The paper introduces the concept of "strong stuck-at codes", which generalize the well-studied problem of coding for "stuck-at" errors. In the traditional stuck-at code framework, a message is encoded into a one-dimensional binary vector, where a certain number of bits are "frozen" and cannot be altered by the encoder. The decoder, aware of the proportion of frozen bits but not their specific positions, is responsible for deciphering the intended message.
The authors consider a more challenging version of this problem where the decoder does not know the fraction of frozen bits. They construct explicit and efficient encoding and decoding algorithms that get arbitrarily close to capacity in this scenario. This is the first fully explicit construction of stuck-at codes that approach capacity.
The key steps are:
An existential result showing that it is possible to encode at virtually the same rate as a conventional stuck-at code even when the size (or a bound on the size) of the set of frozen components is not available to the decoder.
A construction with a clean transmission assumption, where the encoder can transmit a small amount of metadata to the decoder in an errorless manner. This improves upon prior work by reducing the amount of metadata that needs to be transmitted.
The main construction, which removes the clean transmission assumption and encodes the metadata directly into the cover object, allowing for efficient encoding and decoding without any side channel.
The authors prove the correctness and analyze the rate and complexity of their constructions, showing they can approach the information-theoretic capacity with efficient algorithms.
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arxiv.org
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