30,342 research outputs found
Stopping Set Distributions of Some Linear Codes
Stopping sets and stopping set distribution of an low-density parity-check
code are used to determine the performance of this code under iterative
decoding over a binary erasure channel (BEC). Let be a binary
linear code with parity-check matrix , where the rows of may be
dependent. A stopping set of with parity-check matrix is a subset
of column indices of such that the restriction of to does not
contain a row of weight one. The stopping set distribution
enumerates the number of stopping sets with size of with parity-check
matrix . Note that stopping sets and stopping set distribution are related
to the parity-check matrix of . Let be the parity-check matrix
of which is formed by all the non-zero codewords of its dual code
. A parity-check matrix is called BEC-optimal if
and has the smallest number of rows. On the
BEC, iterative decoder of with BEC-optimal parity-check matrix is an
optimal decoder with much lower decoding complexity than the exhaustive
decoder. In this paper, we study stopping sets, stopping set distributions and
BEC-optimal parity-check matrices of binary linear codes. Using finite geometry
in combinatorics, we obtain BEC-optimal parity-check matrices and then
determine the stopping set distributions for the Simplex codes, the Hamming
codes, the first order Reed-Muller codes and the extended Hamming codes.Comment: 33 pages, submitted to IEEE Trans. Inform. Theory, Feb. 201
Stopping Sets of Algebraic Geometry Codes
Abstract — Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let C be an [n, k] linear code over Fq with parity-check matrix H, wheretherowsof H may be dependent. Let [n] ={1, 2,...,n} denote the set of column indices of H. Astopping set S of C with parity-check matrix H is a subset of [n] such that the restriction of H to S does not contain a row of weight 1. The stopping set distribution {Ti (H)} n i=0 enumerates the number of stopping sets with size i of C with parity-check matrix H. Denote H ∗ , the paritycheck matrix, consisting of all the nonzero codewords in the dual code C ⊥. In this paper, we study stopping sets and stopping set distributions of some residue algebraic geometry (AG) codes with parity-check matrix H ∗. First, we give two descriptions of stopping sets of residue AG codes. For the simplest AG codes, i.e., the generalized Reed–Solomon codes, it is easy to determine all the stopping sets. Then, we consider the AG codes from elliptic curves. We use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets. Then, the stopping sets, the stopping set distribution, and the stopping distance of the AG code from an elliptic curve are reduced to the search, counting, and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively. Finally, for some special cases, we determine the stopping set distributions of the AG codes from elliptic curves. Index Terms — Algebraic geometry codes, elliptic curves, stopping distance, stopping sets, stopping set distribution, subset sum problem. I
Low-Density Parity-Check Codes From Transversal Designs With Improved Stopping Set Distributions
This paper examines the construction of low-density parity-check (LDPC) codes
from transversal designs based on sets of mutually orthogonal Latin squares
(MOLS). By transferring the concept of configurations in combinatorial designs
to the level of Latin squares, we thoroughly investigate the occurrence and
avoidance of stopping sets for the arising codes. Stopping sets are known to
determine the decoding performance over the binary erasure channel and should
be avoided for small sizes. Based on large sets of simple-structured MOLS, we
derive powerful constraints for the choice of suitable subsets, leading to
improved stopping set distributions for the corresponding codes. We focus on
LDPC codes with column weight 4, but the results are also applicable for the
construction of codes with higher column weights. Finally, we show that a
subclass of the presented codes has quasi-cyclic structure which allows
low-complexity encoding.Comment: 11 pages; to appear in "IEEE Transactions on Communications
Average Stopping Set Weight Distribution of Redundant Random Matrix Ensembles
In this paper, redundant random matrix ensembles (abbreviated as redundant
random ensembles) are defined and their stopping set (SS) weight distributions
are analyzed. A redundant random ensemble consists of a set of binary matrices
with linearly dependent rows. These linearly dependent rows (redundant rows)
significantly reduce the number of stopping sets of small size. An upper and
lower bound on the average SS weight distribution of the redundant random
ensembles are shown. From these bounds, the trade-off between the number of
redundant rows (corresponding to decoding complexity of BP on BEC) and the
critical exponent of the asymptotic growth rate of SS weight distribution
(corresponding to decoding performance) can be derived. It is shown that, in
some cases, a dense matrix with linearly dependent rows yields asymptotically
(i.e., in the regime of small erasure probability) better performance than
regular LDPC matrices with comparable parameters.Comment: 14 pages, 7 figures, Conference version to appear at the 2007 IEEE
International Symposium on Information Theory, Nice, France, June 200
Good Concatenated Code Ensembles for the Binary Erasure Channel
In this work, we give good concatenated code ensembles for the binary erasure
channel (BEC). In particular, we consider repeat multiple-accumulate (RMA) code
ensembles formed by the serial concatenation of a repetition code with multiple
accumulators, and the hybrid concatenated code (HCC) ensembles recently
introduced by Koller et al. (5th Int. Symp. on Turbo Codes & Rel. Topics,
Lausanne, Switzerland) consisting of an outer multiple parallel concatenated
code serially concatenated with an inner accumulator. We introduce stopping
sets for iterative constituent code oriented decoding using maximum a
posteriori erasure correction in the constituent codes. We then analyze the
asymptotic stopping set distribution for RMA and HCC ensembles and show that
their stopping distance hmin, defined as the size of the smallest nonempty
stopping set, asymptotically grows linearly with the block length. Thus, these
code ensembles are good for the BEC. It is shown that for RMA code ensembles,
contrary to the asymptotic minimum distance dmin, whose growth rate coefficient
increases with the number of accumulate codes, the hmin growth rate coefficient
diminishes with the number of accumulators. We also consider random puncturing
of RMA code ensembles and show that for sufficiently high code rates, the
asymptotic hmin does not grow linearly with the block length, contrary to the
asymptotic dmin, whose growth rate coefficient approaches the Gilbert-Varshamov
bound as the rate increases. Finally, we give iterative decoding thresholds for
the different code ensembles to compare the convergence properties.Comment: To appear in IEEE Journal on Selected Areas in Communications,
special issue on Capacity Approaching Code
Spectral Shape of Doubly-Generalized LDPC Codes: Efficient and Exact Evaluation
This paper analyzes the asymptotic exponent of the weight spectrum for
irregular doubly-generalized LDPC (D-GLDPC) codes. In the process, an efficient
numerical technique for its evaluation is presented, involving the solution of
a 4 x 4 system of polynomial equations. The expression is consistent with
previous results, including the case where the normalized weight or stopping
set size tends to zero. The spectral shape is shown to admit a particularly
simple form in the special case where all variable nodes are repetition codes
of the same degree, a case which includes Tanner codes; for this case it is
also shown how certain symmetry properties of the local weight distribution at
the CNs induce a symmetry in the overall weight spectral shape function.
Finally, using these new results, weight and stopping set size spectral shapes
are evaluated for some example generalized and doubly-generalized LDPC code
ensembles.Comment: 17 pages, 6 figures. To appear in IEEE Transactions on Information
Theor
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