Dimension Augmentation: Enhancing Tile Systems without Resolution Loss
This paper discusses the inherent resolution loss in error-correcting schemes, particularly in compact proofreading methods that require an exponential increase in tile numbers. We introduce the “dimension augmentation” technique as a solution, allowing a linear increase in tile count without resolution loss in original two dimensions. The proposed method is shown to be provably correct for the parity system. However, general correctness remains an open question. Our findings propose a significant advancement in optimizing tile-based error correction.
Dimension Augmentation: Enhancing Tile Systems without Resolution Loss
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Presentation Transcript
Resolution loss • Error-correcting schemes mentioned above always generates a resolution loss. • Compact proofreading [Sahu, Reif, Yin ’04] is proposed to remove the resolution loss. • However, [Soloveichik, Winfree ’05] showed that compact proofreading requires increasing number of tiles exponentially. • We propose “dimension augmentation” technique to solve this problem.
O E O E E O E E E O O O E O O O E O E E E E O O O E E E E E O O O O O E Dimension Augmentation k The original system Odd and even duplicates k duplicates on the third dimension
Dimension Augmentation • No resolution loss in the original two dimensions. • Increase the number of tiles linearly. • Provably correct for an example tile system called “parity system” • Correctness in general remains open.
