78 research outputs found
Diagnosis of weaknesses in modern error correction codes: a physics approach
One of the main obstacles to the wider use of the modern error-correction
codes is that, due to the complex behavior of their decoding algorithms, no
systematic method which would allow characterization of the Bit-Error-Rate
(BER) is known. This is especially true at the weak noise where many systems
operate and where coding performance is difficult to estimate because of the
diminishingly small number of errors. We show how the instanton method of
physics allows one to solve the problem of BER analysis in the weak noise range
by recasting it as a computationally tractable minimization problem.Comment: 9 pages, 8 figure
Worst Configurations (Instantons) for Compressed Sensing over Reals: a Channel Coding Approach
We consider the Linear Programming (LP) solution of the Compressed Sensing
(CS) problem over reals, also known as the Basis Pursuit (BasP) algorithm. The
BasP allows interpretation as a channel-coding problem, and it guarantees
error-free reconstruction with a properly chosen measurement matrix and
sufficiently sparse error vectors. In this manuscript, we examine how the BasP
performs on a given measurement matrix and develop an algorithm to discover the
sparsest vectors for which the BasP fails. The resulting algorithm is a
generalization of our previous results on finding the most probable
error-patterns degrading performance of a finite size Low-Density Parity-Check
(LDPC) code in the error-floor regime. The BasP fails when its output is
different from the actual error-pattern. We design a CS-Instanton Search
Algorithm (ISA) generating a sparse vector, called a CS-instanton, such that
the BasP fails on the CS-instanton, while the BasP recovery is successful for
any modification of the CS-instanton replacing a nonzero element by zero. We
also prove that, given a sufficiently dense random input for the error-vector,
the CS-ISA converges to an instanton in a small finite number of steps. The
performance of the CS-ISA is illustrated on a randomly generated matrix. For this example, the CS-ISA outputs the shortest instanton (error
vector) pattern of length 11.Comment: Accepted to be presented at the IEEE International Symposium on
Information Theory (ISIT 2010). 5 pages, 2 Figures. Minor edits from previous
version. Added a new reference
Polytope of Correct (Linear Programming) Decoding and Low-Weight Pseudo-Codewords
We analyze Linear Programming (LP) decoding of graphical binary codes
operating over soft-output, symmetric and log-concave channels. We show that
the error-surface, separating domain of the correct decoding from domain of the
erroneous decoding, is a polytope. We formulate the problem of finding the
lowest-weight pseudo-codeword as a non-convex optimization (maximization of a
convex function) over a polytope, with the cost function defined by the channel
and the polytope defined by the structure of the code. This formulation
suggests new provably convergent heuristics for finding the lowest weight
pseudo-codewords improving in quality upon previously discussed. The algorithm
performance is tested on the example of the Tanner [155, 64, 20] code over the
Additive White Gaussian Noise (AWGN) channel.Comment: 6 pages, 2 figures, accepted for IEEE ISIT 201
Lowering the Error Floor of LDPC Codes Using Cyclic Liftings
Cyclic liftings are proposed to lower the error floor of low-density
parity-check (LDPC) codes. The liftings are designed to eliminate dominant
trapping sets of the base code by removing the short cycles which form the
trapping sets. We derive a necessary and sufficient condition for the cyclic
permutations assigned to the edges of a cycle of length in the
base graph such that the inverse image of in the lifted graph consists of
only cycles of length strictly larger than . The proposed method is
universal in the sense that it can be applied to any LDPC code over any channel
and for any iterative decoding algorithm. It also preserves important
properties of the base code such as degree distributions, encoder and decoder
structure, and in some cases, the code rate. The proposed method is applied to
both structured and random codes over the binary symmetric channel (BSC). The
error floor improves consistently by increasing the lifting degree, and the
results show significant improvements in the error floor compared to the base
code, a random code of the same degree distribution and block length, and a
random lifting of the same degree. Similar improvements are also observed when
the codes designed for the BSC are applied to the additive white Gaussian noise
(AWGN) channel
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