51 research outputs found
A Lower Bound on the List-Decodability of Insdel Codes
For codes equipped with metrics such as Hamming metric, symbol pair metric or
cover metric, the Johnson bound guarantees list-decodability of such codes.
That is, the Johnson bound provides a lower bound on the list-decoding radius
of a code in terms of its relative minimum distance , list size and
the alphabet size For study of list-decodability of codes with insertion
and deletion errors (we call such codes insdel codes), it is natural to ask the
open problem whether there is also a Johnson-type bound. The problem was first
investigated by Wachter-Zeh and the result was amended by Hayashi and Yasunaga
where a lower bound on the list-decodability for insdel codes was derived.
The main purpose of this paper is to move a step further towards solving the
above open problem. In this work, we provide a new lower bound for the
list-decodability of an insdel code. As a consequence, we show that unlike the
Johnson bound for codes under other metrics that is tight, the bound on
list-decodability of insdel codes given by Hayashi and Yasunaga is not tight.
Our main idea is to show that if an insdel code with a given Levenshtein
distance is not list-decodable with list size , then the list decoding
radius is lower bounded by a bound involving and . In other words, if
the list decoding radius is less than this lower bound, the code must be
list-decodable with list size . At the end of the paper we use such bound to
provide an insdel-list-decodability bound for various well-known codes, which
has not been extensively studied before
Synchronization Strings: Explicit Constructions, Local Decoding, and Applications
This paper gives new results for synchronization strings, a powerful
combinatorial object that allows to efficiently deal with insertions and
deletions in various communication settings:
We give a deterministic, linear time synchronization string
construction, improving over an time randomized construction.
Independently of this work, a deterministic time
construction was just put on arXiv by Cheng, Li, and Wu. We also give a
deterministic linear time construction of an infinite synchronization string,
which was not known to be computable before. Both constructions are highly
explicit, i.e., the symbol can be computed in time.
This paper also introduces a generalized notion we call
long-distance synchronization strings that allow for local and very fast
decoding. In particular, only time and access to logarithmically
many symbols is required to decode any index.
We give several applications for these results:
For any we provide an insdel correcting
code with rate which can correct any fraction
of insdel errors in time. This near linear computational
efficiency is surprising given that we do not even know how to compute the
(edit) distance between the decoding input and output in sub-quadratic time. We
show that such codes can not only efficiently recover from fraction of
insdel errors but, similar to [Schulman, Zuckerman; TransInf'99], also from any
fraction of block transpositions and replications.
We show that highly explicitness and local decoding allow for
infinite channel simulations with exponentially smaller memory and decoding
time requirements. These simulations can be used to give the first near linear
time interactive coding scheme for insdel errors
Improved Asymptotic Bounds for Codes Correcting Insertions and Deletions
This paper studies the cardinality of codes correcting insertions and
deletions. We give an asymptotically improved upper bound on code size. The
bound is obtained by utilizing the asymmetric property of list decoding for
insertions and deletions.Comment: 9 pages, 2 fugure
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