Compact Error-Resilient Self-Assembly of Recursively Defined Patterns
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A limitation to molecular implementations of tile-based self-assembly systems is the high rate of mismatch errors which has been observed to be between 1 and 10 intrinsic error rate ϵ prohibitively slows the growth rate of the system. This has motivated the development of techniques to redundantly encode information in the tiles of a system in such a way that the rate of mismatch errors in the final assembly is reduced even without a reduction in ϵ. Winfree and Bekbolatov, and Chen and Goel, introduced such error-resilient systems that reduce the mismatch error rate to ϵ^2 by replacing each tile in an error-prone system with a k × k block of tiles in the error-resilient system, but this increases the number of tile types used by a factor of k^2, and the scale of the pattern produced by a factor of k. Reif, Sahu and Yin, and Sahu and Reif, introduced compact error-resilient systems for the self-assembly of Boolean arrays that reduce the mismatch error rate to ϵ^2 without increasing the scale of the pattern produced. In this paper, we give a technique to design compact error-resilient systems for the self-assembly of the recursively defined patterns introduced by Kautz and Lathrop. We show that our compact error-resilient systems reduce the mismatch error rate to ϵ^2 by using the independent error model introduced by Sahu and Reif. Surprisingly, our error-resilient systems use the same number of tile types as the error-prone system from which they are constructed.
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