3,946 research outputs found
Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs
A novel technique, based on the pseudo-random properties of certain graphs known as expanders, is used to obtain novel simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling, and then regrouping the code coordinates. For any fixed (small) rate, and for a sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF(2)) as well. Although these concatenated codes lie below the Zyablov bound, they are still superior to previously known explicit constructions in the zero-rate neighborhood
Estimates on the Size of Symbol Weight Codes
The study of codes for powerlines communication has garnered much interest
over the past decade. Various types of codes such as permutation codes,
frequency permutation arrays, and constant composition codes have been proposed
over the years. In this work we study a type of code called the bounded symbol
weight codes which was first introduced by Versfeld et al. in 2005, and a
related family of codes that we term constant symbol weight codes. We provide
new upper and lower bounds on the size of bounded symbol weight and constant
symbol weight codes. We also give direct and recursive constructions of codes
for certain parameters.Comment: 14 pages, 4 figure
Construction of Almost Disjunct Matrices for Group Testing
In a \emph{group testing} scheme, a set of tests is designed to identify a
small number of defective items among a large set (of size ) of items.
In the non-adaptive scenario the set of tests has to be designed in one-shot.
In this setting, designing a testing scheme is equivalent to the construction
of a \emph{disjunct matrix}, an matrix where the union of supports
of any columns does not contain the support of any other column. In
principle, one wants to have such a matrix with minimum possible number of
rows (tests). One of the main ways of constructing disjunct matrices relies on
\emph{constant weight error-correcting codes} and their \emph{minimum
distance}. In this paper, we consider a relaxed definition of a disjunct matrix
known as \emph{almost disjunct matrix}. This concept is also studied under the
name of \emph{weakly separated design} in the literature. The relaxed
definition allows one to come up with group testing schemes where a
close-to-one fraction of all possible sets of defective items are identifiable.
Our main contribution is twofold. First, we go beyond the minimum distance
analysis and connect the \emph{average distance} of a constant weight code to
the parameters of an almost disjunct matrix constructed from it. Our second
contribution is to explicitly construct almost disjunct matrices based on our
average distance analysis, that have much smaller number of rows than any
previous explicit construction of disjunct matrices. The parameters of our
construction can be varied to cover a large range of relations for and .Comment: 15 Page
List Decoding Tensor Products and Interleaved Codes
We design the first efficient algorithms and prove new combinatorial bounds
for list decoding tensor products of codes and interleaved codes. We show that
for {\em every} code, the ratio of its list decoding radius to its minimum
distance stays unchanged under the tensor product operation (rather than
squaring, as one might expect). This gives the first efficient list decoders
and new combinatorial bounds for some natural codes including multivariate
polynomials where the degree in each variable is bounded. We show that for {\em
every} code, its list decoding radius remains unchanged under -wise
interleaving for an integer . This generalizes a recent result of Dinur et
al \cite{DGKS}, who proved such a result for interleaved Hadamard codes
(equivalently, linear transformations). Using the notion of generalized Hamming
weights, we give better list size bounds for {\em both} tensoring and
interleaving of binary linear codes. By analyzing the weight distribution of
these codes, we reduce the task of bounding the list size to bounding the
number of close-by low-rank codewords. For decoding linear transformations,
using rank-reduction together with other ideas, we obtain list size bounds that
are tight over small fields.Comment: 32 page
- …