3,196 research outputs found
An Upper Bound on the Minimum Distance of LDPC Codes over GF(q)
In [1] a syndrome counting based upper bound on the minimum distance of
regular binary LDPC codes is given. In this paper we extend the bound to the
case of irregular and generalized LDPC codes over GF(q). The comparison to the
lower bound for LDPC codes over GF(q) and to the upper bound for non-binary
codes is done. The new bound is shown to lie under the Gilbert-Varshamov bound
at high rates.Comment: 4 pages, submitted to ISIT 201
New constructions of CSS codes obtained by moving to higher alphabets
We generalize a construction of non-binary quantum LDPC codes over \F_{2^m}
due to \cite{KHIS11a} and apply it in particular to toric codes. We obtain in
this way not only codes with better rates than toric codes but also improve
dramatically the performance of standard iterative decoding. Moreover, the new
codes obtained in this fashion inherit the distance properties of the
underlying toric codes and have therefore a minimum distance which grows as the
square root of the length of the code for fixed .Comment: 9 pages, 9 figures, full version of a paper submitted to the IEEE
Symposium on Information Theor
A STUDY OF LINEAR ERROR CORRECTING CODES
Since Shannon's ground-breaking work in 1948, there have been two main development streams
of channel coding in approaching the limit of communication channels, namely classical coding
theory which aims at designing codes with large minimum Hamming distance and probabilistic
coding which places the emphasis on low complexity probabilistic decoding using long codes built
from simple constituent codes. This work presents some further investigations in these two channel
coding development streams.
Low-density parity-check (LDPC) codes form a class of capacity-approaching codes with sparse
parity-check matrix and low-complexity decoder Two novel methods of constructing algebraic binary
LDPC codes are presented. These methods are based on the theory of cyclotomic cosets, idempotents
and Mattson-Solomon polynomials, and are complementary to each other. The two methods
generate in addition to some new cyclic iteratively decodable codes, the well-known Euclidean and
projective geometry codes. Their extension to non binary fields is shown to be straightforward.
These algebraic cyclic LDPC codes, for short block lengths, converge considerably well under iterative
decoding. It is also shown that for some of these codes, maximum likelihood performance may
be achieved by a modified belief propagation decoder which uses a different subset of 7^ codewords
of the dual code for each iteration.
Following a property of the revolving-door combination generator, multi-threaded minimum
Hamming distance computation algorithms are developed. Using these algorithms, the previously
unknown, minimum Hamming distance of the quadratic residue code for prime 199 has been evaluated.
In addition, the highest minimum Hamming distance attainable by all binary cyclic codes
of odd lengths from 129 to 189 has been determined, and as many as 901 new binary linear codes
which have higher minimum Hamming distance than the previously considered best known linear
code have been found.
It is shown that by exploiting the structure of circulant matrices, the number of codewords
required, to compute the minimum Hamming distance and the number of codewords of a given
Hamming weight of binary double-circulant codes based on primes, may be reduced. A means
of independently verifying the exhaustively computed number of codewords of a given Hamming
weight of these double-circulant codes is developed and in coiyunction with this, it is proved that
some published results are incorrect and the correct weight spectra are presented. Moreover, it is
shown that it is possible to estimate the minimum Hamming distance of this family of prime-based
double-circulant codes.
It is shown that linear codes may be efficiently decoded using the incremental correlation Dorsch
algorithm. By extending this algorithm, a list decoder is derived and a novel, CRC-less error detection
mechanism that offers much better throughput and performance than the conventional ORG
scheme is described. Using the same method it is shown that the performance of conventional CRC
scheme may be considerably enhanced. Error detection is an integral part of an incremental redundancy
communications system and it is shown that sequences of good error correction codes,
suitable for use in incremental redundancy communications systems may be obtained using the
Constructions X and XX. Examples are given and their performances presented in comparison to
conventional CRC schemes
Distance Properties of Short LDPC Codes and their Impact on the BP, ML and Near-ML Decoding Performance
Parameters of LDPC codes, such as minimum distance, stopping distance,
stopping redundancy, girth of the Tanner graph, and their influence on the
frame error rate performance of the BP, ML and near-ML decoding over a BEC and
an AWGN channel are studied. Both random and structured LDPC codes are
considered. In particular, the BP decoding is applied to the code parity-check
matrices with an increasing number of redundant rows, and the convergence of
the performance to that of the ML decoding is analyzed. A comparison of the
simulated BP, ML, and near-ML performance with the improved theoretical bounds
on the error probability based on the exact weight spectrum coefficients and
the exact stopping size spectrum coefficients is presented. It is observed that
decoding performance very close to the ML decoding performance can be achieved
with a relatively small number of redundant rows for some codes, for both the
BEC and the AWGN channels
Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights
We present a tree-based construction of LDPC codes that have minimum
pseudocodeword weight equal to or almost equal to the minimum distance, and
perform well with iterative decoding. The construction involves enumerating a
-regular tree for a fixed number of layers and employing a connection
algorithm based on permutations or mutually orthogonal Latin squares to close
the tree. Methods are presented for degrees and , for a
prime. One class corresponds to the well-known finite-geometry and finite
generalized quadrangle LDPC codes; the other codes presented are new. We also
present some bounds on pseudocodeword weight for -ary LDPC codes. Treating
these codes as -ary LDPC codes rather than binary LDPC codes improves their
rates, minimum distances, and pseudocodeword weights, thereby giving a new
importance to the finite geometry LDPC codes where .Comment: Submitted to Transactions on Information Theory. Submitted: Oct. 1,
2005; Revised: May 1, 2006, Nov. 25, 200
Quantum Error Correction beyond the Bounded Distance Decoding Limit
In this paper, we consider quantum error correction over depolarizing
channels with non-binary low-density parity-check codes defined over Galois
field of size . The proposed quantum error correcting codes are based on
the binary quasi-cyclic CSS (Calderbank, Shor and Steane) codes. The resulting
quantum codes outperform the best known quantum codes and surpass the
performance limit of the bounded distance decoder. By increasing the size of
the underlying Galois field, i.e., , the error floors are considerably
improved.Comment: To appear in IEEE Transactions on Information Theor
Weight Distribution for Non-binary Cluster LDPC Code Ensemble
In this paper, we derive the average weight distributions for the irregular
non-binary cluster low-density parity-check (LDPC) code ensembles. Moreover, we
give the exponential growth rate of the average weight distribution in the
limit of large code length. We show that there exist -regular
non-binary cluster LDPC code ensembles whose normalized typical minimum
distances are strictly positive.Comment: 12pages, 6 figures, To be presented in ISIT2013, Submitted to IEICE
Trans. Fundamental
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