Optimal constrained layouts for fixed codes

Determine, for a fixed generator matrix of a linear code, a fixed number of containers, and prescribed container sizes, the partition that minimizes the expected distributed coverage depth; in particular, characterize optimal balanced layouts, identify codes admitting balanced one-round-perfect layouts, determine the minimum total number of reads when one-round perfection is impossible, and construct nearly balanced layouts when the number of strands is not divisible by the number of containers.

Background

The paper derives exact expressions for the expected recovery time associated with a generator matrix and an ordered partition, and it constructs several optimal but generally unbalanced layouts. The staircase construction achieves one-round perfection when the number of containers equals the message dimension, but its container sizes can differ substantially.

The authors state that the optimization problem remains unresolved when the code and container capacities are fixed, including the balanced-container case. This problem concerns the combinatorial organization of a fixed code across containers rather than the selection of the code itself.

References

Even when the partition is required to be balanced, a structural characterization of the optimal layout is not known for a general code.

— The Coverage Depth Problem in Distributed DNA Data Storage  (2610.02931 - Kong et al., 2 Oct 2026) in Section 6, Conclusion and Open Problems, item 1

While this problem is completely settled when MDS codes exist, it remains open for parameters where no MDS code is available (e.g., over small fields where $q < n - 3$).

— The Coverage Depth Problem in Distributed DNA Data Storage  (2610.02931 - Kong et al., 2 Oct 2026) in Section 6, Conclusion and Open Problems, item 2

It remains to determine the optimal distributed coverage depth in the remaining regimes, especially when $M\mid k$, and in the binary cases where $M\mid(k+1)$.

— The Coverage Depth Problem in Distributed DNA Data Storage  (2610.02931 - Kong et al., 2 Oct 2026) in Section 6, Conclusion and Open Problems, item 3

An asynchronous variant, in which a total read budget is allocated is discussed as an open problem in Section~\ref{sec:conclusion}.

— The Coverage Depth Problem in Distributed DNA Data Storage  (2610.02931 - Kong et al., 2 Oct 2026) in Remark 2.1, Section 2.1; referenced again in Section 6