19,631 research outputs found
Constructions of Generalized Concatenated Codes and Their Trellis-Based Decoding Complexity
In this correspondence, constructions of generalized concatenated (GC) codes with good rates and distances are presented. Some of the proposed GC codes have simpler trellis omplexity than Euclidean geometry (EG), Reed–Muller (RM), or Bose–Chaudhuri–Hocquenghem (BCH) codes of approximately the same rates and minimum distances, and in addition can be decoded with trellis-based multistage decoding up to their minimum distances. Several codes of the same length, dimension, and minimum distance as the best linear codes known are constructed
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
Information Theor
List decoding of a class of affine variety codes
Consider a polynomial in variables and a finite point ensemble . When given the leading monomial of with respect to
a lexicographic ordering we derive improved information on the possible number
of zeros of of multiplicity at least from . We then use this
information to design a list decoding algorithm for a large class of affine
variety codes.Comment: 11 pages, 5 table
On the Construction and Decoding of Concatenated Polar Codes
A scheme for concatenating the recently invented polar codes with interleaved
block codes is considered. By concatenating binary polar codes with interleaved
Reed-Solomon codes, we prove that the proposed concatenation scheme captures
the capacity-achieving property of polar codes, while having a significantly
better error-decay rate. We show that for any , and total frame
length , the parameters of the scheme can be set such that the frame error
probability is less than , while the scheme is still
capacity achieving. This improves upon 2^{-N^{0.5-\eps}}, the frame error
probability of Arikan's polar codes. We also propose decoding algorithms for
concatenated polar codes, which significantly improve the error-rate
performance at finite block lengths while preserving the low decoding
complexity
Binary Multilevel Convolutional Codes with Unequal Error Protection Capabilities
Binary multilevel convolutional codes (CCs) with unequal error protection (UEP) capabilities are studied. These codes belong to the class of generalized concatenated (GC) codes. Binary CCs are used as outer codes. Binary linear block codes of short length, and selected subcodes in their two-way subcode partition chain, are used as inner codes. Multistage decodings are presented that use Viterbi decoders operating on trellises with similar structure to that of the constituent binary CCs. Simulation results of example binary two-level CC\u27s are also reported
Feng-Rao decoding of primary codes
We show that the Feng-Rao bound for dual codes and a similar bound by
Andersen and Geil [H.E. Andersen and O. Geil, Evaluation codes from order
domain theory, Finite Fields Appl., 14 (2008), pp. 92-123] for primary codes
are consequences of each other. This implies that the Feng-Rao decoding
algorithm can be applied to decode primary codes up to half their designed
minimum distance. The technique applies to any linear code for which
information on well-behaving pairs is available. Consequently we are able to
decode efficiently a large class of codes for which no non-trivial decoding
algorithm was previously known. Among those are important families of
multivariate polynomial codes. Matsumoto and Miura in [R. Matsumoto and S.
Miura, On the Feng-Rao bound for the L-construction of algebraic geometry
codes, IEICE Trans. Fundamentals, E83-A (2000), pp. 926-930] (See also [P.
Beelen and T. H{\o}holdt, The decoding of algebraic geometry codes, in Advances
in algebraic geometry codes, pp. 49-98]) derived from the Feng-Rao bound a
bound for primary one-point algebraic geometric codes and showed how to decode
up to what is guaranteed by their bound. The exposition by Matsumoto and Miura
requires the use of differentials which was not needed in [Andersen and Geil
2008]. Nevertheless we demonstrate a very strong connection between Matsumoto
and Miura's bound and Andersen and Geil's bound when applied to primary
one-point algebraic geometric codes.Comment: elsarticle.cls, 23 pages, no figure. Version 3 added citations to the
works by I.M. Duursma and R. Pellikaa
Coding for Parallel Channels: Gallager Bounds for Binary Linear Codes with Applications to Repeat-Accumulate Codes and Variations
This paper is focused on the performance analysis of binary linear block
codes (or ensembles) whose transmission takes place over independent and
memoryless parallel channels. New upper bounds on the maximum-likelihood (ML)
decoding error probability are derived. These bounds are applied to various
ensembles of turbo-like codes, focusing especially on repeat-accumulate codes
and their recent variations which possess low encoding and decoding complexity
and exhibit remarkable performance under iterative decoding. The framework of
the second version of the Duman and Salehi (DS2) bounds is generalized to the
case of parallel channels, along with the derivation of their optimized tilting
measures. The connection between the generalized DS2 and the 1961 Gallager
bounds, addressed by Divsalar and by Sason and Shamai for a single channel, is
explored in the case of an arbitrary number of independent parallel channels.
The generalization of the DS2 bound for parallel channels enables to re-derive
specific bounds which were originally derived by Liu et al. as special cases of
the Gallager bound. In the asymptotic case where we let the block length tend
to infinity, the new bounds are used to obtain improved inner bounds on the
attainable channel regions under ML decoding. The tightness of the new bounds
for independent parallel channels is exemplified for structured ensembles of
turbo-like codes. The improved bounds with their optimized tilting measures
show, irrespectively of the block length of the codes, an improvement over the
union bound and other previously reported bounds for independent parallel
channels; this improvement is especially pronounced for moderate to large block
lengths.Comment: Submitted to IEEE Trans. on Information Theory, June 2006 (57 pages,
9 figures
Improved Nearly-MDS Expander Codes
A construction of expander codes is presented with the following three
properties:
(i) the codes lie close to the Singleton bound, (ii) they can be encoded in
time complexity that is linear in their code length, and (iii) they have a
linear-time bounded-distance decoder.
By using a version of the decoder that corrects also erasures, the codes can
replace MDS outer codes in concatenated constructions, thus resulting in
linear-time encodable and decodable codes that approach the Zyablov bound or
the capacity of memoryless channels. The presented construction improves on an
earlier result by Guruswami and Indyk in that any rate and relative minimum
distance that lies below the Singleton bound is attainable for a significantly
smaller alphabet size.Comment: Part of this work was presented at the 2004 IEEE Int'l Symposium on
Information Theory (ISIT'2004), Chicago, Illinois (June 2004). This work was
submitted to IEEE Transactions on Information Theory on January 21, 2005. To
appear in IEEE Transactions on Information Theory, August 2006. 12 page
Wet paper codes and the dual distance in steganography
In 1998 Crandall introduced a method based on coding theory to secretly embed
a message in a digital support such as an image. Later Fridrich et al. improved
this method to minimize the distortion introduced by the embedding; a process
called wet paper. However, as previously emphasized in the literature, this
method can fail during the embedding step. Here we find sufficient and
necessary conditions to guarantee a successful embedding by studying the dual
distance of a linear code. Since these results are essentially of combinatorial
nature, they can be generalized to systematic codes, a large family containing
all linear codes. We also compute the exact number of solutions and point out
the relationship between wet paper codes and orthogonal arrays
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