452 research outputs found
Deterministic Constructions for Large Girth Protograph LDPC Codes
The bit-error threshold of the standard ensemble of Low Density Parity Check
(LDPC) codes is known to be close to capacity, if there is a non-zero fraction
of degree-two bit nodes. However, the degree-two bit nodes preclude the
possibility of a block-error threshold. Interestingly, LDPC codes constructed
using protographs allow the possibility of having both degree-two bit nodes and
a block-error threshold. In this paper, we analyze density evolution for
protograph LDPC codes over the binary erasure channel and show that their
bit-error probability decreases double exponentially with the number of
iterations when the erasure probability is below the bit-error threshold and
long chain of degree-two variable nodes are avoided in the protograph. We
present deterministic constructions of such protograph LDPC codes with girth
logarithmic in blocklength, resulting in an exponential fall in bit-error
probability below the threshold. We provide optimized protographs, whose
block-error thresholds are better than that of the standard ensemble with
minimum bit-node degree three. These protograph LDPC codes are theoretically of
great interest, and have applications, for instance, in coding with strong
secrecy over wiretap channels.Comment: 5 pages, 2 figures; To appear in ISIT 2013; Minor changes in
presentatio
Design of Non-Binary Quasi-Cyclic LDPC Codes by ACE Optimization
An algorithm for constructing Tanner graphs of non-binary irregular
quasi-cyclic LDPC codes is introduced. It employs a new method for selection of
edge labels allowing control over the code's non-binary ACE spectrum and
resulting in low error-floor. The efficiency of the algorithm is demonstrated
by generating good codes of short to moderate length over small fields,
outperforming codes generated by the known methods.Comment: Accepted to 2013 IEEE Information Theory Worksho
Entanglement-assisted quantum low-density parity-check codes
This paper develops a general method for constructing entanglement-assisted
quantum low-density parity-check (LDPC) codes, which is based on combinatorial
design theory. Explicit constructions are given for entanglement-assisted
quantum error-correcting codes (EAQECCs) with many desirable properties. These
properties include the requirement of only one initial entanglement bit, high
error correction performance, high rates, and low decoding complexity. The
proposed method produces infinitely many new codes with a wide variety of
parameters and entanglement requirements. Our framework encompasses various
codes including the previously known entanglement-assisted quantum LDPC codes
having the best error correction performance and many new codes with better
block error rates in simulations over the depolarizing channel. We also
determine important parameters of several well-known classes of quantum and
classical LDPC codes for previously unsettled cases.Comment: 20 pages, 5 figures. Final version appearing in Physical Review
Shortened Array Codes of Large Girth
One approach to designing structured low-density parity-check (LDPC) codes
with large girth is to shorten codes with small girth in such a manner that the
deleted columns of the parity-check matrix contain all the variables involved
in short cycles. This approach is especially effective if the parity-check
matrix of a code is a matrix composed of blocks of circulant permutation
matrices, as is the case for the class of codes known as array codes. We show
how to shorten array codes by deleting certain columns of their parity-check
matrices so as to increase their girth. The shortening approach is based on the
observation that for array codes, and in fact for a slightly more general class
of LDPC codes, the cycles in the corresponding Tanner graph are governed by
certain homogeneous linear equations with integer coefficients. Consequently,
we can selectively eliminate cycles from an array code by only retaining those
columns from the parity-check matrix of the original code that are indexed by
integer sequences that do not contain solutions to the equations governing
those cycles. We provide Ramsey-theoretic estimates for the maximum number of
columns that can be retained from the original parity-check matrix with the
property that the sequence of their indices avoid solutions to various types of
cycle-governing equations. This translates to estimates of the rate penalty
incurred in shortening a code to eliminate cycles. Simulation results show that
for the codes considered, shortening them to increase the girth can lead to
significant gains in signal-to-noise ratio in the case of communication over an
additive white Gaussian noise channel.Comment: 16 pages; 8 figures; to appear in IEEE Transactions on Information
Theory, Aug 200
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