104 research outputs found
Strong Singleton type upper bounds for linear insertion-deletion codes
The insertion-deletion codes was motivated to correct the synchronization
errors. In this paper we prove several Singleton type upper bounds on the
insdel distances of linear insertion-deletion codes, based on the generalized
Hamming weights and the formation of minimum Hamming weight codewords. Our
bound are stronger than some previous known bounds. These upper bounds are
valid for any fixed ordering of coordinate positions. We apply these upper
bounds to some binary cyclic codes and binary Reed-Muller codes with any
coordinate ordering, and some binary Reed-Muller codes and one
algebraic-geometric code with certain special coordinate ordering.Comment: 22 pages, references update
Computationally Relaxed Locally Decodable Codes, Revisited
We revisit computationally relaxed locally decodable codes (crLDCs) (Blocki
et al., Trans. Inf. Theory '21) and give two new constructions. Our first
construction is a Hamming crLDC that is conceptually simpler than prior
constructions, leveraging digital signature schemes and an appropriately chosen
Hamming code. Our second construction is an extension of our Hamming crLDC to
handle insertion-deletion (InsDel) errors, yielding an InsDel crLDC. This
extension crucially relies on the noisy binary search techniques of Block et
al. (FSTTCS '20) to handle InsDel errors. Both crLDC constructions have binary
codeword alphabets, are resilient to a constant fraction of Hamming and InsDel
errors, respectively, and under suitable parameter choices have
poly-logarithmic locality and encoding length linear in the message length and
polynomial in the security parameter. These parameters compare favorably to
prior constructions in the poly-logarithmic locality regime
Efficient Linear and Affine Codes for Correcting Insertions/Deletions
This paper studies \emph{linear} and \emph{affine} error-correcting codes for
correcting synchronization errors such as insertions and deletions. We call
such codes linear/affine insdel codes.
Linear codes that can correct even a single deletion are limited to have
information rate at most (achieved by the trivial 2-fold repetition
code). Previously, it was (erroneously) reported that more generally no
non-trivial linear codes correcting deletions exist, i.e., that the
-fold repetition codes and its rate of are basically optimal
for any . We disprove this and show the existence of binary linear codes of
length and rate just below capable of correcting
insertions and deletions. This identifies rate as a sharp threshold for
recovery from deletions for linear codes, and reopens the quest for a better
understanding of the capabilities of linear codes for correcting
insertions/deletions.
We prove novel outer bounds and existential inner bounds for the rate vs.
(edit) distance trade-off of linear insdel codes. We complement our existential
results with an efficient synchronization-string-based transformation that
converts any asymptotically-good linear code for Hamming errors into an
asymptotically-good linear code for insdel errors. Lastly, we show that the
-rate limitation does not hold for affine codes by giving an
explicit affine code of rate which can efficiently correct a
constant fraction of insdel errors
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
Multiple Packing: Lower Bounds via Infinite Constellations
We study the problem of high-dimensional multiple packing in Euclidean space.
Multiple packing is a natural generalization of sphere packing and is defined
as follows. Let and . A multiple packing is a
set of points in such that any point in lies in the intersection of at most balls of radius around points in . Given a well-known connection
with coding theory, multiple packings can be viewed as the Euclidean analog of
list-decodable codes, which are well-studied for finite fields. In this paper,
we derive the best known lower bounds on the optimal density of list-decodable
infinite constellations for constant under a stronger notion called
average-radius multiple packing. To this end, we apply tools from
high-dimensional geometry and large deviation theory.Comment: The paper arXiv:2107.05161 has been split into three parts with new
results added and significant revision. This paper is one of the three parts.
The other two are arXiv:2211.04408 and arXiv:2211.0440
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