238 research outputs found
Single-Deletion Single-Substitution Correcting Codes
Correcting insertions/deletions as well as substitution errors simultaneously
plays an important role in DNA-based storage systems as well as in classical
communications. This paper deals with the fundamental task of constructing
codes that can correct a single insertion or deletion along with a single
substitution. A non-asymptotic upper bound on the size of single-deletion
single-substitution correcting codes is derived, showing that the redundancy of
such a code of length has to be at least . The bound is presented
both for binary and non-binary codes while an extension to single deletion and
multiple substitutions is presented for binary codes. An explicit construction
of single-deletion single-substitution correcting codes with at most redundancy bits is derived. Note that the best known construction for this
problem has to use 3-deletion correcting codes whose best known redundancy is
roughly .Comment: Paper submitted to International Symposium on Information Theory
(ISIT) 202
Rank error-correcting pairs
Error-correcting pairs were introduced independently by Pellikaan and
K\"otter as a general method of decoding linear codes with respect to the
Hamming metric using coordinatewise products of vectors, and are used for many
well-known families of codes. In this paper, we define new types of vector
products, extending the coordinatewise product, some of which preserve symbolic
products of linearized polynomials after evaluation and some of which coincide
with usual products of matrices. Then we define rank error-correcting pairs for
codes that are linear over the extension field and for codes that are linear
over the base field, and relate both types. Bounds on the minimum rank distance
of codes and MRD conditions are given. Finally we show that some well-known
families of rank-metric codes admit rank error-correcting pairs, and show that
the given algorithm generalizes the classical algorithm using error-correcting
pairs for the Hamming metric
Efficient Majority-Logic Decoding of Short-Length Reed--Muller Codes at Information Positions
Short-length Reed--Muller codes under majority-logic decoding are of
particular importance for efficient hardware implementations in real-time and
embedded systems. This paper significantly improves Chen's two-step
majority-logic decoding method for binary Reed--Muller codes ,
, if --- systematic encoding assumed --- only errors at information
positions are to be corrected. Some general results on the minimal number of
majority gates are presented that are particularly good for short codes.
Specifically, with its importance in applications as a 3-error-correcting,
self-dual code, the smallest non-trivial example, of dimension
16 and length 32, is investigated in detail. Further, the decoding complexity
of our procedure is compared with that of Chen's decoding algorithm for various
Reed--Muller codes up to length .Comment: 8 pages; to appear in "IEEE Transactions on Communications
An Improvement of Non-binary Code Correcting Single b-Burst of Insertions or Deletions
This paper constructs a non-binary code correcting a single -burst of
insertions or deletions with a large cardinality. This paper also proposes a
decoding algorithm of this code and evaluates a lower bound of the cardinality
of this code. Moreover, we evaluate an asymptotic upper bound on the
cardinality of codes which correct a single burst of insertions or deletions.Comment: 7 pages, accepted to ISITA 201
An Improvement to Levenshtein's Upper Bound on the Cardinality of Deletion Correcting Codes
We consider deletion correcting codes over a q-ary alphabet. It is well known
that any code capable of correcting s deletions can also correct any
combination of s total insertions and deletions. To obtain asymptotic upper
bounds on code size, we apply a packing argument to channels that perform
different mixtures of insertions and deletions. Even though the set of codes is
identical for all of these channels, the bounds that we obtain vary. Prior to
this work, only the bounds corresponding to the all insertion case and the all
deletion case were known. We recover these as special cases. The bound from the
all deletion case, due to Levenshtein, has been the best known for more than
forty five years. Our generalized bound is better than Levenshtein's bound
whenever the number of deletions to be corrected is larger than the alphabet
size
Silence is Golden: exploiting jamming and radio silence to communicate
Jamming techniques require just moderate resources to be deployed, while
their effectiveness in disrupting communications is unprecedented. In this
paper we introduce several contributions to jamming mitigation. In particular,
we introduce a novel adversary model that has both (unlimited) jamming reactive
capabilities as well as powerful (but limited) proactive jamming capabilities.
Under this powerful but yet realistic adversary model, the communication
bandwidth provided by current anti-jamming solutions drops to zero. We then
present Silence is Golden (SiG): a novel anti jamming protocol that,
introducing a tunable, asymmetric communication channel, is able to mitigate
the adversary capabilities, enabling the parties to communicate. For instance,
with SiG it is possible to deliver a 128 bits long message with a probability
greater than 99% in 4096 time slots in the presence of a jammer that jams all
the on-the-fly communications and the 74% of the silent radio spectrum---while
competing proposals simply fail. The provided solution enjoys a thorough
theoretical analysis and is supported by extensive experimental results,
showing the viability of our proposal
The Error-Pattern-Correcting Turbo Equalizer
The error-pattern correcting code (EPCC) is incorporated in the design of a
turbo equalizer (TE) with aim to correct dominant error events of the
inter-symbol interference (ISI) channel at the output of its matching Viterbi
detector. By targeting the low Hamming-weight interleaved errors of the outer
convolutional code, which are responsible for low Euclidean-weight errors in
the Viterbi trellis, the turbo equalizer with an error-pattern correcting code
(TE-EPCC) exhibits a much lower bit-error rate (BER) floor compared to the
conventional non-precoded TE, especially for high rate applications. A
maximum-likelihood upper bound is developed on the BER floor of the TE-EPCC for
a generalized two-tap ISI channel, in order to study TE-EPCC's signal-to-noise
ratio (SNR) gain for various channel conditions and design parameters. In
addition, the SNR gain of the TE-EPCC relative to an existing precoded TE is
compared to demonstrate the present TE's superiority for short interleaver
lengths and high coding rates.Comment: This work has been submitted to the special issue of the IEEE
Transactions on Information Theory titled: "Facets of Coding Theory: from
Algorithms to Networks". This work was supported in part by the NSF
Theoretical Foundation Grant 0728676
Quickest Sequence Phase Detection
A phase detection sequence is a length- cyclic sequence, such that the
location of any length- contiguous subsequence can be determined from a
noisy observation of that subsequence. In this paper, we derive bounds on the
minimal possible in the limit of , and describe some sequence
constructions. We further consider multiple phase detection sequences, where
the location of any length- contiguous subsequence of each sequence can be
determined simultaneously from a noisy mixture of those subsequences. We study
the optimal trade-offs between the lengths of the sequences, and describe some
sequence constructions. We compare these phase detection problems to their
natural channel coding counterparts, and show a strict separation between the
fundamental limits in the multiple sequence case. Both adversarial and
probabilistic noise models are addressed.Comment: To appear in the IEEE Transactions on Information Theor
Tutorial on algebraic deletion correction codes
The deletion channel is known to be a notoriously diffcult channel to design
error-correction codes for. In spite of this difficulty, there are some
beautiful code constructions which give some intuition about the channel and
about what good deletion codes look like. In this tutorial we will take a look
at some of them. This document is a transcript of my talk at the coding theory
reading group on some interesting works on deletion channel. It is not intended
to be an exhaustive survey of works on deletion channel, but more as a tutorial
to some of the important and cute ideas in this area. For a comprehensive
survey, we refer the reader to the cited sources and surveys.
We also provide an implementation of VT codes that correct single
insertion/deletion errors for general alphabets at
https://github.com/shubhamchandak94/VT_codes/
Canonical form of linear subspaces and coding invariants: the poset metric point of view
In this work we introduce the concept of a sub-space decomposition, subject
to a partition of the coordinates. Considering metrics determined by partial
orders in the set of coordinates, the so called poset metrics, we show the
existence of maximal decompositions according to the metric. These
decompositions turns to be an important tool to obtain the canonical form for
codes over any poset metrics and to obtain bounds for important invariants such
as the packing radius of a linear subspace. Furthermore, using maximal
decompositions, we are able to reduce and optimize the full lookup table
algorithm for the syndrome decoding process.Comment: 28 page
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