57,869 research outputs found
Asymmetric Lee Distance Codes for DNA-Based Storage
We consider a new family of codes, termed asymmetric Lee distance codes, that
arise in the design and implementation of DNA-based storage systems and systems
with parallel string transmission protocols. The codewords are defined over a
quaternary alphabet, although the results carry over to other alphabet sizes;
furthermore, symbol confusability is dictated by their underlying binary
representation. Our contributions are two-fold. First, we demonstrate that the
new distance represents a linear combination of the Lee and Hamming distance
and derive upper bounds on the size of the codes under this metric based on
linear programming techniques. Second, we propose a number of code
constructions which imply lower bounds
Semidefinite programming bounds for Lee codes
For , let denote the maximum cardinality
of a code with minimum Lee distance at least ,
where denotes the cyclic group of order . We consider a
semidefinite programming bound based on triples of codewords, which bound can
be computed efficiently using symmetry reductions, resulting in several new
upper bounds on . The technique also yields an upper bound on the
independent set number of the -th strong product power of the circular graph
, which number is related to the Shannon capacity of . Here
is the graph with vertex set , in which two vertices
are adjacent if and only if their distance (mod ) is strictly less than .
The new bound does not seem to improve significantly over the bound obtained
from Lov\'asz theta-function, except for very small .Comment: 14 pages. arXiv admin note: text overlap with arXiv:1703.0517
Correcting Charge-Constrained Errors in the Rank-Modulation Scheme
We investigate error-correcting codes for a the
rank-modulation scheme with an application to flash memory
devices. In this scheme, a set of n cells stores information in the
permutation induced by the different charge levels of the individual
cells. The resulting scheme eliminates the need for discrete
cell levels, overcomes overshoot errors when programming cells (a
serious problem that reduces the writing speed), and mitigates the
problem of asymmetric errors. In this paper, we study the properties
of error-correcting codes for charge-constrained errors in the
rank-modulation scheme. In this error model the number of errors
corresponds to the minimal number of adjacent transpositions required
to change a given stored permutation to another erroneous
one—a distance measure known as Kendall’s τ-distance.We show
bounds on the size of such codes, and use metric-embedding techniques
to give constructions which translate a wealth of knowledge
of codes in the Lee metric to codes over permutations in Kendall’s
τ-metric. Specifically, the one-error-correcting codes we construct
are at least half the ball-packing upper bound
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