705 research outputs found
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
Iterative Algebraic Soft-Decision List Decoding of Reed-Solomon Codes
In this paper, we present an iterative soft-decision decoding algorithm for
Reed-Solomon codes offering both complexity and performance advantages over
previously known decoding algorithms. Our algorithm is a list decoding
algorithm which combines two powerful soft decision decoding techniques which
were previously regarded in the literature as competitive, namely, the
Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation
based on adaptive parity check matrices, recently proposed by Jiang and
Narayanan. Building on the Jiang-Narayanan algorithm, we present a
belief-propagation based algorithm with a significant reduction in
computational complexity. We introduce the concept of using a
belief-propagation based decoder to enhance the soft-input information prior to
decoding with an algebraic soft-decision decoder. Our algorithm can also be
viewed as an interpolation multiplicity assignment scheme for algebraic
soft-decision decoding of Reed-Solomon codes.Comment: Submitted to IEEE for publication in Jan 200
Iterative Soft Input Soft Output Decoding of Reed-Solomon Codes by Adapting the Parity Check Matrix
An iterative algorithm is presented for soft-input-soft-output (SISO)
decoding of Reed-Solomon (RS) codes. The proposed iterative algorithm uses the
sum product algorithm (SPA) in conjunction with a binary parity check matrix of
the RS code. The novelty is in reducing a submatrix of the binary parity check
matrix that corresponds to less reliable bits to a sparse nature before the SPA
is applied at each iteration. The proposed algorithm can be geometrically
interpreted as a two-stage gradient descent with an adaptive potential
function. This adaptive procedure is crucial to the convergence behavior of the
gradient descent algorithm and, therefore, significantly improves the
performance. Simulation results show that the proposed decoding algorithm and
its variations provide significant gain over hard decision decoding (HDD) and
compare favorably with other popular soft decision decoding methods.Comment: 10 pages, 10 figures, final version accepted by IEEE Trans. on
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