n3. Supplementary Tables 6–8 show that set-partitioning (SP) yields much lower error rates for standard bases (A/T/C/G) but higher error rates for composite letters (R, Y, M, K, etc.), which dominate the overall error budget, compared to KL-divergence (e.g. at 15×, error ≈4.8% for composites under SP vs 4.2% under KL). The authors should conduct end-to-end decoding performance comparison, using the same outer error-correction scheme (e.g., Reed-Solomon or LDPC), with detection via either set-partitioning or KL-divergence. This will clarify real gains (if any) in net data recovery. Furthermore, Xu et al. (2024) demonstrated that a MAP-based decoder can achieve >99% letter accuracy at 19× coverage—and still 98.3% accuracy even at 15×—clearly outperforming SP (demonstrated in Fig 5d and Supplementary Tables 6–8). The authors should therefore include MAP in their performance comparison to elucidate SP’s relative strengths and limitations.\n4. While the conceptual use of set‑partitioning is sound, its current entropy‑based implementation appears suboptimal for composite letters. The results suggest the proposed method reduces within‑subset confusion at the expense of mis‑grouping between subsets. The authors appear to have directly ported the telecommunications partitioning approach without adaptation to composite letter characteristics. I suggest to develop a more tailored partitioning approach to accurately separate subsets.\n5. The manuscript reports discarding ~23% of reads during length filtering (Fig. 5g), a shocking large fraction for Illumina sequencing (which typically has very low indel rates). The authors should explain the source of these indels and verify that filtering did not bias the results. Additionally, it is unclear whether the stated coverage thresholds (e.g. 18× for error-free recovery and in Fig. 5e-f) refer to pre- or post-filter read counts. Since coverage directly translates to DNA-storage readout cost, it should be defined on the basis of raw sequencing depth rather than the subset surviving filtering.\n6. The manuscript lacks critical detail regarding the synthesis methodology for composite letters, which is a key practical challenge in composite DNA storage. In particular, it is not clear whether each oligo encoding a composite letter is individually designed (deterministic assignment of A/T/C/G at each oligo) or whether the synthesis process uses probabilistic base mixing at degenerate positions, resulting in random incorporation across molecules. This distinction impacts scalability, and decoding accuracy. Also, discuss scalability implications for large-scale oligo pools.\nMinor reversions\nLine 141: The Galois field definition for NB‑LDPC code appears incorrect—please correct the GF notation."}, "author_response": "We sincerely thank both reviewers for their insightful comments on our manuscript and greatly appreciate their constructive remarks and suggestions. We have revised the manuscript accordingly based on their valuable feedback and provide detailed point-by-point responses below. All changes in the revised manuscript are highlighted in yellow.\nResponse to the Reviewers #1\nThis work discussed an interesting approach, named DNA diamond, which a composite letter constellation model for encoding in DNA data storage. This model expands the letters available for DNA data storage from 4 (simply natural bases A, T, C, G) to 15, considering the probability distribution of natural bases. Compared with other composite letter approaches, this method achieved the highest bit-to-letter coding efficiency with a relatively low sequencing coverage required to fully recover the data. The authors also developed methods to enable accurate and lossless recovery of data, including set partitioning, maximum likelihood estimation (MLE), appending double-end indices and perform length filtering.\nThe concept of DNA diamond is great. However, in practice, one of the key challenges of DNA data storage is the synthesis cost, which is measured by how many bits are encoded in one nucleotide. Although this work did a great job in improving the letter efficiency, I am not able to find any elaborations on the actual coding potential of this coding strategy. How many nucleotides, in total, are synthesized to encode each experimental (7,560-bit and 18,900-bit data) and simulation (780,713-byte image)? In my understanding, as shown in Supplementary Figure 2, N strands of 60-nt DNA only yielded 60 letters. If 1 letter can encode 3 bits in their wet experiments, then these strands just encoded 180 bits of data. I suggest the authors to calculate and clearly state the coding potential (not considering the indices, PCR primers and error correction codes) and the net information density (all synthesized nucleotides considered) using DNA diamond, with a unit of “bits/nt”.\nResponse: Thank you for your insightful comments. We are sorry for this confusion.\nWe provide a detailed clarification below.\n(1) Stored data volume and encoded strands (Supplementary Table 4)\nIn our prior small-scale experiments (before this revision), 7,560 bits and 18,900 bits were encoded into 7,560 composite letters, which were partitioned into short fragments of 60 letters, yielding a total of 126 composite strands (Supplementary Fig. 2). We compared different eight-letter alphabets using four experiments. Therefore, 30,240 composite letters were synthesized and tested in all four experiments.\nDuring manuscript revision, we further added validation on large-scale oligonucleotide pools. In our new experiments, two image files of 187,500 bytes and 234,375 bytes were encoded into 600,000 composite letters, which were then partitioned into 10,000 strands of 60 letters (Supplementary Fig. 3). We tested two different index schemes. Therefore, 2,400,000 composite letters were synthesized and tested during manuscript revision, with 1,200,000 using an eight-letter alphabet and 1,200,000 using a 15-letter alphabet.\nThe comparison with existing works can be found in the updated Supplementary Table 1. A summary can be found in Table R1 below.\nTable R1. Data volume in ‘wet’ experiments (practical synthesis & sequencing)\n(2) Workflow of composite DNA storage (How many nucleotides, in total, are synthesized?)\nIn order to clarify the data volume and synthesized nucleotides, we illustrate the basic workflow of our scheme with an eight-letter composite DNA storage process as an example (Supplementary Figs. 4 and 5).\nIn our scheme, digital data were first converted into binary bits. Every three bits were then mapped to a single composite DNA letter. For example, in the mapping table, ‘000’ corresponds to ‘A’, ‘001’ to ‘T’, ‘010’ to ‘G’, etc. Using this rule, 22,680 bits of data were converted into 7,560 composite letters. Next, these letters were divided into 126 strands, each strand carrying 60 composite letters (Supplementary Fig. 5).\nDuring synthesis, each composite letter was realized as a programmed mixture of the four natural bases (A, T, G, C). For instance, the composite letter ‘Y’ represents a 50% mixture of C and T, while ‘M’ represents a 50% mixture of A and C. At each synthesis cycle, a large number of ordinary DNA strands (Supplementary Table 3) were added a base in parallel within a single CPG column or a single synthesis well (Supplementary Fig. 4). Each CPG column or synthesis well was corresponded to a single composite strand. For column-based synthesis, we used 126 columns in each experiment. We performed four experiments. For array-based inkjet printing synthesis, we used 10,000 wells to generate 10,000 composite strands. We also performed four experiments. Two used Alphabet 1 (8 letters), and the other two used Alphabet 4 (15 letters).\nAs shown in Supplementary Fig. 8 (Supplementary Fig. 2 in previous version), during data readout, many noisy ordinary reads are collected for each composite strand. A total number of n sequencing reads corresponding to a unique index were used to reconstruct the original composite strand.\nTherefore, when we synthesis only a single composite strand, we practically synthesized a very large number of ordinary DNA strands (real oligos). When we read out a single composite strand, we require to read out n ordinary DNA strands (real oligos) to get n sequencing reads for recovery of this specific composite strand.\n(3) Theoretical coding potential and the net information density\nIn four-base DNA storage systems, information density is usually expressed in bits per nucleotide (bits/nt). In composite-letter systems, each synthesis cycle produces one composite letter that represents a mixture of multiple nucleotides. Therefore, bits/letter may be more proper to describe the information density of composite alphabetsR1, R2.\n For the eight-letter system, the theoretical coding potential is log (8) = 3 bits/letter. In our column-based synthesis experiment, 18,900 bits were stored in 7,560 letters, giving a payload logical density of 2.5 bits/letter (using RS(1890, 1575) code), excluding indices and primers. The logical density including indices and primers is 1.29 bits/letter. In our array-based DNA synthesis experiments, 187,500 bytes were stored in 600,000 composite letters. The payload logical density is 2.5 bits/letter. The logical density including indices and primers is 1.21 bits/letter (strand length: 124 letters) and 1.34 bits/letter (strand length: 112 letters), respectively.\n For the 15-letter system, the theoretical coding potential increases to log (15) ≈ 3.9 bits/letter. We achieved a density of 3.125 bits/letter in the payload segment (due to ECC), excluding indices and primers. The logical density including indices and primers is 1.51 bits/letter (strand length: 124 letters) and 1.67 bits/letter (strand length: 112 letters).\nDetails are presented in Supplementary Table 4.\nReferences\n[R1]. Anavy, L., Vaknin, I., Atar, O., Amit, R. & Yakhini, Z. Data storage in DNA with fewer synthesis cycles using composite DNA letters. Nat. Biotechnol. 37, 1229–1236 (2019).\n[R2]. Xu, Y., Ding, L., Wu, S. & Ruan, J. Overcoming the high error rate of composite DNA letters-based digital storage through soft-decision decoding. Adv. Sci. 11, 2402951 (2024).\nWe revised main manuscript and supplementary information as follows.\nLines 83–86:\nTo verify the proposed scheme, we designed error correction codes tailored to different alphabets and conducted experiments on both column-based and array-based synthesis platforms, along with simulation tests (Supplementary Figs. 2, 3 and 4).\nLines 152–166:\nThen, we validated the applicability of discrete-entropy composite letters by testing multiple storage schemes across different codes and alphabets (Fig. 2c, Supplementary Figs. 10 and 11, Supplementary Table 4). The encoding and writing process of the composite DNA storage system included three stages: data encoding, composite letter mapping, and DNA synthesis.\nFor data encoding, in the eight-letter system, 945 bytes of text were first encoded using NB-LDPC(3780,1260) code (R=0.33) over Galois Field (GF(64)), with interleaving to mitigate burst erasures39 (Supplementary Fig. 12). To validate high logical density, 2,355 bytes of text data were encoded with RS(1890,1575) code (R=0.83) over GF(212), achieving 2.5 bits/letter (Supplementary Fig. 13). We further extended the eight-letter scheme to large-scale pools, where a 187,500-byte image file was encoded using RS(3750,3125) code (Supplementary Fig. 14). In the 15-letter system, both experiments and simulations were performed. A 234,375-byte image file was encoded with RS(30000,25000) code (R=0.83, 3.125 bits/letter) for experimental verification (Supplementary Fig. 15). A 780,713-byte image file was encoded with RS(32767,27767) code (R=0.85, 3.18 bits/letter) for simulations (Supplementary Note 2, Supplementary Fig. 16).\nIn Supplementary information:\n(1) Supplementary Table 4 is added to provide detailed information on the amount of stored data, the number of synthesized composite letters, and the information density.\n(2) Supplementary Note 2 is added to provide the logical density computation of the composite letter DNA storage.\n(3) Supplementary Figs. 2–5 are added to illustrate our experimental and simulation setups as well as the workflow of composite letter DNA storage.\nSupplementary Table 4. Coding potential and the net information density.\nScheme Column-based DNA synthesis experiments Array-based DNA synthesis experiments Simulation\nData volume (bytes) 945 2,355 187,500 234,375 780,713\nBit information (bits) 7,560 18,900 1,500,000 1,875,000 6,247,575\nPayload 126*60 126*60 10000*60 10000*60 32767*60\n(M*L) =7560 letters =7560 letters =600,000 letters =600,000 letters =1,966,020 letters\nAlphabet ∑ ∑ ∑ ∑ ∑\n8 8 8 15 15\nCoding potential\nlog (8)=3 log (8)=3 log (8)=3 log (15)=3.9 log (15)=3.9\n2 2 2 2 2\n(bits/letter)\nCoding scheme NB-LDPC(3780, 1260) RS(1890, 1575) RS(3750, 3125) RS(30000, 25000) RS(32767, 27767)\nCode rate R=1/3 R=5/6 R=5/6 R=5/6 R=17/20\nLogical density\n(payload) 1 bit/letter 2.5 bits/letter 2.5 bits/letter 3.125 bits/letter 3.18 bits/letter\n(bits/letter)\nSynthesized nucleotides 126*116 126*116 10000*124 10000*112 10000*124 10000*112\n/\n(including primers/indices) =14,616 =14,616 =1,240,000 =1,120,000 =1,240,000 =1,120,000\nNet information density\n1.21 1.34 1.51 1.67\n(synthesized nucleotides) 0.52 bits/letter 1.29 bits/letter /\nbits/letter bits/letter bits/letter bits/letter\n(bits/letter)\nSupplementary Figure 2. Experimental verification using column-based synthesis and simulation. a, Four independent experiments were performed using column-based phosphoramidite chemistry. Two code rates, 1/3 and 5/6, were employed in combination with three distinct eight-letter alphabet schemes. In each experiment, 126 designed sequences with a payload of 60 composite letters and a total length of 116 nucleotides were synthesized. Each of the 126 sequences was individually synthesized in a separate column well. b, Computer simulation using the USC-SIPI image dataset (780,713 bytes). Data were encoded with an RS code (R=0.847), mapped to the 15-letter alphabet (15-bit to 4-letter) and formed 32,767 composite payload strands of 60 letters.\nSupplementary Figure 3. Experimental verification using array-based DNA synthesis. a, Two composite oligo pools based on the eight-letter alphabet were synthesized using low-cost array-based inkjet phosphoramidite chemistry. Each experiment encoded a 187,500-byte image file into 600,000 composite letters, segmented into 10,000 strands of 60 letters. After adding 20-nt primers and indices of 12 nt or 6 nt, the resulting oligonucleotides were 124 letters and 112 letters, respectively.\nb, Two composite oligo pools based on the 15-letter alphabet were synthesized. Each experiment encoded a 234,375-byte image file into 600,000 composite letters, segmented into 10,000 strands of 60 letters. All four pools were synthesized using array-based inkjet DNA synthesis.\nSupplementary Figure 4. Illustration of the composite strand synthesis with column-based and array-based synthesizers. a, Column-based synthesis of composite DNA strands. Eight reagent channels correspond to eight composite bases. Three composite alphabets were chosen and 126 composite strands were synthesized for each alphabet. b, Each designed composite strand, composed of composite letters (e.g., B, G, T, D, A, etc.), is individually synthesized in a dedicated well using column-based DNA synthesis. c, Array-based inkjet synthesis of large-scale composite-letter pools. Four printheads deliver monomer droplets onto array chips following predefined base mixing ratios.\nSupplementary Figure 5. Writing workflow of composite DNA data storage. First, the encoded bits are mapped to composite DNA letters (e.g., mapping 3 bits to 1 letter), where each composite letter represents a predefined mixture of nucleotides (e.g., A, T, G, C, R, Y, M, K). Second, 126 composite strands with a payload length of 60 letters are generated. Third, these sequences are synthesized in parallel using column-based phosphoramidite chemistry, in which different nucleotide mixtures are deposited in each synthesis cycle. Finally, ordinary DNA strands are produced, forming clusters that reflect the designed composite letters.\nExcept the major concern above, I also have other concerns in terms of the feasibility of this coding strategy in real settings:\n1. In practice, the existence of homopolymers (3 or more) of natural DNA bases is detrimental to sequencing-based data recovery. However, by mapping 1 letter to 3-bit of data, it is impossible to avoid the appearance of homopolymers in synthesized DNA strands. How would the authors overcome this issue?\nResponse: Thank you for raising this important point about the homopolymers.\nWe agree with the reviewer that it is impossible to avoid the appearance of homopolymers in synthesized DNA strands. To evaluate this concern, we performed a detailed analysis of the homopolymer length distribution with raw sequencing data.\nHomopolymers are present in both conventional and composite letter storage systems. We analyzed the proportion of homopolymers. The sequencing reads from our composite strands show nearly identical distributions to those of randomly generated DNA sequences (Supplementary Fig. 54).\nThe probability of long homopolymers decreases sharply as the homopolymer length increases. Therefore, we did not control the homopolymer length dedicatedly. We exploited the low probability to avoid long homopolymers. This method has also been verified in [R1].\nThen, we used two schemes to overcome the potential effects due to homopolymers.\n(1) Frequency-based composite letter detection alleviates the effects of homopolymers. Composite letter readout depends on measuring base frequency distributions at each synthesis site (Supplementary Fig. 55). Even when sequencing errors (e.g., indels due to homopolymers) occur within homopolymer regions, with multiple copies available, the model infers each composite letter from the observed base frequencies across all aligned reads. The effects of an error in a single sequencing read (including indels due to homopolymer) can be further reduced.\n(2) Length filtering. When indel errors occur within homopolymer regions and cause local shifts, misaligned bases can distort frequency observations and reduce detection accuracy. Our workflow filters out reads with abnormal lengths to further suppress indel noise (Supplementary Fig. 58). As detailed in Section 5 of the manuscript, length filtering improves per-letter detection accuracy and lowers the sequencing coverage required for error-free recovery (Fig. 5e, f).\nThe revised main text in the manuscript is as follows:\nLines 296–300:\nNotably, homopolymers are unavoidable in sequencing reads and long homopolymers may introduce more indels. Fortunately, long homopolymers occurred rarely. Homopolymer length distributions in composite systems are comparable to conventional four-letter storage and random sequences (Supplementary Figs. 54 and 55).\nIn the Supplementary information:\nSupplementary Figs. 54, 55, and 58 are added to show homopolymer occurrence in composite letter storage system and how frequency-based inference combined with length filtering reduces their impact on letter detection.\n[R1] Weindel, F., Gimpel, A. L., Grass, R. N. & Heckel, R. Embracing errors is more effective than avoiding them through constrained coding for DNA data storage. in 2023 59th Annual Allerton Conference on Communication, Control, and Computing (Allerton) 1–8 (2023). doi:10.1109/Allerton58177.2023.10313494.\nSupplementary Figure 54. Homopolymer length distribution. a, Sequencing reads from composite letter strands (measured in bases). The proportions of single-base homopolymers are 75.84% in the 8-letter system and 75.46% in the 15-letter system. b, Sequencing reads from a Fountain coding–based natural DNA system (measured in bases), with 75.90% single-base homopolymers. c, Randomly generated sequences. The proportion of single-base homopolymers is 75.39% in the natural-DNA system (measured in bases).\nSupplementary Figure 55. Frequency-based inference of composite letters. a, Multiple sequencing reads with the same indices are aligned, and the observed base frequencies at each synthesis site are used to infer the most probable composite letter. b, Indel, Indel errors in homopolymer regions cause subsequent bases to misalign. c, Composite letters are detected based on frequency distributions from multiple copies, which offset the effect of local errors.\nSupplementary Figure 58. Readout workflow with length filtering. Sequencing reads are first aligned to double-end primers to identify payload and index segments (Primer alignment). Reads with abnormal length are discarded, ensuring equal-length, aligned sequences for downstream readout (Length filtering). Reads are then grouped into different clusters according to the index sequences (Index identification). Within each cluster, base frequency distributions at each position are calculated across multiple copies, and the most probable composite letter is inferred (Letter detection).\n2. In their simulation experiment, the mapping between 15 binary bits (2^15 = 32,768) and 4 letters (15^4 = 50625) left so many combinations of letters not used in encoding and therefore is not efficient. This origins from the fact that this encoding strategy is very sensitive to the total bits of information to be coded, and the coding efficiency is not always high for different data size. In addition, I noticed that for different sizes of data, a different size of block should be carefully selected (6-bit in wet experiment and 15-bit in simulation). Did the authors investigate the impact of the size of block on coding potential?\nResponse: Thank you for this insightful question.\nWhen mapping 15 bits to 4 letters of the 15-letter alphabet, not all possible letter combinations are used, which inevitably leads to some redundancy. The theoretical density of the 15-letter alphabet is log (15)≈3.9069, which is not an integer. Each letter can maximally store about 3.9069 bits.\nWhen grouping 4 letters together, the storage capacity is 4 × 3.9069 ≈ 15.6 bits, slightly higher than 15 bits and lower than 16 bits. Therefore, if we choose 15 bits as a group, it can be represented by 4 letters, with an efficiency of 15/15.6 ≈ 0.96.\nIf 16 bits are chosen as one block, 5 letters are required, and 5 × 3.9069 ≈ 19.5 bits, which is greater than 16 bits. However, 216 = 65,536, while the total number of combinations of 5 letters is 155 = 759,375, leading to lower efficiency.\nTherefore, in our storage system, mapping efficiency is used as a key metric to evaluate encoding efficiency. The detailed calculation is as follows.\n(1) Mapping efficiency of 15-letter and eight-letter systems\nFor the 15-letter alphabet, the theoretical density is log (15)≈3.9069 bits per letter. When the block size is set to 15, the required number of letters is 15bits/log (15)≈3.84, rounded up to 4 letters. The mapping efficiency is defined as the ratio of stored binary bits to the maximum capacity of the letters used. Therefore, the efficiency of 15 binary bits is 15/(4*log (15))≈0.96 (Supplementary Fig. 17).\nFor the eight-letter alphabet, any 3 bits can be directly mapped to one letter. Given log (8)=3 bits/letter, the mapping efficiency is 100%. In our current composite storage system, limited by synthesis feasibility, we validated both the 15-letter and eight-letter, as these represent the practical design choices.\n(2) Relationship between mapping efficiency and bit group size\nIn our design, we presented a proper and high-efficiency scheme. We are sorry that we did not present the detailed explanation on this point. In this revised version, we have added the details on how to choose the conversion block size.\nThe mapping efficiency depends on both the chosen block size and the alphabet size. We derived a formula to calculate mapping efficiency as a function of block size, which can be expressed as:\nlog (2)×n\nη= 2 ,\n(log (k)×n/log (k))\n2 2\nwhere n is the size of the binary data block (in bits), k is the size of the composite alphabet (e.g., k =8 or k =15), and  denotes the ceiling function to calculate the number of composite letters required to represent n bits of data. The efficiency fluctuates with block size (Supplementary Fig. 18). When the block size is set to 2N −1 (e.g., n=31), the efficiency is relatively high (≈0.99), but each block requires multiple composite letters (e.g., it is eight for n = 31). In this case, a single error in one letter would corrupt many bits in the 31-bit block, causing error propagation across all the bit group (e.g., n=31). To balance efficiency and error control, we selected a block size of 15 bits (n=15 ), which achieves an efficiency of 0.96 while limiting error propagation to within 15 bits.\nFurthermore, we chose n=15, also because it is quite compatible with our RS codes defined in GF(215). For our RS code, a single RS symbol also contains 15 bits.\nWe added Supplementary Figs. 17 and 18 to illustrate the mapping process of the bit-to-letter. We also added Supplementary Note 3 to further clarify the mapping efficiency between binary bits and composite letters (please find in the supplementary information).\nThe revised text has been added in the manuscript:\nLines 167–173:\nFor composite letter mapping, we chose four composite letter alphabets (Fig. 2d) and analyzed the impact of different bit group sizes on mapping efficiency (Supplementary Figs. 17 and 18). The eight-letter alphabet was implemented by converting every three bits into a single composite letter. The mapping efficiency is 100%. For the 15-letter alphabet, a group of 15 bits was converted into four composite letters, corresponding to an optimized efficiency of 96%. We chose the group size of 15 for satisfactory efficiency and error propagation within the group. Details can be found in Supplementary Note 3.\nIn the Supplementary information:\n(1) Supplementary Figs. 17, and 18 are added to illustrate the mapping process of the bit-to-letter.\n(2) Supplementary Note 3 is added to provide a textual explanation of the mapping efficiency between binary bits and composite letters.\nSupplementary Figure 17. Mapping efficiency of bits to composite letters. Binary blocks of n bits are mapped to composite letters from an alphabet of size k, requiring n/log (k) letters. Examples show that for an eight-letter alphabet, 3 bits can be perfectly represented by a single composite letter, achieving a mapping efficiency of 100%. In contrast, for a 15-letter alphabet, 15 bits require 4 letters, and the mapping efficiency is 96%.\nSupplementary Figure 18. Bit-to-alphabet mapping efficiency as a function of bit group size. a, Mapping efficiency from binary bits to the eight-letter alphabets. b, Mapping efficiency from binary bits to 15-letter alphabets. Mapping efficiency is defined as the ratio between the number of input bits in each block and the amount of information that can be represented using a fixed number of multi-valued symbols. The red stars indicate the configurations used in our study. Every three bits are mapped to a single octal symbol in the “wet” experiment, and a 15-bit group is mapped to four base-15 symbols. For the 15-letter system, this mapping achieves a mapping efficiency of 96%. c, Example of letter mapping and de-mapping in the 15-letter composite encoding scheme. During sequencing or synthesis, a letter error may occur (for example, “H” is misread as “V”), leading to an incorrect symbol after de-mapping. Each 15-bit symbol is protected by Reed–Solomon coding (RS(30000, 25000) over GF(215)), enabling correction of such symbol-level errors during decoding.\n3. I am not clear on how the authors can accurately control the read count of each DNA strand through sequencing. I assume this was achieved through adjustment of copy numbers of each strand. However, this significantly complicates experimental procedure and is apparently not practical in commercial DNA data storage settings. In addition, the efficiency of PCR is different for different amplicon sequences (the payload in this study). For example, the amplification may be difficult to proceed when there are homopolymers in the amplicon. When this encoding strategy scale up to 10,000 or more strands in one oligo pool (which is very typical in real DNA data storage applications), can the probability distribution of each natural DNA base still be guaranteed?\nResponse: We thank the reviewer for this important question. In our study, we did not attempt to control the read count of each DNA strand during sequencing. Instead, we relied on pooled DNA manipulation and statistical sampling.\nSpecifically, we performed downsampling of raw sequencing reads on pooled all DNA strands and evaluated the recovery performance at different average coverages.\nTo avoid ambiguity, we revised all mentions of “coverage” in the manuscript to “average coverage”.\n(1) Experimental validation of small-scale oligo pools (126 composite oligos).\nIn Part 4 of the Results, we analyzed strand copy number variation across different average coverages, with a particular focus on low-coverage conditions.\nAll composite oligos were pooled together for amplification and sequencing.\nTherefore, we did not control the copies of each composite strand. Actually, we could not control the copies. Due to stochastic sampling, bias in PCR amplification efficiency and sequencing across DNA strands, the observed read counts were found to follow a negative binomial distribution (Supplementary Figs. 29 and 30). The probability distribution of natural bases at composite positions varies across strands and is subject to random sampling noise as well as synthesis, amplification, and sequencing stochasticity. At lower sequencing coverages, this manifests more clearly in the form of dropout (i.e., erasures) or uneven base frequencies (Figure 4a and 4b). Despite this, most composite letters maintain the desired probability distribution of each natural DNA base (Supplementary Figs. 19–21) and also maintain separable entropy characteristic (Fig. 4a, b).\nFurthermore, our decoding strategy incorporates both erasure correction and substitution recovery, enabling accurate data readout even under low-coverage conditions (Supplementary Figs. 42 and 43).\n(2) Experimental validation of large-scale oligo pools (10,000 composite oligos).\nIn response to the reviewer’s suggestion, we further validated our encoding scheme in large-scale oligo pools, considering that large pools show significant bias.\nSpecifically, we designed four oligo pools. Each contained 10,000 composite strands based on the eight-letter or 15-letter alphabets (Supplementary Figs. 14 and 15). The pools were synthesized using a high-throughput inkjet-based oligonucleotide synthesizer and then sequenced with NGS. We evaluated read count distributions, letter detection error rates, and recovery performance under different average coverages. The results showed that even in pools containing 10,000 oligonucleotides, the observed read counts followed a negative binomial distribution (Supplementary Figs. 31–33). When the composite-letter DNA storage system is scaled to a pool of 10,000 strands, the probability distribution for each composite letter remains valid and retain entropy discriminability (Supplementary Figs. 22–26).\nWith increasing coverage, composite letters can be reliably inferred, and the encoded data can be recovered (Supplementary Figs. 44–47).\nWe updated Part 4 of the Results. The revised manuscript is as follows:\nLines 211–217:\nThe coverage parameter in the model was obtained by fitting experimental copy number distributions at different average sequencing depths. For column-based synthesis pools, the observed read counts per strand followed a negative binomial distribution (Supplementary Figs. 29 and 30). Consistently, in array-based synthesis pools containing up to 10,000 oligonucleotides, the read count distributions also conformed to the negative binomial model (Supplementary Figs. 31–33). This reflects stochastic sampling and strand-specific differences in PCR amplification efficiency.\nIn the Supplementary information:\nSupplementary Figs. 25–28 are added to present read-count distributions under different average sequencing coverages, highlighting non-uniform strand representation and extending the analysis to 10,000-strand pools for different alphabets.\nSupplementary Figure 30. Read count distribution of composite DNA strands at different average coverages (126 composite strands). a, Distribution of sequencing read counts per composite strand synthesized by the column-based method (NB-LDPC(3780, 1260) code with an eight-letter alphabet). The left panel shows that the read count distributions at different coverages follow a negative binomial distribution. The parameter p denotes the fitted success probability of the negative binomial model. The right panel shows the read count distribution across 126 composite strands at 9× average coverage, where error-free data recovery was achieved. The read counts exhibited noticeable non-uniformity. b, Distribution of sequencing read counts from the composite pool encoded with the RS(1890, 1575) code using the eight-letter alphabet. The left panel shows the fitted negative binomial distributions at different coverages. The right panel presents the read-count distribution of individual composite strands (126 in total) at 14× average coverage, where error-free recovery was obtained at this coverage.\nSupplementary Figure 31. Read count distributions at different sequencing coverages (10,000 composite strands). a, Read count distribution of 10,000 composite strands with an eight-letter alphabet (array-based inkjet synthesis). The observed distributions at different average coverages follow negative binomial statistics. b, Distribution of read counts for 10,000 composite strands (124 letter length, 15-letter alphabet). In all cases, PCR amplification bias and random sequencing sampling produced negative binomial-like strand copy distributions, and the fitted parameter p quantifies the degree of non-uniformity across strands.\nSupplementary Figure 32. Read count distribution of 10,000 individual composite strands with a chosen eight-letter alphabet. Read count distribution of 10,000 composite strands obtained from the pool experiment using the eight-letter composite alphabet at an average sequencing coverage of 19×. Each bar represents the number of sequencing reads corresponding to a single composite strand.\nSupplementary Figure 33. Read count distribution of 10,000 individual composite strands with a chosen 15-letter alphabet. The read count distribution of 10,000 composite strands synthesized with the 15-letter composite alphabet at an average sequencing coverage of 33×. The copy number of each strand is non-uniform after sequencing.\nSupplementary Figure 22. Base frequency distribution (1,000×, 10,000 composite oligos, eight-letter alphabet). a–h, Distributions of four constituent bases (A, T, G, C) for each of the eight composite letters at 1,000× average coverage at each synthesis position. The results are derived from sequencing data of 10,000 composite strands encoded with an eight-letter alphabet {A, T, G, C, R, Y, M, K}.\nSupplementary Figure 23. Base frequency distribution (19×, 10,000 composite oligos, eight-letter alphabet). a–h, Distributions of four constituent bases (A, T, G, C) for each of the eight composite letters at 19× average coverage, representing the minimum coverage at which error-free data recovery was achieved. The results are derived from sequencing data of 10,000 composite strands encoded with the eight-letter alphabet {A, T, G, C, R, Y, M, K}.\nSupplementary Figure 24. Base frequency distribution (1,000×, 10,000 composite oligos, 15-letter alphabet). a–o, Distributions of four constituent bases (A, T, G, C) for each of the 15 composite letters at 1,000× average coverage. The results are derived from sequencing data of 10,000 composite strands encoded with the 15-letter alphabet {A, T, G, C, R, Y, S, W, M, K, H, B, V, D, N}.\nSupplementary Figure 25. Base frequency distribution (33×, 10,000 composite oligos, 15-letter alphabet). a–o, Distributions of four constituent bases (A, T, G, C) for each of the 15 composite letters at 33× average coverage, representing the minimum coverage at which error-free data recovery was achieved. The results are derived from raw sequencing data of 10,000 composite strands with the 15-letter alphabet {A, T, G, C, R, Y, S, W, M, K, H, B, V, D, N}.\nSupplementary Figure 26. Entropy distribution under practical synthesis and sequencing. a and b, Entropy distributions of the 15-letter alphabet at high coverage (1,000×) and the minimum coverage for error-free decoding (33×). c and d, Entropy distributions of the eight-letter alphabet at high coverage (1,000×) and the minimum coverage for error-free decoding (19×). Each curve represents the entropy distribution for individual composite letters under noisy sequencing conditions.\n4. The degradation of DNA is also a concern in practice, and the degradation rate of DNA strands with different sequences may also be non-homogeneous, which cause distortion of the assumed probability distribution of natural bases. I suggest the authors to perform accelerated aging experiments on their synthesized DNA pools (elevated temperatures, acidic and basic conditions, repeated freeze-thaw cycles) to investigate if this encoding strategy is less resistant to DNA degradations compared with conventional encoding strategies.\nResponse: We thank the reviewer for this valuable suggestion. To assess the stability and robustness of our composite DNA strands under degradation, we performed accelerated thermal aging experiments and analyzed the effect of degradation on both physical strand integrity and data retrieval accuracy. We added Supplementary Figs. 60–66.\n(1) Double indices enabling readout under DNA degradation (Supplementary Fig. 60)\nOur composite strand incorporates index and primer sequences at both ends of the payload. During sequencing, any read containing at least one intact primer–index pair can still be correctly clustered and aligned according to its positional index. When strand breakage occurs, the remaining fragments with a single-end primer are sufficient to reconstruct missing positions by aggregating multiple overlapping copies, allowing the original composite letters to be inferred. This design inherently mitigates the impact of physical strand degradation on data decoding.\n(2) Accelerated aging experiments.\nWe conducted gradient thermal degradation experiments on the small-scale pool, which consisting of 126 composite strands of 116 letters. Samples were incubated at 80 °C (2.5 h), 85 °C (2 h), 90 °C (2 h), and 95 °C (2 h), with an unheated control group. DNA breaks are induced frequently during DNA degradation, consequently leading to increased sequence dropout R1, R2.\n(3) Sequencing and degradation analysis.\nEach degraded sample was sequenced using the paired-end 150-bp mode on an NGS platform, and the sequencing reads were assembled from paired-end reads to reconstruct full-length fragments. The cumulative read-length distributions show that higher incubation temperatures lead to broader distributions, reflecting increased degradation (Supplementary Fig. 61). Quantitatively, the fraction of correctly assembled reads (116 nt) decreased with temperature, while truncated reads increased (Supplementary Figs. 62 and 63). Furthermore, primer-based classification of the reads showed that the proportion of reads containing both paired-end primers declined with increasing temperature. At 95 °C, approximately 14% of the reads carried only a single-end primer (Supplementary Fig. 64), which can be attributed to strand breakage.\n(4) Data recovery and error analysis\nWe evaluated data readout performance using the degraded sequencing datasets.\nWe added Supplementary Figs. 65 and 66 to illustrate the recovery performance. Although higher temperatures increased the inferred-letter errors, the overall recovery performance remained high (Supplementary Fig. 65). Specifically, the letter error rate rose moderately with temperature, but all degraded samples achieved complete data recovery. Furthermore, considering that the degradation was not severe, we performed a stress test using selected short degraded fragments. The original file was still recovered with a relatively high sequencing coverage (Supplementary Fig. 66).\nThese results demonstrate that our scheme is resilient to strand degradation and that increasing sequencing coverage effectively compensates for DNA damage.\n(5) On the repeated freeze-thaw cycles\nWe have also performed repeated freeze-thaw cycles on synthesized DNA. However, we did not observe obvious degradation during limited freeze-thaw cycles (Figure R1). Considering that most of the degradations can be modeled as strand breaks R1, R2, we used the gradient thermal degradation to verify our proposed scheme. Therefore, we expect the reviewer to allow us not to include these experiments in our modified manuscript.\nFigure R1. Analysis of degradation due to repeated freeze-thaw cycles. a and c, Cumulative read-length distributions of assembled reads from two small-scale composite pools (126 strands, 116 letters) subjected to ten freeze–thaw cycles. Each cycle consisted of freezing at –20 °C for 30 min and thawing at room temperature for 6 min. The read lengths remained concentrated around the full 116 nt. b and d, Proportions of reads containing paired-end primers, only 5’ primers, only 3’ primers, or invalid primer structures. Over 96% of reads retained both primer sequences, confirming that repeated freeze–thaw treatment did not lead to strand breaks.\nReferences\n[R1]. Song, L. et al. Robust data storage in DNA by de Bruijn graph-based de novo strand assembly. Nat. Commun. 13, 5361 (2022).\n[R2]. Gimpel, A.L., Stark, W.J., Heckel, R. et al. A digital twin for DNA data storage based on comprehensive quantification of errors and biases. Nat. Commun. 14, 6026 (2023).\nTherefore, we added a new paragraph at the end of Results, Part 5 describing the resilience of composite DNA strands under thermal degradation, as detailed below.\nLines 312–326:\nThen, the double-end strand structure enables robust data retrieval even when partial degradation occurs, for example, strand break. Any fragment retaining at least one intact primer–index pair can still be correctly clustered and aligned according to its index (Supplementary Fig. 60). To evaluate the performance on strand break, we performed accelerated thermal aging experiments. The composite pool containing 126 strands of 116 letters was incubated at 80 °C (2.5 h), 85 °C (2 h), 90 °C (2 h), and 95 °C (2 h), with an unheated sample as the control. Under higher temperatures, the read lengths showed a pronounced shift toward shorter fragments. Consistently, the proportion of correctly assembled 116 nt reads decreased, while truncated fragments accumulated (Supplementary Figs. 61–63). At 95 °C, nearly 14% of reads contained only a single-end primer (Supplementary Fig. 64). Despite these degradation, data recovery remained highly robust. Letter error rates increased moderately with degradation temperature. Error-free recovery was still achieved at sufficient sequencing coverage (Supplementary Fig. 65). Stress tests using short fragments revealed that the proposed scheme could recover the original error free, though the required sequencing coverage increased (Supplementary Fig. 66).\nIn the Supplementary information:\nSupplementary Figs. 60-66 are added to present the composite DNA architecture, read length under strand degradation, data recovery performance, and stress tests using degraded reads.\nSupplementary Figure 60. Design of composite DNA strand structure enabling degradation-tolerant decoding. a, Schematic of composite strand architecture with double-end primers and indices. Three sequencing scenarios are shown: (1) reads containing both paired-end primers, (2) reads containing only the 5’ primer, and (3) reads containing only the 3’ primer. b, Multiple sequencing reads sharing the same index are aggregated to compute position-wise observed frequencies and infer composite letters. Even if one end of the strand is degraded, the remaining indexed fragment contributes to composite letter detection.\nSupplementary Figure 61. Cumulative distributions of read length under different degradation conditions. a, Cumulative frequency curves of assembled reads for the pool encoded with RS(1890, 1575). Increasing temperature led to a gradual shift toward shorter read lengths and a broader cumulative distribution, indicating progressive degradation of DNA strands. b, Magnified view of the 60–110 nt region. The original full-length reads were 116 nt in design, and the detailed view highlights the emergence of shorter degraded fragments at higher temperatures (95 °C, 2 h).\nSupplementary Figure 62. Proportion of read length under different thermal degradation conditions (only payload). The y-axis is payload length (nt), and the x-axis is the proportion of reads when strands are sorted by payload length from long to short; i.e., for each proportion on the x-axis, the curve shows the corresponding payload length. Higher temperatures (95 °C, 2 h) shift the curves downward, indicating a larger fraction of shorter payloads.\nSupplementary Figure 63. Magnified view of proportion of read length under different thermal degradation conditions (only payload). Detailed view of the payload ratio range 0.9–1.0. The payload exhibited a gradual shortening trend at higher temperatures. Degradation becomes more pronounced at elevated temperatures, with partial fractures observed within the payload region.\nSupplementary Figure 64. Classification of sequencing reads according to primers under different thermal degradation conditions (126 composite strands, 116 nt). Proportions of reads containing paired-end primers, only 5’ primer, only 3’ primer, or invalid primers. The percentage of paired-end reads gradually decreases with increasing temperature, while single-end reads become more prevalent. At 95 °C, nearly 14% of reads exhibited fragmentation.\nSupplementary Figure 65. Data recovery performance of degraded samples (126 composite strands, 116 nt). a, Relationship between valid base coverage and average coverage. Valid base coverage is defined as the ratio of the number of bases used for composite-letter detection to the total number of encoded letters. b, Index identification accuracy. The payload regions identified were used for composite-letter detection. c, Letter error rate as a function of average coverage for each condition (mean ± s.d.). The non-heated control sample showed the lowest error rate. d, Erasure rate at different average coverages. e, Percentage of successful (error-free) recoveries across 100 independent trials using the RS(1890, 1575) coding scheme. f, Composition of valid payloads at 14× coverage by index identification, including double-end indices and those identified by only 5’-end or 3’-end indices.\nSupplementary Figure 66. Recovery performance using severely-degraded sequencing reads. a, Length distribution of payload regions identified based on double-end indices. Approximately 40% of the fragments showed degradation within the payload region. b, Proportion of degraded reads containing double-end indices (38.71%), only 5’-end index (27.17%), or only 3’-end index (34.12%). c, Length distributions of payload regions identified by single-end indices, where sequences shorter than 60 nt indicate degradation within the payload region. d and e, Data recovery performance based on degraded reads, showing letter error rate and successful recovery rate as functions of average base coverage. f, Distribution of 491,147 degraded sequencing reads extracted from the aging experiment at 95 °C for 2 h.\nResponse to the Reviewers #2\nThis manuscript introduces a “diamond” composite-letter alphabet for DNA storage and an entropy-guided, two-stage detection pipeline: composite letters are first partitioned into entropy-defined subsets and subsequently identified within subsets by maximum-likelihood inference. Wet-lab experiments on an eight-letter alphabet report achieving ~2.5 bits per letter and error‑free recovery at 18× Illumina coverage. Although these proof-of-concept results are promising, the study currently exhibits several empirical, methodological, and contextual shortcomings.\nResponse: We thank the reviewer for their detailed review of our manuscript. Below are our point-by-point responses and the modifications detailed in the manuscript. We expect the revised version should address the reviewer’s concerns.\n1. The manuscript contains several instances of imprecise descriptions and inaccurate data. In particular, the authors should more accurately clarify the challenges of composite-letter storage and carefully contextualize prior studies.\n(1) Lines 42-47: While the authors identify key manifestations of the composite-DNA readout challenge—namely, difficulty in discriminating similar base mixtures and obscuration of constituent components by sequencing noise—the discussion remains somewhat surface-level and does not sufficiently articulate the fundamental underlying causes. In fact, the elevated error rates in composite DNA storage arise from two interacting sources: (i) synthesis and sequencing errors, which perturb the intended base mixture proportions; and (ii) statistical sampling bias, especially under low coverage, which leads to fluctuations in observed base‑frequency distributions.\nResponse: We thank the reviewer for this comment. In Part 4 of the Results (“Performance evaluation on constituent base frequency variation at low average coverage”), we analyzed how statistical sampling bias under low sequencing coverage affected base-frequency distributions. We quite agree with the reviewer.\nWe revised the relevant text in the manuscript as follows.\nLines 41–50:\nDespite these advantages, composite DNA storage faces challenges during readout, particularly elevated error rates and the requirement of high sequencing coverage. These issues stem from two fundamental and interacting sources. First, synthesis and sequencing errors perturb the intended base mixture proportions. As the mixture complexity increases, this perturbation leads to higher inference error rates for composite letters at a given sequencing depth (Fig. 1c). Second, statistical sampling bias at low sequencing coverage leads to fluctuations in the observed base-frequency distributions, making it difficult to accurately infer the original composition. Together, these factors reduce the discriminability of similar base mixtures, thereby increasing error rates and necessitating higher coverage to ensure reliable readout (Fig. 1d).\n(2) Lines 54–61: The current summary of prior work on composite-letter DNA storage is incomplete and may inadvertently mislead readers about the actual performance trade-offs.\n(i) Six-letter experiments (Σ₆). Anavy et al. (2019) demonstrated data recovery at 29× coverage using Kullback–Leibler (KL) divergence inference followed by Fountain and Reed–Solomon coding. More recently, Xu et al. (2024) showed that a soft-decision decoder (Derrick-cp) applied to the same six-letter alphabet reduced the practical coverage threshold to 17×. Including this update will provide a realistic comparison point.\n(ii) Large alphabets (256 letters). The early estimates of 2000× coverage from Anavy et al. (2019)’s simulations with a 256-letter composite alphabet stemmed from high-density aims (~8 bits/letter). Xu et al. (2024)’s Derrick-cp soft-decision decoding algorithm reduced requirements to 490×. Importantly, these elevated coverage needs are not signs of irrational design, but rather inherent consequences of pursuing high‑density information encoding. The key question is whether advanced strategies (KL-divergence, soft-decision, or the present set-partitioning) can offset this cost effectively.\nResponse: We thank the reviewer for highlighting these important developments. In response, we revised the relevant text in the manuscript (Lines 57–65) to incorporate the recent results by Xu et al. (2024).\nWe also clarified the context of large-alphabet encoding schemes, emphasizing that the expanded composite alphabets (e.g., 256-letter systems) are the valid approach to achieving higher information density. Furthermore, we stressed the critical role of robust letter detection and decoding algorithms, particularly those incorporating soft information, in effectively reducing coverage requirements and lowering sequencing costs. According to the suggestion above, we revised main text in the manuscript as follows.\nLines 57–65:\nIn prior work with a six-letter composite alphabet (Σ ), Anavy et al.9 demonstrated data recovery at 29× sequencing coverage using Kullback–Leibler (KL) divergence inference, Fountain coding, and Reed–Solomon (RS) coding26–28. More recently, Xu et al.29 applied a soft-decision decoding algorithm (Derrick-cp) to the same alphabet, reducing the practical coverage threshold to 17×. For larger composite alphabets, such as a 256-letter scheme targeting ultra-high logical density (theoretically up to 8 bits/letter), simulations by Anavy et al. showed that approximately 2000× coverage would be required. The recent Derrick-cp algorithm reduced this requirement to 490×, highlighting the potential of advanced decoding methods29.\n(3) Lines 80-82: The current Fig. 1f and Extended Data Table 1 inaccurately report Xu et al.’s metrics: Their six-letter performance derived from in vitro experiments following Anavy et al., not simulations; The value of 2.06 bits/cycle includes primer and index regions, while Extended Data Table 1 consistently omits these overheads. Excluding primers/indexes, the correct density is 2.39 bits/cycle. The revised methods and corresponding numerical values are listed in bellowed table.\nTherefore, in direct comparison with Xu et al.—considering both logical density and coverage—the performance of the current eight-letter system is comparable, but does not demonstrate notable advantages. I recommend updating Fig. 1f and Extended Data Table 1 to correct these values and revising the accompanying text to reflect that actual performance.\nResponse: We sincerely thank the reviewer for the detailed and constructive comment. We apologize for the inaccuracy in our previous reporting of Xu et al.’s work.\nWe have updated both Fig. 1f and Supplementary Table 1 to reflect the accurate value for the six-letter system, and clarified that this metric is derived from in vitro experimental data. Correspondingly, we have revised the main text in the manuscript (Lines 89–101) to provide a more accurate and balanced comparison. During revision, we further conducted synthesis experiments with a 15-letter system, achieving a logical density of 3.125 bits per letter (excluding primers/indices) and error-free data recovery at 33× coverage.\nAccording to the suggestions above, we revised main text in the manuscript as follows. The modified figures and tables are also listed below.\nLines 89–101:\nCompared with the previous six-letter composite systems18,19, our eight-letter storage system demonstrated advantages in logical density and physical redundancy. In comparison with state-of-the-art soft-decision inference algorithms, our system achieved comparable performance in terms of density and sequencing coverage (Fig. 1f and Supplementary Table 1). Specifically, Xu et al.29 reported a logical density of 2.39 bits per letter (excluding primers/indices) at a coverage of 17× using a soft-decision decoder, while our system achieved 2.5 bits per letter at a similar coverage level. We further designed large-scale oligonucleotide pools containing 10,000 composite strands. Both the eight-letter and 15-letter alphabets were experimentally validated using array-based synthesis. For the 15-letter system, our composite letter detection method enabled error-free recovery at 33× sequencing coverage. The system achieved an information density of 3.125 bits per letter. Overall, the proposed 15-letter storage system demonstrated advantages in the number of composite letters, experimental synthesis scale, storage density, and coverage required for data recovery.\nIn the main manuscript, Fig. 1f is revised to update the comparison with prior work.\nIn the Supplementary Information, Supplementary Table 1 is revised to provide a detailed comparison, including the coding scheme, total data volume, alphabet size, logical density, and the sequencing coverage required for error-free recovery.\nFigure 1f. Sequencing coverage and logical density of the present work compared with published composite alphabets.\nSupplementary Table 1. Comparison of published composite letter data storage schemes.\nTotal Logical density (composite\nStudy Error correction Alphabet Data volume Average coverage\ncomposite letters letters, bits/letter)\nRS(45, 43)\n∑ 2.12 MB 7,830,000 2.16 29×\n+Fountain (1.1) 6\nRS(45, 43)\n∑ 6.42 MB 23,490,000 2.18 29×\n+Fountain (1.08) 6\nAnavy et\nRS(45, 43)\nal.1\n+Fountain (1.08)\n∑\n5\n6.42 MB 26,055,000 1.97 29×\nBinary Huffman ∑ 19.25 Bytes 42 3.67 100×\nExp. 15\nChoi et al.2\nRS ∑ 135 Bytes 499,833 2.16 250×\n∑ 945 Bytes*3 7,560*3 1 9×\nGF(26) 8\nRS(1890, 1575) over GF(212) ∑ 2,355 Bytes 7,560 2.5 14×\nRS(3750, 3125) over GF(212) ∑ 187,500 Bytes*2 600,000*2 2.5 16×\nAnavy 256\net al.1\nRS(68, 65) ∑ 2.12 MB 3,354,480 5.05 500×\nXu\nRS(45, 41) over GF(27) ∑ 20.8 MB 27,776,250 6.37 300×\net al.6 128\nRS(45, 41) over GF(28) ∑ 20.8 MB 24,273,000 7.28 490×\n15\n2. The main motivation for composite alphabets is to reduce synthesis cycles, which dominate the costs in DNA Storage system. However, the current manuscript does not quantify this critical trade-off. A comparative cost analysis aligned with prior studies would clarify the practical benefits of the proposed scheme.\nResponse: We thank the reviewer for this valuable suggestion.\nThe costs in the DNA storage system associated with our eight-letter and 15-letter schemes were estimated by quantifying total synthesis and sequencing cost. We conducted the comparative cost analysis aligned with prior studies from two perspectives.\n(1) Comparison of the total synthesis and sequencing costs. Synthesis costs are primarily determined by logical density, while sequencing cost depends on the required coverage for successful decoding.\nWe estimated the number of synthesized nucleotides needed to store 1 MB of data using the reported logical density of each method R1. Sequencing cost was calculated as the product of coverage and nucleotide count R2. For consistency, we assumed a sequencing cost of $0.0000012/100 nt and a synthesis cost of $0.05/100 nt. To account for index and amplification overhead, we considered both “payload-only” and “all synthesized nucleotides” cases (Supplementary Fig. 68). Compared with the standard four-base design of Organick et al.R3, our eight- and 15-letter designs reduce synthesis cost by 56% and 64%, respectively. Compared with recent six-letter systems, the eight-letter design lowers synthesis cost by 13% (Anavy et al.R4) and 4% (Xu et al.R5), and the 15-letter design by 30% and 23%, respectively. Even after accounting for indexing and primer overheads, our design remains more cost-effective than the traditional four-base approach, with a slightly higher cost resulting from the use of a more robust double-end indices scheme. Detailed calculations are provided in Supplementary Note 6.\n(2) Normalized cost under different synthesis-to-sequencing cost ratios. Under our cost model, the synthesis-to-sequencing cost ratio (C :C ) can be as high as 40,000:1, though this ratio is expected to decline as synthesis technology advances. We further normalized total costs under different synthesis-to-sequencing cost ratios (C :C ), using the standard four-base scheme of Organick et al.R3. (requiring 5× sequencing depth) as the baseline. Representative ratios of 1000:1 and 500:1 were evaluated, and the costs of different schemes were scaled relative to the baseline work (Supplementary Table 10). Across all prior studies, composite-letter systems outperform the conventional four-base system. When considering only the composite-letter payload, the total cost of our system was reduced by 56% (eight-letter) and 64% (15-letter) at a 1000:1 ratio; at 500:1, the reductions remained 55% and 63%, respectively. Details are included in Supplementary Table 10.\nIn the main manuscript, Part 6 of the Results (“Cost analysis on composite letter DNA storage at low coverage”) is added to describe the synthesis of our composite letters and to provide an analysis of the associated costs.\nIn the Supplementary information:\n(1) Supplementary Table 10 is added to present a detailed comparison of normalized costs.\n(2) Supplementary Fig. 68 is added to illustrate the overall cost, including synthesis and sequencing, as well as a cost analysis reflecting the “write once, read many” characteristic of DNA storage.\n(3) Supplementary Note 6, titled “Cost analysis on composite letter DNA storage at low coverage” is added to provide a detailed description of the cost calculation methods.\nReferences\n[R1]. Erlich, Y. & Zielinski, D. DNA Fountain enables a robust and efficient storage architecture. Science 355, 950–954 (2017).\n[R2]. Wetterstrand KA. DNA sequencing costs: data from the NHGRI genome sequencing program (GSP). Available at: www.genome.gov/sequencingcostsdata (2019).\n[R3]. Organick, L. et al. Random access in large-scale DNA data storage. Nat. Biotechnol. 36, 242–248 (2018).\n[R4]. Anavy, L., Vaknin, I., Atar, O., Amit, R. & Yakhini, Z. Data storage in DNA with fewer synthesis cycles using composite DNA letters. Nat. Biotechnol. 37, 1229–1236 (2019).\n[R5]. Xu, Y., Ding, L., Wu, S. & Ruan, J. Overcoming the high error rate of composite DNA letters-based digital storage through soft-decision decoding. Adv. Sci. 11, 2402951 (2024).\nSupplementary Figure 68. Cost analysis of different DNA storage schemes.\nEstimated costs for writing (left) and reading (right) 1 MB of data, assuming DNA synthesis and sequencing costs of $0.05/100 nt and $0.0000012/100 nt, respectively.\nTwo conditions are considered: payload sequences (light bars) and all synthesized nucleotides including primers and indices (dark bars).\nSupplementary Table 10. Comparative cost analysis aligned with prior studies.\nLogical density Normalized cost\n(bits/nt) C :C (500:1) C :C (1000:1)\nsyn seq syn seq\nStudy Alphabet Cov.\nIncluding Including Including\nPayload Payload Payload\nprimers + indices primers + indices primers + indices\nOrganick et al.15 ∑ 5× 1.10 0.81 1.00 1.00 1.00 1.00\n4\n∑ 29× 1.97 1.37 0.58 0.62 0.57 0.61\n5\n∑ 29× 2.18 1.52 0.53 0.56 0.52 0.55\n∑ 100× 3.67 1.56 0.36 0.62 0.33 0.57\n15\n∑ 100× 4.29 1.82 0.30 0.53 0.28 0.49\n20\n∑ 250× 2.16 1.50 0.76 0.80 0.63 0.67\n∑ 250× 3.61 1.79 0.45 0.67 0.38 0.56\n6\n∑ 14× 2.50 1.29 0.45 0.64 0.44 0.63\n∑ 35× 3.125 1.67 0.37 0.51 0.36 0.50\n15\n3. Supplementary Tables 6–8 show that set-partitioning (SP) yields much lower error rates for standard bases (A/T/C/G) but higher error rates for composite letters (R, Y, M, K, etc.), which dominate the overall error budget, compared to KL-divergence (e.g. at 15×, error ≈4.8% for composites under SP vs 4.2% under KL). The authors should conduct end-to-end decoding performance comparison, using the same outer error-correction scheme (e.g., Reed-Solomon or LDPC), with detection via either set-partitioning or KL-divergence. This will clarify real gains (if any) in net data recovery. Furthermore, Xu et al. (2024) demonstrated that a MAP-based decoder can achieve >99% letter accuracy at 19× coverage—and still 98.3% accuracy even at 15×—clearly outperforming SP (demonstrated in Fig 5d and Supplementary Tables 6–8). The authors should therefore include MAP in their performance comparison to elucidate SP’s relative strengths and limitations.\nResponse: We thank the reviewer for the constructive comments. In response, we have incorporated the state-of-the-art maximum a posteriori (MAP) detection scheme into the performance analysis. We conducted end-to-end decoding comparisons of SP, KL and MAP, using the same outer error-correction codes.\n(1) End-to-end validation with practical raw sequencing data.\nBased on synthesis and sequencing experiments, we performed six sets of experimental comparisons, including two eight-letter designs with 126 composite strands each and four large-scale designs (two eight-letter and two 15-letter systems with 10,000 composite strands each). For SP, KL and MAP methods, using raw sequencing data, we evaluated both letter detection accuracy and end-to-end decoding performance under different average coverages (Supplementary Figs. 42–46).\n(2) Letter detection accuracy and required coverage for error-free recovery\nAt low coverage, sampling bias leads to insufficient strand copies for some sequences. To mitigate this, we applied a copy threshold during data readout: sequences with fewer than four copies were treated as erasures and corrected by the outer code, thereby improved overall detection accuracy.\nIn the two small-scale pool experiments with eight-letter alphabets, one composite pool was encoded using a high-reliability LDPC code and the other using a high-rate RS code. In both experiments, the MAP method achieved 99% detection accuracy at 12× and 13× sequencing coverage, respectively, while our SP method reached approximately 98.9% accuracy at 13× coverage (Supplementary Table 7). When using the same RS code with a coding rate of 0.83, MAP achieved error-free recovery at 12× coverage, whereas SP and KL required 14× coverage (Supplementary Fig. 42). In contrast, when using a low-rate LDPC code (R=1/3), erasures became the dominant factor affecting recovery at low sequencing depths. Although SP and KL exhibited slightly lower detection accuracy than MAP, all three schemes achieved error-free recovery at 9× coverage (Supplementary Fig. 43).\nIn the two large-scale pool experiments with eight-letter alphabets, each pool contained 10,000 composite strands synthesized via inkjet chemistry. When combined with an outer RS code (R=0.83), the results were consistent with the small-scale validation: the MAP algorithm achieved the highest detection accuracy and therefore required the lowest sequencing coverage (18× and 15× for the two pools), followed by SP (19× and 16×, respectively) (Supplementary Figs. 44 and 45).\nFor the two 15-letter systems, MAP remained the most accurate method, reaching >98% detection accuracy at approximately 26× coverage and achieving error-free recovery at 26× and 27× coverage (Supplementary Figs. 46 and 47; Supplementary Table 8). In contrast, SP suffered from partitioning errors near set boundaries, leading to slightly reduced accuracy (98%) and requiring 33× and 35× coverage for error-free recovery. The KL approach was more affected, requiring approximately 42× and 48× coverage. Furthermore, we established a simulation model to evaluate recovery performance for 32,767 composite strands under a simulated base error rate of 0.3%. Consistently, MAP required the lowest sequencing coverage, followed by SP and KL (Fig. 3d, e).\n(3) Advantages and limitations of the proposed SP scheme\nUnlike MAP, which directly compares posterior probabilities across all candidate letters, our set-partitioning (SP) approach reduces complexity by first grouping letters and then applying maximum likelihood estimation within each subset. This substantially narrows the candidate space and lowers computational cost. Across all six experiments, SP consistently demonstrated lower time complexity compared with MAP and KL, confirming its computational efficiency advantage (Supplementary Figs. 42–46).\nWe acknowledge that with larger alphabets, SP exhibited reduced detection accuracy relative to MAP due to boundary effects in set partitioning. In this contribution, we limited to 15-letter schemes.\nWe revised the manuscript as follows.\n(3) Advantages and limitations of the proposed SP scheme\nUnlike MAP, which directly compares posterior probabilities across all candidate letters, our set-partitioning (SP) approach reduces complexity by first grouping letters and then applying maximum likelihood estimation within each subset. This substantially narrows the candidate space and lowers computational cost. Across all six experiments, SP consistently demonstrated lower time complexity compared with MAP and KL, confirming its computational efficiency advantage.\nWe acknowledge that with larger alphabets, SP may exhibit reduced detection accuracy relative to MAP due to boundary effects in set partitioning. In this contribution, we limited to 15-letter schemes.\nThe revised manuscript as follows.\nLines 226–236:\nFinally, we compared the end-to-end recovery performance of our set partitioning (SP) method with the KL method19 and the maximum a posteriori probability (MAP) method29 with computer simulations (Fig. 3d, e). MAP performs global inference and shows the highest detection accuracy but the longest runtime. SP reduces the candidate space through partitioning, resulting in a slight degradation in accuracy, also a reduction in computational complexity. Both SP and MAP methods achieved lower error rates than the KL method. With 1,000 independent trials using 16 threads on the same server (Intel Xeon Gold 5220R CPU@2.20 GHz and 256 GB RAM), SP consumed only 25% of the time compared to the MAP method. In the simulation settings, MAP achieved error-free recovery at an average coverage of 25×, SP at 30×, whereas KL required 45× coverage. The performance in vitro experiments will be further verified in the following sections.\nLines 260–277:\nThen, the minimum sequencing coverage for data recovery was evaluated with 1,000 independent trials on two sequencing datasets. Both the erasure error rate and the letter error rate progressively decreased as the average coverage increased (Fig. 4d, e). At low sequencing coverage, erasures dominated, while with increasing average coverage, erasures were suppressed, enabling reliable recovery. Using the NB-LDPC code with a code rate of 0.33, error-free recovery was achieved at an average coverage of 9×, even with an erasure error rate as high as 7% (Fig. 4f). Using RS code with a code rate of 0.83, at least 14× coverage was required for error-free recovery.\nThird, we benchmarked SP method against the KL and MAP approaches at low sequencing coverages. In the column-based eight-letter experiments and the array-based experiments of 10,000 strands (including two eight-letter and two 15-letter systems), MAP consistently achieved the highest detection accuracy (Supplementary Tables 7 and 8). SP, as the second-best in accuracy, showed clear advantages in runtime over both MAP and KL (Supplementary Figs. 42 and 43). For the eight-letter pool of 10,000 strands, MAP enabled error-free recovery at 15× coverage, compared with 16× for SP. Executed with 10 threads, SP completed 300 validation runs in ~10 s (Supplementary Figs. 44 and 45). For the 15-letter system, MAP also provided the highest accuracy (Supplementary Figs. 46 and 47), while SP showed boundary-related precision loss yet still achieved error-free recovery at 33× coverage, consistent with simulations.\nIn the Supplementary information:\n(1) Supplementary Table 7 and Table 8 are added to present the detailed error rates of each composite letter.\n(2) Supplementary Figs. 42–47 are added to show the data recovery performance of the SP method and its comparison with the KL and MAP methods.\nSupplementary Figure 42. End-to-end decoding performance of the eight-letter system (126 strands, 116 letters, RS code, R=5/6). a, Relationship between raw sequencing coverage and valid base coverage used for data recovery. b, 94.86% of reads passed length filtering and 99.96% were successfully identified by double-end indices. c, 99% accuracy at 13× for MAP and 98.9% for SP. d, Erasure rate at different average coverages. e, Percentage of successful error-free recovery across 1,000 independent trials per coverage. All three methods achieved error-free recovery at 14× coverage. f, Runtime comparison of SP, KL, and MAP methods in single-thread mode.\nSupplementary Figure 43. End-to-end decoding performance of the eight-letter system (126 strands, 116 letters, NB-LDPC code, R=1/3). a, Relationship between raw sequencing coverage and valid base coverage used for data recovery. b, 94.95% of reads passed length filtering, and 99.96% were successfully identified by indices. c, Letter error rate at different average coverages. d, Erasure errors were the major limiting factor for recovery, with the erasure rate exceeding 20% at 6× coverage and remaining as high as 5% at 10×. e, Percentage of successful recoveries across 1,000 independent trials. With the strong error-correction capability of the outer code (R=1/3), all three methods achieved successful recovery at 9× coverage. f, Runtime comparison in single-thread mode, where SP maintained a ~25% runtime advantage over MAP.\nSupplementary Figure 44. End-to-end decoding performance of the eight-letter system (10,000 strands, 124 letters). a, Relationship between raw sequencing coverage and valid base coverage used for data recovery. b, 85.72% of reads passed length filtering and 99.82% were successfully identified by indices. c, Letter error rates show that SP closely matches MAP, with both outperforming KL in accuracy. d, At the coverage of 17×, the erasure rate was very low (<1%). e, Percentage of successful recoveries across 300 independent trials at different coverages. MAP achieved error-free recovery at 18× coverage, while SP, with a slight loss in accuracy, achieved error-free recovery at 19×. f, Runtime comparison of SP, KL, and MAP in multi-threaded mode (10 threads), values represent the cumulative runtime of 300 trials. SP requires less than 15 s, compared with 20 s for KL and 25 s for MAP.\nSupplementary Figure 45. End-to-end decoding performance of the eight-letter system (10,000 strands, 112 letters). a, Relationship between raw sequencing coverage and valid base coverage used for data recovery. b, 87.53% of reads passed length filtering and 99.94% were successfully identified by indices. c, Detection accuracy was comparable among the three methods, with MAP performing the best. d, Erasure rates decreased below 0.5% by 16× coverage. e, Error-free recovery was achieved by MAP at 15×, and by SP and KL at 16× coverage, based on 300 independent trials. f, SP completed 300 trials in ~12 s, whereas MAP required ~25 s.\nSupplementary Figure 46. End-to-end decoding performance of the 15-letter system (10,000 strands, 124 letters). a, Relationship between raw sequencing coverage and valid base coverage used for data recovery. b, 86.38% of reads passed length filtering and 99.93% were successfully identified by indices. c, With larger alphabets, SP showed higher error rates than MAP. d, At the coverage ≥25×, erasure errors were nearly absent. e, Percentage of successful recoveries across 300 independent trials at each coverage. Error-free recovery was achieved at 26× coverage using MAP, whereas 33× coverage was required for SP. f, Runtime comparison for 300 trials: SP completed letter detection in ~18 s, compared with ~35 s for the KL method and >60 s for the MAP method.\nSupplementary Figure 47. End-to-end decoding performance of the 15-letter system (10,000 strands, 112 letters). a, Relationship between raw sequencing coverage and valid base coverage used for data recovery. b, 86.66% of reads passed length filtering and 99.96% were successfully identified by indices. c, With larger alphabets, SP showed higher error rates than MAP. d, At the coverage ≥26×, erasure errors were nearly absent. e, Percentage of successful recoveries across 300 independent trials at each coverage. Error-free recovery was achieved at 27× coverage using MAP, whereas 35× coverage was required using SP and 48× with the KL method. f, Runtime comparison for 300 trials: SP completed letter detection in ~20 s, compared with ~35 s for the KL method and ~60 s for the MAP method.\nSupplementary Table 7. Letter detection error rates under different detection methods with an eight-letter alphabet {A, T, G, C, R, Y, M, K} and RS(1890, 1575).\nMethod Cov. A(%) T(%) G(%) C(%) R(%) Y(%) M(%) K(%) Total\n10× 0.129 0.089 0.138 0.093 0.294 0.454 0.293 0.264 1.754\n11× 0.123 0.082 0.126 0.087 0.243 0.390 0.246 0.223 1.519\n12× 0.117 0.075 0.116 0.081 0.204 0.341 0.206 0.187 1.327\n13× 0.113 0.070 0.108 0.076 0.172 0.298 0.175 0.159 1.170\n14× 0.110 0.066 0.100 0.072 0.145 0.262 0.150 0.137 1.042\n15× 0.105 0.061 0.093 0.068 0.122 0.229 0.131 0.117 0.926\nSP\n16× 0.103 0.058 0.089 0.065 0.104 0.200 0.112 0.104 0.834\n17× 0.101 0.055 0.085 0.064 0.090 0.176 0.097 0.091 0.758\n18× 0.098 0.052 0.081 0.061 0.077 0.156 0.085 0.082 0.691\n19× 0.096 0.050 0.079 0.060 0.067 0.138 0.074 0.073 0.636\n20× 0.096 0.049 0.078 0.059 0.060 0.123 0.066 0.066 0.597\n21× 0.095 0.047 0.076 0.059 0.054 0.109 0.058 0.060 0.558\n10× 0.215 0.153 0.276 0.174 0.235 0.339 0.227 0.207 1.825\n11× 0.222 0.160 0.289 0.183 0.182 0.268 0.177 0.164 1.644\n12× 0.225 0.164 0.296 0.186 0.142 0.213 0.139 0.130 1.493\n13× 0.225 0.165 0.298 0.186 0.113 0.170 0.111 0.104 1.372\n14× 0.219 0.163 0.295 0.185 0.091 0.139 0.090 0.086 1.268\n15× 0.213 0.159 0.287 0.183 0.073 0.114 0.074 0.072 1.175\nKL\n16× 0.206 0.154 0.279 0.178 0.060 0.094 0.062 0.063 1.096\n17× 0.197 0.148 0.267 0.172 0.051 0.080 0.052 0.056 1.022\n18× 0.188 0.141 0.255 0.166 0.045 0.068 0.046 0.051 0.959\n19× 0.183 0.136 0.243 0.161 0.039 0.059 0.040 0.046 0.907\n20× 0.178 0.131 0.232 0.155 0.035 0.054 0.036 0.044 0.864\n21× 0.173 0.126 0.220 0.149 0.032 0.047 0.033 0.041 0.820\n10× 0.126 0.088 0.161 0.108 0.250 0.377 0.243 0.219 1.572\n11× 0.116 0.082 0.149 0.102 0.200 0.312 0.196 0.179 1.336\n12× 0.109 0.076 0.139 0.096 0.162 0.261 0.161 0.146 1.150\n13× 0.102 0.071 0.127 0.090 0.135 0.222 0.135 0.121 1.002\n14× 0.098 0.066 0.117 0.086 0.113 0.193 0.116 0.104 0.892\n15× 0.094 0.062 0.108 0.082 0.096 0.169 0.102 0.090 0.801\nMAP\n16× 0.092 0.060 0.102 0.078 0.083 0.148 0.089 0.082 0.735\n17× 0.089 0.058 0.097 0.076 0.074 0.132 0.081 0.075 0.680\n18× 0.086 0.053 0.090 0.072 0.067 0.119 0.074 0.070 0.631\n19× 0.084 0.051 0.086 0.071 0.061 0.108 0.069 0.065 0.594\n20× 0.082 0.049 0.083 0.069 0.058 0.101 0.066 0.063 0.572\n21× 0.082 0.048 0.081 0.068 0.055 0.092 0.063 0.061 0.550\nSupplementary Table 8. Letter detection error rates under different detection methods with a 15-letter alphabet (10,000 composite strands, 124 letters).\nMethod Cov. A(%) T(%) G(%) C(%) R(%) Y(%) M(%) K(%) S(%) W(%) H(%) B(%) V(%) D(%) N(%) Total\n26× 0.037 0.044 0.073 0.032 0.245 0.129 0.109 0.225 0.240 0.151 0.272 0.375 0.184 0.381 0.809 3.305\n30× 0.033 0.041 0.063 0.029 0.203 0.098 0.088 0.171 0.198 0.114 0.198 0.285 0.156 0.291 0.633 2.602\n34× 0.030 0.040 0.056 0.027 0.167 0.078 0.072 0.135 0.160 0.090 0.148 0.220 0.140 0.228 0.502 2.090\nSP 38× 0.028 0.038 0.051 0.025 0.141 0.066 0.061 0.110 0.132 0.074 0.113 0.173 0.127 0.181 0.402 1.722\n42× 0.026 0.037 0.048 0.024 0.126 0.059 0.054 0.094 0.116 0.064 0.088 0.137 0.115 0.146 0.324 1.457\n46× 0.024 0.037 0.045 0.023 0.119 0.054 0.051 0.084 0.106 0.058 0.070 0.110 0.103 0.119 0.263 1.265\n50× 0.023 0.036 0.042 0.022 0.114 0.051 0.049 0.077 0.101 0.054 0.056 0.089 0.090 0.098 0.215 1.119\n26× 0.174 0.166 0.415 0.146 0.461 0.155 0.196 0.324 0.447 0.228 0.111 0.137 0.192 0.185 0.266 3.431\n30× 0.145 0.138 0.325 0.120 0.493 0.159 0.208 0.336 0.479 0.236 0.087 0.108 0.197 0.164 0.155 3.349\n34× 0.128 0.122 0.263 0.104 0.490 0.155 0.205 0.330 0.477 0.230 0.079 0.097 0.211 0.160 0.091 3.141\nKL 38× 0.119 0.112 0.225 0.095 0.453 0.144 0.189 0.306 0.441 0.212 0.077 0.093 0.226 0.162 0.055 2.909\n42× 0.113 0.108 0.202 0.089 0.396 0.129 0.164 0.270 0.384 0.187 0.077 0.092 0.237 0.164 0.034 2.647\n46× 0.108 0.104 0.187 0.085 0.335 0.114 0.139 0.234 0.323 0.161 0.076 0.091 0.238 0.164 0.021 2.379\n50× 0.105 0.101 0.174 0.082 0.284 0.101 0.118 0.203 0.269 0.141 0.073 0.087 0.230 0.158 0.014 2.138\n26× 0.040 0.046 0.067 0.035 0.108 0.102 0.050 0.182 0.104 0.115 0.197 0.251 0.145 0.264 0.448 2.152\n30× 0.036 0.043 0.058 0.031 0.079 0.074 0.036 0.132 0.075 0.082 0.138 0.182 0.097 0.192 0.356 1.611\n34× 0.033 0.040 0.051 0.029 0.064 0.060 0.029 0.103 0.059 0.064 0.097 0.131 0.065 0.140 0.278 1.243\nMAP 38× 0.030 0.039 0.046 0.027 0.054 0.051 0.025 0.086 0.050 0.054 0.069 0.097 0.045 0.105 0.210 0.990\n42× 0.028 0.037 0.043 0.026 0.048 0.046 0.023 0.075 0.045 0.047 0.052 0.074 0.033 0.082 0.156 0.815\n46× 0.027 0.037 0.040 0.025 0.044 0.044 0.021 0.068 0.041 0.043 0.040 0.058 0.025 0.066 0.116 0.694\n50× 0.026 0.036 0.038 0.024 0.041 0.042 0.020 0.063 0.039 0.040 0.033 0.047 0.021 0.055 0.087 0.611\n5. The manuscript reports discarding ~23% of reads during length filtering (Fig. 5g), a shocking large fraction for Illumina sequencing (which typically has very low indel rates). The authors should explain the source of these indels and verify that filtering did not bias the results. Additionally, it is unclear whether the stated coverage thresholds (e.g. 18× for error-free recovery and in Fig. 5e-f) refer to pre- or post-filter read counts. Since coverage directly translates to DNA-storage readout cost, it should be defined on the basis of raw sequencing depth rather than the subset surviving filtering.\nResponse: We thank the reviewer for this insightful comment. We updated the computation of sequencing coverage. We updated all the results with raw sequencing data. We thank the reviewer for pointing out this point. In this situation, we observed the ratios of the filtered reads reduced. Reads removed by length filtering primarily arose from induced insertion/deletion errors in the synthesis step, which caused deviations from the designed payload length.\n(1) On the read filtering ratio of length filtering\nWe have updated the data. In our prior version of manuscript, we used clean data with quality control (QC). The QC may induce bias on the reads. Therefore, we filter about 23% reads during length filtering. We thank the reviewer. According to the comments, we have updated the results on raw data. The filtered ratio is about 5%.\nHowever, it is still quite high considering the relatively low indel errors of Illumina sequencing. We conjectured that indels may stem from the synthesis process. For the column-based synthesis, the indel error rate was low, and the filtered ratio was also low. For the low-cost synthesis using inkjet printing, the indel rate was relatively high, and the filtered ratio was higher (Supplementary Figs. 56 and 57). The filtered ratios were quite identical to the indel error ratio. Correspondingly, the filtered ratios increased to greater than 10% (Supplementary Table 9). For the low-cost synthesis using inkjet printing, low-cost protocols were adopted for cost and no rigorous purification was performed. Thus, the indel especially deletion rate was higher.\nUnfortunately, unlike conventional DNA storage schemes, composite letter DNA storage currently lacks effective strategies for handling indels. We incorporated a length filtering step into our data recovery workflow (Supplementary Fig. 58) to prevent such errors from interfering composite letter detection.\nWe down-sampled the raw reads to target coverages (e.g., 10×, 12×, 14×), and then ran the complete workflow with and without length filtering. For each coverage, 1,000 independent random trials were conducted. Results from both coding schemes (coding rate of 5/6 and 1/3) consistently demonstrated that removing off-length reads improved base detection accuracy (Fig. 5e, f and Supplementary Fig. 67). For instance, at 14× coverage, length filtering reduced the composite-letter error rate from an average of 2.8% to 1.0% (accuracy > 99%), and when combined with RS decoding (R=5/6), achieved error-free recovery in 1000/1000 trials. In contrast, without length filtering, only ~30% of trials achieved error-free recovery under the same (pre-filter) coverage condition.\n(2) On the sequencing coverage\nIn the prior version of our manuscript, we used clean data. According to the reviewer’ kind comments, we have modified all the results with the raw sequencing data. In this new version, the coverage values reported in this study are all defined based on the raw sequencing depth after paired-end assembly, not on the subset of filtered reads.\nWe updated Part 5 of the Results with a revised performance analysis of length filtering. The revised text in the manuscript is as follow.\nLines 328–337:\nUnder the same average coverage, we evaluated the effectiveness of length filtering in reducing the symbol error rate by comparing two schemes. Across both coding schemes (R=0.83 and R=0.33), filtering consistently improved letter detection accuracy (Fig. 5e, f and Supplementary Fig. 67). At the 14× coverage, filtering reduced the average composite-letter error rate from 2.8% to 1.0% (accuracy > 99%). With RS code (R=0.83), length filtering enabled error-free recovery in all trials. Without filtering, only ~30% of trials achieved error-free recovery under the same coverage. Although filtering decreased the number of usable reads, 94.39% were retained after primer identification, length filtering, and index identification, ensuring high-quality input for downstream detection (Fig. 5g). Supplementary Table 9 presents the results of all pools, including small-scale pools and large-scale pools.\nSupplementary Table 9. Length filtering of the different experiments.\nExp.1 Exp.2 Exp.3 Exp.4 Exp.5 Exp.6\nComposite strands 126 126 10,000 10,000 10,000 10,000\nAlphabet 8 8 8 8 15 15\nLength (Primer+index) 28 nt 28 nt 32 nt 26 nt 32 nt 26 nt\nTotal 0.0059 0.0062 0.0102 0.0030 0.0087 0.0134\nIns. 0.0001 0.0001 0.0004 0.0001 0.0004 0.0001\nError rate (Forward\nprimer+index)\nDel. 0.0053 0.0056 0.0085 0.0027 0.0071 0.0129\nSub. 0.0005 0.0005 0.0013 0.0002 0.0012 0.0004\nTotal 0.0020 0.0020 0.0040 0.0009 0.0036 0.0036\nIns. 0.0001 0.0001 0.0003 0.0001 0.0003 0.0001\nError rate (Reverse\nprimer+index)\nDel. 0.0013 0.0013 0.0025 0.0005 0.0021 0.0031\nSub. 0.0006 0.0006 0.0012 0.0003 0.0012 0.0004\nLength (Index+payload) 76 76 84 72 84 72\nLength filtering (Valid reads) 94.9% 94.9% 85.9% 87.6% 86.5% 86.7%\nLength filtering (Invalid reads) 5.1% 5.1% 14.1% 12.4% 13.5% 13.3%\nSupplementary Figure 56. Length distribution of assembled sequencing reads. a and b, Length distributions of 126 composite strands. The strands were encoded using either the NB-LDPC(3780, 1260) code or the RS(1890, 1575) code, with 79.6% and 78.5% of sequencing reads, respectively, reaching the designed full length of 116 nt. c and d, Large-scale composite pools using the eight-letter alphabet, where 56.0% and 73.4% of reads, respectively, reached the designed full length after paired-end assembly. e and f, Large-scale composite pools using the fifteen-letter alphabet, where 63.0% and 55.9% of reads, respectively, reached the designed full length after paired-end assembly.\nSupplementary Figure 57. Length distributions of index and payload regions after primer identification. a and b, Both datasets exhibited nearly identical distributions, with 94.9% of reads retaining the designed full length (76 nt) after primer trimming. c and d, Two large-scale pools using the eight-letter alphabet, with 85.9% (84 nt) and 87.6% (72 nt) of reads, respectively, retaining the designed payload length. e and f, Two large-scale pools using the fifteen-letter alphabet, with 86.5% (84 nt) and 86.7% (72 nt) of reads, respectively, retaining the designed payload length.\nSupplementary Figure 58. Readout workflow with length filtering. Sequencing reads are first aligned to double-end primers to identify payload and index segments (Primer alignment). Reads with abnormal length are discarded, ensuring equal-length, aligned sequences for downstream readout (Length filtering). Reads are then grouped into different clusters according to the index sequences (Index identification). Within each cluster, base frequency distributions at each position are calculated across multiple copies, and the most probable composite letter is inferred (Letter detection).\nFig. 5 | Performance evaluation considering interference and indels. Panels a–d are shown in the main text and omitted here for clarity. e, The length filtering step prevents error propagation and improves letter detection accuracy. Results are based on 1,000 independent trials at each coverage, with error bars indicating the standard deviation. f, The length filtering step reduces the sequencing coverage required for data recovery. Plots show the number of successful recoveries out of 1,000 independent trials at each coverage. g, The proportion of effective reads after primer identification, length filtering, and index identification.\nSupplementary Figure 67. Letter detection accuracy and recovery performance with length filtering. a, Letter error rates as a function of average coverage, comparing workflows with (red) and without (blue) length filtering. Length filtering effectively reduces the error rate across all coverages. b, Successful recovery rate as a function of average coverage. For each coverage, 1,000 independent evaluations were performed, and the successful recovery rate represents the proportion of tests achieving error-free recovery. Although the omission of length filtering leads to higher error rates, error-free recovery can still be achieved using NB-LDPC coding with a rate of 1/3. c, Proportion of valid reads at each processing step of the recovery workflow. Starting from 6,066,534 raw paired-end assembled reads (100%), 99.23% were retained after primer identification, 94.18% after length filtering, and 94.15% after index identification.\n6. The manuscript lacks critical detail regarding the synthesis methodology for composite letters, which is a key practical challenge in composite DNA storage. In particular, it is not clear whether each oligo encoding a composite letter is individually designed (deterministic assignment of A/T/C/G at each oligo) or whether the synthesis process uses probabilistic base mixing at degenerate positions, resulting in random incorporation across molecules. This distinction impacts scalability, and decoding accuracy. Also, discuss scalability implications for large-scale oligo pools.\nResponse:\nWe thank the reviewer for raising this valuable comment. We quite agree with the reviewer. The specific synthesis of composite letters is indeed a key practical aspect of composite DNA storage. In all our design, we did not individually design each strand. During the chemical synthesis process, we used mixed monomers at each degenerate positions.\nWe also agree with the reviewer that there may be some concerns on the scalability and decoding accuracy. We performed several practical verifications on 10,000 composite strands using eight-letter and 15-letter alphabets.\nWe provide detailed clarification below.\n1. Synthesis methodology for composite letters\nIn our work, each composite DNA strand was synthesized independently in a dedicated well using column-based or array-based phosphoramidite chemistry.\nFor column-based synthesis, during each coupling cycle, composite letters were generated by mixing different bases, i.e., preprogrammed volumetric blending of A, T, C, G monomers at specified ratios (Supplementary Fig. 4a and Supplementary Table 2). This approach implements composite letters through controlled degeneracy rather than by designing deterministic single-base sequences.\nMoreover, in our scheme, we used eight or 15 letters. Each composite letter was indeed mixed at the equal ratio (Supplementary Table 2). For column-based synthesis, we only needed to pre-mix two or three or four monomers with the equal molecule quantities. It is quite convenient. The synthesizer can only support eight inputs for monomers. Therefore, we used this synthesizer platform to support Alphabets 1–3.\nFor inkjet array-based synthesis, the mixture process was implemented by controlling the equal jetting volumes of different monomers according to the specific composite letter within one coupling cycle. Therefore, this platform can at least support 15 different composite letters. We tested the Alphabets 1 and 4. That is, Alphabet 1, {A, T, G, C, R, Y, M, K} and Alphabet 4, {A, T, G, C, R, M, Y, K, S, W, H, B, V, D, N}.\n2. Practical column-based synthesis and data recovery.\nWe performed four synthesis experiments based on eight different composite-letter alphabets. In each experiment, 126 composite strands were independently synthesized across 126 CPG columns and subsequently stored in 126 individual tubes. These composite strands were synthesized by Hippobio Co., Ltd. An eight-channel oligo synthesizer was used, which can precisely control volumetric delivery of each channel (Supplementary Fig. 4b). Eight channels delivered the pure or mixed monomers, corresponding to eight composite letters. Successful data recovery was achieved directly on raw sequencing reads on the pooled oligos (Supplementary Figs. 42 and 43).\n3. Scalability implications for large-scale oligo pools\nAccording to the kind comments, we performed four oligonucleotide pool experiments, each synthesizing 10,000 composite strands. The strand lengths were 112 letters and 124 letters, respectively. The length of composite letter length in each strand was 60. Two different alphabets (Alphabets 1 and 4, eight-letter and 15-letter) were employed (Supplementary Fig. 4c). The other parts were primers and indices with the length of 60nt using standard four bases.\nThese pools were synthesized by Dynegene Technologies, using array-based inkjet oligonucleotide synthesis. The pools were sequenced using NGS in PE150 mode. Successful data recovery was achieved directly on raw reads (Supplementary Figs. 44–47).\nTherefore, we expect that the scalability for large-scale oligo pools is promising, even for larger synthesized pools.\nWe added Part 6 of Results Section to include a description of the experimental scheme.\nLines 176–185:\nFor practical writing into DNA, two synthesis platforms were used to test different composite alphabet implementations (Supplementary Fig. 4). First, 126-strand pools were constructed with eight-letter alphabets using column-based phosphoramidite synthesis. Each strand was synthesized in an individual well, with composite letters implemented by volumetric mixing of A, T, C, and G phosphoramidite monomers. Second, large-scale pools containing 10,000 composite strands were synthesized by array-based inkjet phosphoramidite chemistry. In all experiments, payloads were uniformly 60 letters, flanked with double-end indices and PCR primer sequences (Fig. 2e). For the column-based pools of 126 strands, each carried two indices (8 nt) and two primers (20 nt) at the 5’ and 3’ ends. For the array-based pools of 10,000 strands, the indices were 6 nt or 12 nt at the two ends, respectively.\nWe updated the Methods section to include a description of the synthesis of composite-letter oligonucleotide pools at different scales.\nLines 473–481:\nSynthesis of small-scale pools (column-based synthesis)\nA total of four small-scale composite-letter oligonucleotide pools were prepared. For each pool, 126 composite DNA sequences were synthesized using Oligo-192 Oligo Synthesizer by Hippobio (Huzhou, China). Each strand was synthesized independently in a dedicated column, with composite letters implemented through volumetric mixing of A, T, G, and C phosphoramidite monomers. HPLC Purification was performed. All 126 samples were then manually combined in equimolar amounts to generate one master pool. Four such master pools were obtained, with final DNA concentrations of 2,482 ng/μL, 2,515 ng/μL, 301 ng/μL, and 459 ng/μL.\nLines 483–491:\nSynthesis of large-scale pools (array-based inkjet printing synthesis)\nFour large-scale oligonucleotide pools were designed, each comprising 10,000 composite strands: two based on the eight-letter alphabet and two based on the 15-letter alphabet. These pools were synthesized by Dynegene Technologies (Shanghai, China) using array-based inkjet printing synthesizer. During synthesis, four inkjet nozzles dispensed monomer droplets onto array chips according to predefined base-mixing ratios. The eight-letter system used the alphabet {A, T, G, C, R, Y, M, K}, and the 15-letter system used the alphabet {A, T, G, C, R, M, Y, K, S, W, H, B, V, D, N}. Each large-scale pool was delivered as a lyophilized master pool containing 3 µg of DNA.\nSupplementary Figure 4. Illustration of the composite strand synthesis with column-based and array-based synthesizers. a, Column-based synthesis of composite DNA strands. Eight reagent channels correspond to eight composite bases. Three composite alphabets were chosen and 126 composite strands were synthesized for each alphabet. b, Each designed composite strand, composed of composite letters (e.g., B, G, T, D, A, etc.), is individually synthesized in a dedicated well using column-based DNA synthesis. c, Array-based inkjet synthesis of large-scale composite-letter pools. Four printheads deliver monomer droplets onto array chips following predefined base mixing ratios.\nMinor reversions\nLine 141: The Galois field definition for NB‑LDPC code appears incorrect—please correct the GF notation.\nResponse: Thank you for your valuable comments. We are sorry for this error and have corrected this error. The manuscript has been revised as follows:\nLines 156–158:\nFor data encoding, in the eight-letter system, 945 bytes of text were first encoded using NB-LDPC(3780,1260) code (R=0.33) over Galois Field (GF(64)), with interleaving to mitigate burst erasures39.\nResponse to the Reviewers’ Comments\nWe sincerely thank both reviewers for their valuable comments on our manuscript and greatly appreciate their constructive remarks and suggestions. We have revised the manuscript accordingly based on their valuable feedback and provide detailed point-by-point responses below. All changes in the revised manuscript are highlighted in yellow.\nResponse to the Reviewers #1\nI appreciate the authors for their detailed responses, which have addressed most of my earlier concerns. However, one remaining issue relates to the statement in the revised manuscript that “homopolymer length distributions in composite systems are comparable to conventional four-letter storage.” In fact, conventional DNA data storage systems do not generally use unconstrained four-letter encoding. A widely adopted strategy (Goldman et al., Nature, 2013) is to convert the binary data to base-3 digits (trits), and then convert to DNA code by replacement of each trit with one of the three nucleotides different from the previous one used. This strategy explicitly prevents homopolymers in every DNA strand. Thus, the authors may clarify that conventional systems can avoid homopolymers through appropriate encoding algorithms rather than simply employing 4-letter representations. More details can be found in this paper:\nGoldman, N., Bertone, P., Chen, S. et al. Towards practical, high-capacity, low-maintenance information storage in synthesized DNA. Nature 494, 77–80 (2013).\nResponse: We thank the reviewer for point this problem. We completely agree with the reviewer. We have revised this point. We clarified that our comparison refers specifically to four-letter storage without homopolymer optimization. Several existing DNA storage systems avoided homopolymers through appropriate encoding algorithms, e.g., rotation coding or homopolymer filtering.\nThe revised main text in the manuscript is as follows:\nLines 294–302:\nHomopolymer length distributions in composite systems are similar to four-letter DNA storage without homopolymer optimization or random sequences (Supplementary Fig. 55). Several DNA storage systems guaranteed homopolymer length through appropriate encoding, e.g., rotation coding4 or homopolymer filtering9. We did not used any optimization on homopolymer, balancing the efficiency and performance40. To address these challenges, we incorporated two tailored strategies into the readout pipeline: robust double-end indices to mitigate strand crosstalk and DNA degradation; length filtering to reduce the impact of indel errors during synthesis and sequencing.\nResponse to the Reviewers #2\nI thank the authors for their considerable efforts in revising this manuscript. The manuscript is strengthened by the inclusion of 15-letter in vitro experiments, data written through two composite-letter synthesis platforms tailored to different oligo scales, and additional degradation experiments assessing DNA stability. With Illumina sequencing for readout, all decoding experiments achieved 100% data recovery, demonstrating the robustness of the proposed framework. Overall, the concept of “DNA diamond” is original and appealing, offering a useful entropy-based perspective on composite-letter codec for DNA storage.\nFor further clarification and fairness in reporting, I have a few minor suggestions:\n1. When comparing overall performance (lines 86–100), please pair sequencing coverage with a net information density that includes all synthesized nucleotides (including primers and indices), or at least add the index overhead. In this design, the double-ended indices are intentionally introduced to mitigate strand displacement/oligo breaks and degradation (lines 301–302) and therefore form part of the error-correction mechanism that enables lower recovery depth; omitting them can overstate payload efficiency. By contrast, in prior studies such as Xu et al. and Anavy et al., indices primarily serve addressing, so payload-only metric would be reasonable within those contexts. The cost analysis should follow the consistent accounting and incorporate index costs to ensure fair comparison.\nResponse: We thank the reviewer for the kind suggestion. Following this advice, we have recalculated the logical densities explicitly including index and payload. Our encoded indices have two functions. First, they served as reliable addresses avoiding strand crosstalk. Reliable read addressing is important to achieve low recovery coverage. Second, they can identify degraded oligos due to break. The revised values are now reported in Supplementary Table 1 and Supplementary Table 4. We also revised the cost analysis considering the index cost, and accordingly updated Supplementary Table 10 and Supplementary Figure 68.\nThe revised main text in the manuscript is as follows:\nLines 85–97:\nSpecifically, we implemented the eight-letter alphabet using column-based synthesis. Error-free data recovery was achieved at 14× sequencing coverage. This scheme increased the information density from the canonical 2 bits per letter to 2.5 bits per letter (excluding primers and indices). The logical density of payload and indices was 1.97 bits per letter. In comparison with state-of-the-art schemes, our system achieved comparable performance in terms of density and sequencing coverage (Fig. 1f and Supplementary Table 1). We further designed large-scale oligonucleotide pools containing 10,000 composite strands. Both the eight-letter and 15-letter alphabets were experimentally validated using array-based synthesis. For the 15-letter system, our composite letter detection method enabled error-free recovery at 33× sequencing coverage. The system achieved a logical density of 3.125 bits per letter for payload, corresponding to a density of 2.23 bits per letter for payload and indices. Our work validated the feasibility of using a 15-letter alphabet for DNA data storage (10,000 strands synthesized in a single pool) and demonstrated that error-free recovery can be achieved at relatively low sequencing coverage.\nIn the Supplementary information:\n(1) Supplementary Table 1 was updated to include the logical density that considering index overhead;\n(2) Supplementary Table 4 was revised to incorporate the logical density that considering index overhead;\n(3) Supplementary Table 10 was revised to present the cost analysis considering index overhead;\n(4) Supplementary Fig. 68 was also revised to incorporate the synthesis and sequencing cost calculations that account for index overhead.\nSupplementary Table 1. Comparison of published composite letter data storage schemes.\nLogical density\nTotal Logical density Average\nStudy Error correction codes Alphabet Data volume (composite\ncomposite letters (including indices) coverage\nletters, bits/letter)\nRS(45, 43)\n∑ 2.12 MB 7,830,000 2.16 1.93 29×\n+Fountain (1.1) 6\nRS(45, 43)\n∑ 6.42 MB 23,490,000 2.19 1.96 29×\n+Fountain (1.08) 6\nAnavy et RS(45, 43)\n∑ 6.42 MB 26,055,000 1.97 1.76 29×\nal.1 +Fountain (1.08) 5\n3.67 (only one strand, no\nBinary Huffman ∑ 19.25 Bytes 42 3.67 100×\n15 index)\n4.29 (only one strand, no\nBinary Huffman ∑ 22.5 Bytes 42 4.29 100×\n20 index)\nRS ∑ 854 Bytes 1,890 3.61 3.37 250×\nExp. Choi et 15\nal.2\nRS ∑ 135 Bytes 499,833 2.16 2.0 250×\n∑ 945 Bytes*3 7,560*3 1 0.79 9×\nGF(26) 8\nRS(1890, 1575) over GF(212) ∑ 2,355 Bytes 7,560 2.5 1.97 14×\n2.08 (112 letters) / 16× /\nRS(3750, 3125) over GF(212) ∑ 187,500 Bytes*2 600,000*2 2.5\n8 1.79 (124 letters) 19×\n2.60 (112 letters) / 35× /\nRS(30000, 25000) over GF(215) ∑ 234,375 Bytes*2 600,000*2 3.125\n15 2.23 (124 letters) 33×\nRS(136, 130) ∑ 2.12 MB 2,373,975 7.13 / 2000×\nAnavy 256\net al.1\nRS(68, 65) ∑ 2.12 MB 3,354,480 5.05 / 500×\nSim. 64\nXu\nRS(45, 41) over GF(27) ∑ 20.8 MB 27,776,250 6.37 / 300×\net al.6 128\nRS(45, 41) over GF(28) ∑ 20.8 MB 24,273,000 7.28 / 490×\nScheme Column-based DNA synthesis experiments Array-based DNA synthesis experiments Simulation\nData volume (bytes) 945 2,355 187,500 234,375 780,713\nBit information (bits) 7,560 18,900 1,500,000 1,875,000 6,247,575\nPayload 126*60 126*60 10000*60 10000*60 32767*60\n(M*L) =7560 letters =7560 letters =600,000 letters =600,000 letters =1,966,020 letters\nAlphabet ∑ ∑ ∑ ∑ ∑\n8 8 8 15 15\nCoding potential\nlog (8)=3 log (8)=3 log (8)=3 log (15)=3.9 log (15)=3.9\n2 2 2 2 2\n(bits/letter)\nCoding scheme NB-LDPC(3780, 1260) RS(1890, 1575) RS(3750, 3125) RS(30000, 25000) RS(32767, 27767)\nCode rate R=1/3 R=5/6 R=5/6 R=5/6 R=17/20\nLogical density\n1 bit/letter 2.5 bits/letter 2.5 bits/letter 3.125 bits/letter 3.18 bits/letter\n(payload)\nSynthesized nucleotides 126*116 126*116 10000*124 10000*112 10000*124 10000*112\n/\n(including primers/indices) =14,616 =14,616 =1,240,000 =1,120,000 =1,240,000 =1,120,000\nLogical density 2.60\n0.79 bit/letter 1.97 bits/letter 1.79 bits/letter 2.08 bits/letter 2.23 bits/letter /\n(payload+index) bits/letter\nNet information density\n1.67\n(synthesized nucleotides) 0.52 bits/letter 1.29 bits/letter 1.21 bits/letter 1.34 bits/letter 1.51 bits/letter /\nbits/letter\n(bits/letter)\nSupplementary Table 10. Comparative cost analysis aligned with prior studies.\nLogical density Normalized cost\nAlphabet Study Cov.\n(bits/letter) (excluding primers) C syn:C seq (500:1) C syn:C seq (1000:1)\nOrganick et al.15 5× 1.10 1.00 1.00\n∑ Goldman et al.16 51× 0.29 4.14 3.97\n∑ Anavy et al.1 29× 1.76 0.65 0.64\n∑ Choi et al.2 250× 2.16 0.82 0.67\n∑ This work 16× 2.08 0.54 0.53\n∑ Choi et al.2 250× 3.37 0.48 0.41\n∑ Anavy et al.1 100× 4.29 0.30 0.28\nEstimated costs for writing (left) and reading (right) 1 MB of data, assuming DNA synthesis and sequencing costs of $0.05/100 nt and $0.0000012/100 nt, respectively.\nTwo conditions are considered: Only payload part (light bars) and the region including payload and indices (dark bars).\n2. When positioning against prior studies (lines 351–354), please compare like-for-like alphabet sizes (e.g., 15-letter vs earlier 15-letter systems; for 8-letter vs 6-letter.).\nResponse: We thank the reviewer for this valuable suggestion. We have revised the synthesis cost comparison. We modified the comparison in Supplementary Fig. 68 and Supplementary Table 10.\nThe revised text in the main manuscript is as follows:\nLines 348–355:\nOn the one hand, only considering payload part, our eight-letter and 15-letter systems reduced the synthesis cost by 47% and 57%, respectively, compared with the conventional four-base DNA storage system25. On the other hand, considering the index overhead (excluding primers), for the similar alphabet size of six or eight, we obtained the similar cost reductions. For the 15-letter alphabet, we achieved large-scale validation using 10,000 strands also with a slight increase on synthesis cost (Supplementary Fig. 68). This was primarily due to the adoption of a more robust double-end index scheme that introduced additional redundancy to improve reliability.\nLines 358–361:\nOnly considering the region excluding primers at a 1000:1 ratio, the eight-letter and 15-letter systems reduced the total storage cost by 47% and 56%, respectively. When the ratio was reduced to 500:1, the total cost reductions remained at 46% and 55%, respectively.\n3.The statements near lines 99–101 about advantages of the 15-letter system should be calibrated to match the evidence shown in Fig. 1f and Supplementary Table 10, where—under comparable 15-letter conditions—the storage density appears lower than that in Anavy et al.\nResponse: We thank the reviewer for pointing out this problem. Our scheme indeed did not show superiority in storage density over that of Anavy et al., for we used more overhead for large-scale verification. In order to support a practical large-scale verification experiment (10,000 strands in a single pool), more overhead was allocated to indices and reliable error-correcting codes with more redundancy were adopted for the payload part. We advanced in experiment scale for composite DNA storage with 15-letter alphabet.\nThe revised text in the main manuscript is as follows:\nLines: 95–97:\nOur work validated the feasibility of using a 15-letter alphabet for DNA data storage (10,000 strands synthesized in a single pool) and demonstrated that error-free recovery can be achieved at relatively low sequencing coverage.\nResponse to the Reviewers’ Comments\nWe sincerely thank the reviewer for the valuable comments and constructive suggestions provided during the review of our manuscript. The manuscript has been carefully revised according to the reviewer’s minor suggestions, and the main text has been further improved. All changes in the revised manuscript are highlighted in yellow.\nResponse to the Reviewers #2\nI appreciate the authors' detailed response and the rigorous revisions made to the manuscript, which have adequately addressed my previous concerns. Before publication, I have just two minor suggestions to harmonize the text with the supplementary data.\n1. Lines 93–97: The revised text reports a density of 1.51 bits/letter for \"payload and indices\", which appears to reference the net information density (including primers). According to Supplementary Tables 1 and 4, the correct logical density for “payload and indices” is 2.23 bits/letter. Please correct the main text to align with your tabular data.\nResponse: We thank the reviewer for carefully checking and kind comment. We are very sorry for this obvious mistake. When considering “Payload and Indices” together, the logical density was 2.23 bits/letter. We misused the logical density of “Payload + Indices + Primers”, which was 1.51 bits/letter.\nIn the revised manuscript, the statement has been corrected so that the logical density for “payload and indices” is now reported as 2.23 bits/letter, in agreement with Supplementary Tables 1 and 4.\nLines 95–96:\nThe system achieved a logical density of 3.125 bits per letter for payload, corresponding to a density of 2.23 bits per letter for payload and indices.\n2. Lines 22–25 and 379–381: When highlighting the system's overall performance, pairing the sequencing coverage with the logical density of \"payload and indices\" (e.g., 33× sequencing coverage with 2.23 bits/letter) is more appropriate. This ensures the reader fully appreciates the efficiency of the error-correction mechanism (the indices) that enables such low recovery depth.\nResponse: We thank the reviewer for this constructive suggestion. In the revised manuscript, the overall system performance described in both the Abstract (Lines 24–26) and the Discussion (Lines 379–381) has been updated to pair the sequencing coverage with the logical density of “payload and indices”. In Table S4, we honestly listed all the three logical densities in detail: Payload only (3.125 bits/letter); Payload + Indices (2.23 bits/letter); Payload + indices + Primers (1.51 bits/letter).\nThe revised main text in the manuscript is as follows:\nLines 24–26:\nThe full 15-letter constellation enables 3.125 bits per letter for payload with error-free recovery at 33× coverage, corresponding to a density of 2.23 bits per letter for payload plus indices.\nLines 381–383:\nNotably, the 15-letter alphabet was experimentally realized at the 10,000-strand scale, achieving 3.125 bits per letter for payload (or 2.23 bits per letter for payload plus indices) and error-free recovery at 33× coverage."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I appreciate the authors for their detailed responses, which have addressed most of my earlier concerns. However, one remaining issue relates to the statement in the revised manuscript that “homopolymer length distributions in composite systems are comparable to conventional four-letter storage.” In fact, conventional DNA data storage systems do not generally use unconstrained four-letter encoding. A widely adopted strategy (Goldman et al., Nature, 2013) is to convert the binary data to base-3 digits (trits), and then convert to DNA code by replacement of each trit with one of the three nucleotides different from the previous one used. This strategy explicitly prevents homopolymers in every DNA strand. Thus, the authors may clarify that conventional systems can avoid homopolymers through appropriate encoding algorithms rather than simply employing 4-letter representations. More details can be found in this paper:\nGoldman, N., Bertone, P., Chen, S. et al. Towards practical, high-capacity, low-maintenance information storage in synthesized DNA. Nature 494, 77–80 (2013).", "reviewer_2": "I thank the authors for their considerable efforts in revising this manuscript. The manuscript is strengthened by the inclusion of 15-letter in vitro experiments, data written through two composite-letter synthesis platforms tailored to different oligo scales, and additional degradation experiments assessing DNA stability. With Illumina sequencing for readout, all decoding experiments achieved 100% data recovery, demonstrating the robustness of the proposed framework. Overall, the concept of “DNA diamond” is original and appealing, offering a useful entropy-based perspective on composite-letter codec for DNA storage.\n\nFor further clarification and fairness in reporting, I have a few minor suggestions:\n1.When comparing overall performance (lines 86–100), please pair sequencing coverage with a net information density that includes all synthesized nucleotides (including primers and indices), or at least add the index overhead. In this design, the double-ended indices are intentionally introduced to mitigate strand displacement/oligo breaks and degradation (lines 301–302) and therefore form part of the error-correction mechanism that enables lower recovery depth; omitting them can overstate payload efficiency. By contrast, in prior studies such as Xu et al. and Anavy et al., indices primarily serve addressing, so payload-only metric would be reasonable within those contexts. The cost analysis should follow the consistent accounting and incorporate index costs to ensure fair comparison.\n2.When positioning against prior studies (lines 351–354), please compare like-for-like alphabet sizes (e.g., 15-letter vs earlier 15-letter systems; for 8-letter vs 6-letter.).\n3.The statements near lines 99–101 about advantages of the 15-letter system should be calibrated to match the evidence shown in Fig. 1f and Supplementary Table 10, where—under comparable 15-letter conditions—the storage density appears lower than that in Anavy et al."}, "author_response": "We sincerely thank the reviewer for the valuable comments and constructive suggestions provided during the review of our manuscript. The manuscript has been carefully revised according to the reviewer’s minor suggestions, and the main text has been further improved. All changes in the revised manuscript are highlighted in yellow.\nResponse to the Reviewers #2\nI appreciate the authors' detailed response and the rigorous revisions made to the manuscript, which have adequately addressed my previous concerns. Before publication, I have just two minor suggestions to harmonize the text with the supplementary data.\n1. Lines 93–97: The revised text reports a density of 1.51 bits/letter for \"payload and indices\", which appears to reference the net information density (including primers). According to Supplementary Tables 1 and 4, the correct logical density for “payload and indices” is 2.23 bits/letter. Please correct the main text to align with your tabular data.\nResponse: We thank the reviewer for carefully checking and kind comment. We are very sorry for this obvious mistake. When considering “Payload and Indices” together, the logical density was 2.23 bits/letter. We misused the logical density of “Payload + Indices + Primers”, which was 1.51 bits/letter.\nIn the revised manuscript, the statement has been corrected so that the logical density for “payload and indices” is now reported as 2.23 bits/letter, in agreement with Supplementary Tables 1 and 4.\nLines 95–96:\nThe system achieved a logical density of 3.125 bits per letter for payload, corresponding to a density of 2.23 bits per letter for payload and indices.\n2. Lines 22–25 and 379–381: When highlighting the system's overall performance, pairing the sequencing coverage with the logical density of \"payload and indices\" (e.g., 33× sequencing coverage with 2.23 bits/letter) is more appropriate. This ensures the reader fully appreciates the efficiency of the error-correction mechanism (the indices) that enables such low recovery depth.\nResponse: We thank the reviewer for this constructive suggestion. In the revised manuscript, the overall system performance described in both the Abstract (Lines 24–26) and the Discussion (Lines 379–381) has been updated to pair the sequencing coverage with the logical density of “payload and indices”. In Table S4, we honestly listed all the three logical densities in detail: Payload only (3.125 bits/letter); Payload + Indices (2.23 bits/letter); Payload + indices + Primers (1.51 bits/letter).\nThe revised main text in the manuscript is as follows:\nLines 24–26:\nThe full 15-letter constellation enables 3.125 bits per letter for payload with error-free recovery at 33× coverage, corresponding to a density of 2.23 bits per letter for payload plus indices.\nLines 381–383:\nNotably, the 15-letter alphabet was experimentally realized at the 10,000-strand scale, achieving 3.125 bits per letter for payload (or 2.23 bits per letter for payload plus indices) and error-free recovery at 33× coverage."}, {"round": 3, "reviewer_comments": {"reviewer_2": "I appreciate the authors' detailed response and the rigorous revisions made to the manuscript, which have adequately addressed my previous concerns. Before publication, I have just two minor suggestions to harmonize the text with the supplementary data.\n1. Lines 93–97: The revised text reports a density of 1.51 bits/letter for \"payload and indices\", which appears to reference the net information density (including primers). According to Supplementary Tables 1 and 4, the correct logical density for “payload and indices” is 2.23 bits/letter. Please correct the main text to align with your tabular data.\n2. Lines 22–25 and 379–381: When highlighting the system's overall performance, pairing the sequencing coverage with the logical density of \"payload and indices\" (e.g., 33× sequencing coverage with 2.23 bits/letter) is more appropriate. This ensures the reader fully appreciates the efficiency of the error-correction mechanism (the indices) that enables such low recovery depth."}, "author_response": ""}]

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