Rank modulation codes for DNA storage

Synthesis of DNA molecules offers unprecedented advances in storage technology. Yet, the microscopic world in which these molecules reside induces error patterns that are fundamentally different from their digital counterparts. Hence, to maintain reliability in reading and writing, new coding schemes must be developed. In a reading technique called shotgun sequencing, a long DNA string is read in a sliding window fashion, and a profile vector is produced. It was recently suggested by Kiah et al. that such a vector can represent the permutation which is induced by its entries, and hence a rank-modulation scheme arises. Although this interpretation suggests high error tolerance, it is unclear which permutations are feasible, and how to produce a DNA string whose profile vector induces a given permutation. In this paper, by observing some necessary conditions, an upper bound for the number of feasible permutations is given. Further, a technique for deciding the feasibility of a permutation is devised. By using insights from this technique, an algorithm for producing a considerable number of feasible permutations is given, which applies to any alphabet size and any window length.

💡 Research Summary

The paper investigates the use of rank‑modulation coding for DNA‑based data storage, where the output of shotgun sequencing is a profile vector that counts the occurrences of each ℓ‑mer in a DNA string. Instead of using the absolute counts, the authors propose to encode information solely in the relative ordering (the permutation) of these counts, a scheme known as rank‑modulation.

Key contributions are as follows:

1. Modeling with De Bruijn graphs – For an alphabet Σ of size q and a window length ℓ, the ℓ‑order De Bruijn graph Gℓ,q has vertices Σℓ and directed edges that correspond to (ℓ+1)‑mers. A circular DNA string corresponds to a closed walk in this graph, and its profile vector pₓ is the multiset of edge multiplicities.

2. Necessary feasibility conditions – A permutation π∈S_{q^ℓ} can be realized only if (i) all entries of the profile vector are distinct (so the permutation is well‑defined) and (ii) the flow‑conservation constraints hold: for every (ℓ‑1)‑mer w, the total count of edges entering w equals the total count of edges leaving w. This condition follows directly from the fact that each occurrence of w as a prefix must be matched by an occurrence as a suffix in a closed walk.

3. Upper bound on the number of feasible permutations – By coloring each edge of Gℓ,q green when π orders its tail before its head and red otherwise, the authors show that if any induced subgraph G(v) (where v∈Σ^{ℓ‑1}) contains an all‑red or all‑green perfect matching, then π violates flow‑conservation and is infeasible. Using collections of mutually independent (ℓ‑1)‑mers, they count how many permutations contain at least one such forbidden matching. This yields a combinatorial upper bound:
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