36 research outputs found
On Coset Leader Graphs of LDPC Codes
Our main technical result is that, in the coset leader graph of a linear
binary code of block length n, the metric balls spanned by constant-weight
vectors grow exponentially slower than those in .
Following the approach of Friedman and Tillich (2006), we use this fact to
improve on the first linear programming bound on the rate of LDPC codes, as the
function of their minimal distance. This improvement, combined with the
techniques of Ben-Haim and Lytsin (2006), improves the rate vs distance bounds
for LDPC codes in a significant sub-range of relative distances
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
On Universal Properties of Capacity-Approaching LDPC Ensembles
This paper is focused on the derivation of some universal properties of
capacity-approaching low-density parity-check (LDPC) code ensembles whose
transmission takes place over memoryless binary-input output-symmetric (MBIOS)
channels. Properties of the degree distributions, graphical complexity and the
number of fundamental cycles in the bipartite graphs are considered via the
derivation of information-theoretic bounds. These bounds are expressed in terms
of the target block/ bit error probability and the gap (in rate) to capacity.
Most of the bounds are general for any decoding algorithm, and some others are
proved under belief propagation (BP) decoding. Proving these bounds under a
certain decoding algorithm, validates them automatically also under any
sub-optimal decoding algorithm. A proper modification of these bounds makes
them universal for the set of all MBIOS channels which exhibit a given
capacity. Bounds on the degree distributions and graphical complexity apply to
finite-length LDPC codes and to the asymptotic case of an infinite block
length. The bounds are compared with capacity-approaching LDPC code ensembles
under BP decoding, and they are shown to be informative and are easy to
calculate. Finally, some interesting open problems are considered.Comment: Published in the IEEE Trans. on Information Theory, vol. 55, no. 7,
pp. 2956 - 2990, July 200
Quickest Sequence Phase Detection
A phase detection sequence is a length- cyclic sequence, such that the
location of any length- contiguous subsequence can be determined from a
noisy observation of that subsequence. In this paper, we derive bounds on the
minimal possible in the limit of , and describe some sequence
constructions. We further consider multiple phase detection sequences, where
the location of any length- contiguous subsequence of each sequence can be
determined simultaneously from a noisy mixture of those subsequences. We study
the optimal trade-offs between the lengths of the sequences, and describe some
sequence constructions. We compare these phase detection problems to their
natural channel coding counterparts, and show a strict separation between the
fundamental limits in the multiple sequence case. Both adversarial and
probabilistic noise models are addressed.Comment: To appear in the IEEE Transactions on Information Theor