1,500 research outputs found
Asymptotic bounds for spherical codes
The set of all error-correcting codes C over a fixed finite alphabet F of
cardinality q determines the set of code points in the unit square with
coordinates (R(C), delta (C)):= (relative transmission rate, relative minimal
distance). The central problem of the theory of such codes consists in
maximizing simultaneously the transmission rate of the code and the relative
minimum Hamming distance between two different code words. The classical
approach to this problem explored in vast literature consists in the inventing
explicit constructions of "good codes" and comparing new classes of codes with
earlier ones. Less classical approach studies the geometry of the whole set of
code points (R,delta) (with q fixed), at first independently of its
computability properties, and only afterwords turning to the problems of
computability, analogies with statistical physics etc. The main purpose of this
article consists in extending this latter strategy to domain of spherical
codes.Comment: 34 pages amstex, 3 figure
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
Mutually Uncorrelated Primers for DNA-Based Data Storage
We introduce the notion of weakly mutually uncorrelated (WMU) sequences,
motivated by applications in DNA-based data storage systems and for
synchronization of communication devices. WMU sequences are characterized by
the property that no sufficiently long suffix of one sequence is the prefix of
the same or another sequence. WMU sequences used for primer design in DNA-based
data storage systems are also required to be at large mutual Hamming distance
from each other, have balanced compositions of symbols, and avoid primer-dimer
byproducts. We derive bounds on the size of WMU and various constrained WMU
codes and present a number of constructions for balanced, error-correcting,
primer-dimer free WMU codes using Dyck paths, prefix-synchronized and cyclic
codes.Comment: 14 pages, 3 figures, 1 Table. arXiv admin note: text overlap with
arXiv:1601.0817
Constructions and bounds for codes with restricted overlaps
Non-overlapping codes have been studied for almost 60 years. In such a code,
no proper, non-empty prefix of any codeword is a suffix of any codeword. In
this paper, we study codes in which overlaps of certain specified sizes are
forbidden. We prove some general bounds and we give several constructions in
the case of binary codes. Our techniques also allow us to provide an
alternative, elementary proof of a lower bound on non-overlapping codes due to
Levenshtein in 1964.Comment: 25 pages. Extra citations, typos corrected and explanations expande
Constructions and bounds for codes with restricted overlaps
Non-overlapping codes have been studied for almost 60 years. In such a code, no proper, non-empty prefix of any codeword is a suffix of any codeword. In this paper, we study codes in which over-laps of certain specified sizes are forbidden. We prove some general bounds and we give several constructions in the case of binary codes. Our techniques also allow us to provide an alternative, elementary proof of a lower bound on non-overlapping codes due to Levenshtein [9] in 1964
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