28,117 research outputs found
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
Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes
We introduce the notion of the stopping redundancy hierarchy of a linear
block code as a measure of the trade-off between performance and complexity of
iterative decoding for the binary erasure channel. We derive lower and upper
bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and
Bonferroni-type inequalities, and specialize them for codes with cyclic
parity-check matrices. Based on the observed properties of parity-check
matrices with good stopping redundancy characteristics, we develop a novel
decoding technique, termed automorphism group decoding, that combines iterative
message passing and permutation decoding. We also present bounds on the
smallest number of permutations of an automorphism group decoder needed to
correct any set of erasures up to a prescribed size. Simulation results
demonstrate that for a large number of algebraic codes, the performance of the
new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on
Information Theor
On the size of identifying codes in triangle-free graphs
In an undirected graph , a subset such that is a
dominating set of , and each vertex in is dominated by a distinct
subset of vertices from , is called an identifying code of . The concept
of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in
1998. For a given identifiable graph , let \M(G) be the minimum
cardinality of an identifying code in . In this paper, we show that for any
connected identifiable triangle-free graph on vertices having maximum
degree , \M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}. This bound is
asymptotically tight up to constants due to various classes of graphs including
-ary trees, which are known to have their minimum identifying code
of size . We also provide improved bounds for
restricted subfamilies of triangle-free graphs, and conjecture that there
exists some constant such that the bound \M(G)\le n-\tfrac{n}{\Delta}+c
holds for any nontrivial connected identifiable graph
A single-photon sampling architecture for solid-state imaging
Advances in solid-state technology have enabled the development of silicon
photomultiplier sensor arrays capable of sensing individual photons. Combined
with high-frequency time-to-digital converters (TDCs), this technology opens up
the prospect of sensors capable of recording with high accuracy both the time
and location of each detected photon. Such a capability could lead to
significant improvements in imaging accuracy, especially for applications
operating with low photon fluxes such as LiDAR and positron emission
tomography.
The demands placed on on-chip readout circuitry imposes stringent trade-offs
between fill factor and spatio-temporal resolution, causing many contemporary
designs to severely underutilize the technology's full potential. Concentrating
on the low photon flux setting, this paper leverages results from group testing
and proposes an architecture for a highly efficient readout of pixels using
only a small number of TDCs, thereby also reducing both cost and power
consumption. The design relies on a multiplexing technique based on binary
interconnection matrices. We provide optimized instances of these matrices for
various sensor parameters and give explicit upper and lower bounds on the
number of TDCs required to uniquely decode a given maximum number of
simultaneous photon arrivals.
To illustrate the strength of the proposed architecture, we note a typical
digitization result of a 120x120 photodiode sensor on a 30um x 30um pitch with
a 40ps time resolution and an estimated fill factor of approximately 70%, using
only 161 TDCs. The design guarantees registration and unique recovery of up to
4 simultaneous photon arrivals using a fast decoding algorithm. In a series of
realistic simulations of scintillation events in clinical positron emission
tomography the design was able to recover the spatio-temporal location of 98.6%
of all photons that caused pixel firings.Comment: 24 pages, 3 figures, 5 table
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