8,962 research outputs found
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
Synchronization Strings: Codes for Insertions and Deletions Approaching the Singleton Bound
We introduce synchronization strings as a novel way of efficiently dealing
with synchronization errors, i.e., insertions and deletions. Synchronization
errors are strictly more general and much harder to deal with than commonly
considered half-errors, i.e., symbol corruptions and erasures. For every
, synchronization strings allow to index a sequence with an
size alphabet such that one can efficiently transform
synchronization errors into half-errors. This powerful new
technique has many applications. In this paper, we focus on designing insdel
codes, i.e., error correcting block codes (ECCs) for insertion deletion
channels.
While ECCs for both half-errors and synchronization errors have been
intensely studied, the later has largely resisted progress. Indeed, it took
until 1999 for the first insdel codes with constant rate, constant distance,
and constant alphabet size to be constructed by Schulman and Zuckerman. Insdel
codes for asymptotically large or small noise rates were given in 2016 by
Guruswami et al. but these codes are still polynomially far from the optimal
rate-distance tradeoff. This makes the understanding of insdel codes up to this
work equivalent to what was known for regular ECCs after Forney introduced
concatenated codes in his doctoral thesis 50 years ago.
A direct application of our synchronization strings based indexing method
gives a simple black-box construction which transforms any ECC into an equally
efficient insdel code with a slightly larger alphabet size. This instantly
transfers much of the highly developed understanding for regular ECCs over
large constant alphabets into the realm of insdel codes. Most notably, we
obtain efficient insdel codes which get arbitrarily close to the optimal
rate-distance tradeoff given by the Singleton bound for the complete noise
spectrum
Guess & Check Codes for Deletions and Synchronization
We consider the problem of constructing codes that can correct
deletions occurring in an arbitrary binary string of length bits.
Varshamov-Tenengolts (VT) codes can correct all possible single deletions
with an asymptotically optimal redundancy. Finding similar codes
for deletions is an open problem. We propose a new family of
codes, that we call Guess & Check (GC) codes, that can correct, with high
probability, a constant number of deletions occurring at uniformly
random positions within an arbitrary string. The GC codes are based on MDS
codes and have an asymptotically optimal redundancy that is . We provide deterministic polynomial time encoding and decoding schemes for
these codes. We also describe the applications of GC codes to file
synchronization.Comment: Accepted in ISIT 201
Spectrum of Sizes for Perfect Deletion-Correcting Codes
One peculiarity with deletion-correcting codes is that perfect
-deletion-correcting codes of the same length over the same alphabet can
have different numbers of codewords, because the balls of radius with
respect to the Levenshte\u{\i}n distance may be of different sizes. There is
interest, therefore, in determining all possible sizes of a perfect
-deletion-correcting code, given the length and the alphabet size~.
In this paper, we determine completely the spectrum of possible sizes for
perfect -ary 1-deletion-correcting codes of length three for all , and
perfect -ary 2-deletion-correcting codes of length four for almost all ,
leaving only a small finite number of cases in doubt.Comment: 23 page
Non-asymptotic Upper Bounds for Deletion Correcting Codes
Explicit non-asymptotic upper bounds on the sizes of multiple-deletion
correcting codes are presented. In particular, the largest single-deletion
correcting code for -ary alphabet and string length is shown to be of
size at most . An improved bound on the asymptotic
rate function is obtained as a corollary. Upper bounds are also derived on
sizes of codes for a constrained source that does not necessarily comprise of
all strings of a particular length, and this idea is demonstrated by
application to sets of run-length limited strings.
The problem of finding the largest deletion correcting code is modeled as a
matching problem on a hypergraph. This problem is formulated as an integer
linear program. The upper bound is obtained by the construction of a feasible
point for the dual of the linear programming relaxation of this integer linear
program.
The non-asymptotic bounds derived imply the known asymptotic bounds of
Levenshtein and Tenengolts and improve on known non-asymptotic bounds.
Numerical results support the conjecture that in the binary case, the
Varshamov-Tenengolts codes are the largest single-deletion correcting codes.Comment: 18 pages, 4 figure
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