15,566 research outputs found
Optimal Codes Detecting Deletions in Concatenated Binary Strings Applied to Trace Reconstruction
Consider two or more strings that are
concatenated to form . Suppose that up to deletions occur in each of the
concatenated strings. Since deletions alter the lengths of the strings, a
fundamental question to ask is: how much redundancy do we need to introduce in
in order to recover the boundaries of
? This boundary problem is equivalent to the
problem of designing codes that can detect the exact number of deletions in
each concatenated string. In this work, we answer the question above by first
deriving converse results that give lower bounds on the redundancy of
deletion-detecting codes. Then, we present a marker-based code construction
whose redundancy is asymptotically optimal in among all families of
deletion-detecting codes, and exactly optimal among all block-by-block
decodable codes. To exemplify the usefulness of such deletion-detecting codes,
we apply our code to trace reconstruction and design an efficient coded
reconstruction scheme that requires a constant number of traces.Comment: Accepted for publication in the IEEE Transactions on Information
Theory. arXiv admin note: substantial text overlap with arXiv:2207.05126,
arXiv:2105.0021
Lower Bounds on the Redundancy of Huffman Codes with Known and Unknown Probabilities
In this paper we provide a method to obtain tight lower bounds on the minimum
redundancy achievable by a Huffman code when the probability distribution
underlying an alphabet is only partially known. In particular, we address the
case where the occurrence probabilities are unknown for some of the symbols in
an alphabet. Bounds can be obtained for alphabets of a given size, for
alphabets of up to a given size, and for alphabets of arbitrary size. The
method operates on a Computer Algebra System, yielding closed-form numbers for
all results. Finally, we show the potential of the proposed method to shed some
light on the structure of the minimum redundancy achievable by the Huffman
code
- …