1,402 research outputs found
Quantum Low-Density Parity-Check Codes
Quantum error correction is an indispensable ingredient for scalable quantum computing. In this Perspective we discuss a particular class of quantum codes called “quantum low-density parity-check (LDPC) codes.” The codes we discuss are alternatives to the surface code, which is currently the leading candidate to implement quantum fault tolerance. We introduce the zoo of quantum LDPC codes and discuss their potential for making quantum computers robust with regard to noise. In particular, we explain recent advances in the theory of quantum LDPC codes related to certain product constructions and discuss open problems in the field
A Class of Quantum LDPC Codes Constructed From Finite Geometries
Low-density parity check (LDPC) codes are a significant class of classical
codes with many applications. Several good LDPC codes have been constructed
using random, algebraic, and finite geometries approaches, with containing
cycles of length at least six in their Tanner graphs. However, it is impossible
to design a self-orthogonal parity check matrix of an LDPC code without
introducing cycles of length four.
In this paper, a new class of quantum LDPC codes based on lines and points of
finite geometries is constructed. The parity check matrices of these codes are
adapted to be self-orthogonal with containing only one cycle of length four.
Also, the column and row weights, and bounds on the minimum distance of these
codes are given. As a consequence, the encoding and decoding algorithms of
these codes as well as their performance over various quantum depolarizing
channels will be investigated.Comment: 5pages, 2 figure
Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes
Linear programming (LP) decoding approximates maximum-likelihood (ML)
decoding of a linear block code by relaxing the equivalent ML integer
programming (IP) problem into a more easily solved LP problem. The LP problem
is defined by a set of box constraints together with a set of linear
inequalities called "parity inequalities" that are derived from the constraints
represented by the rows of a parity-check matrix of the code and can be added
iteratively and adaptively. In this paper, we first derive a new necessary
condition and a new sufficient condition for a violated parity inequality
constraint, or "cut," at a point in the unit hypercube. Then, we propose a new
and effective algorithm to generate parity inequalities derived from certain
additional redundant parity check (RPC) constraints that can eliminate
pseudocodewords produced by the LP decoder, often significantly improving the
decoder error-rate performance. The cut-generating algorithm is based upon a
specific transformation of an initial parity-check matrix of the linear block
code. We also design two variations of the proposed decoder to make it more
efficient when it is combined with the new cut-generating algorithm. Simulation
results for several low-density parity-check (LDPC) codes demonstrate that the
proposed decoding algorithms significantly narrow the performance gap between
LP decoding and ML decoding
On Constructing Low-Density Parity-Check Codes
This thesis focuses on designing Low-Density Parity-Check (LDPC)
codes for forward-error-correction. The target application is
real-time multimedia communications over packet networks. We
investigate two code design issues, which are important in the
target application scenarios, designing LDPC codes with low
decoding latency, and constructing capacity-approaching LDPC codes
with very low error probabilities.
On designing LDPC codes with low decoding latency, we present a
framework for optimizing the code parameters so that the decoding
can be fulfilled after only a small number of iterative decoding
iterations. The brute force approach for such optimization is
numerical intractable, because it involves a difficult discrete
optimization programming. In this thesis, we show an asymptotic
approximation to the number of decoding iterations. Based on this
asymptotic approximation, we propose an approximate optimization
framework for finding near-optimal code parameters, so that the
number of decoding iterations is minimized. The approximate
optimization approach is numerically tractable. Numerical results
confirm that the proposed optimization approach has excellent
numerical properties, and codes with excellent performance in terms
of number of decoding iterations can be obtained. Our results show
that the numbers of decoding iterations of the codes by the proposed
design approach can be as small as one-fifth of the numbers of
decoding iterations of some previously well-known codes. The
numerical results also show that the proposed asymptotic
approximation is generally tight for even non-extremely limiting
cases.
On constructing capacity-approaching LDPC codes with very low error
probabilities, we propose a new LDPC code construction scheme based
on -lifts. Based on stopping set distribution analysis, we
propose design criteria for the resulting codes to have very low
error floors. High error floors are the main problems of previously
constructed capacity-approaching codes, which prevent them from
achieving very low error probabilities. Numerical results confirm
that codes with very low error floors can be obtained by the
proposed code construction scheme and the design criteria. Compared
with the codes by the previous standard construction schemes, which
have error floors at the levels of to , the codes
by the proposed approach do not have observable error floors at the
levels higher than . The error floors of the codes by the
proposed approach are also significantly lower compared with the
codes by the previous approaches to constructing codes with low
error floors
EXTREMAL ABSORBING SETS IN LOW-DENSITY PARITY-CHECK CODES
Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of b for a given a for which an (a, b)-absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms
Challenges and Some New Directions in Channel Coding
Three areas of ongoing research in channel coding are surveyed, and recent developments are presented in each area: spatially coupled Low-Density Parity-Check (LDPC) codes, nonbinary LDPC codes, and polar coding.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/JCN.2015.00006
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