77 research outputs found
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
Deterministic Constructions of Binary Measurement Matrices from Finite Geometry
Deterministic constructions of measurement matrices in compressed sensing
(CS) are considered in this paper. The constructions are inspired by the recent
discovery of Dimakis, Smarandache and Vontobel which says that parity-check
matrices of good low-density parity-check (LDPC) codes can be used as
{provably} good measurement matrices for compressed sensing under
-minimization. The performance of the proposed binary measurement
matrices is mainly theoretically analyzed with the help of the analyzing
methods and results from (finite geometry) LDPC codes. Particularly, several
lower bounds of the spark (i.e., the smallest number of columns that are
linearly dependent, which totally characterizes the recovery performance of
-minimization) of general binary matrices and finite geometry matrices
are obtained and they improve the previously known results in most cases.
Simulation results show that the proposed matrices perform comparably to,
sometimes even better than, the corresponding Gaussian random matrices.
Moreover, the proposed matrices are sparse, binary, and most of them have
cyclic or quasi-cyclic structure, which will make the hardware realization
convenient and easy.Comment: 12 pages, 11 figure
New Classes of Partial Geometries and Their Associated LDPC Codes
The use of partial geometries to construct parity-check matrices for LDPC
codes has resulted in the design of successful codes with a probability of
error close to the Shannon capacity at bit error rates down to . Such
considerations have motivated this further investigation. A new and simple
construction of a type of partial geometries with quasi-cyclic structure is
given and their properties are investigated. The trapping sets of the partial
geometry codes were considered previously using the geometric aspects of the
underlying structure to derive information on the size of allowable trapping
sets. This topic is further considered here. Finally, there is a natural
relationship between partial geometries and strongly regular graphs. The
eigenvalues of the adjacency matrices of such graphs are well known and it is
of interest to determine if any of the Tanner graphs derived from the partial
geometries are good expanders for certain parameter sets, since it can be
argued that codes with good geometric and expansion properties might perform
well under message-passing decoding.Comment: 34 pages with single column, 6 figure
Low-Density Arrays of Circulant Matrices: Rank and Row-Redundancy Analysis, and Quasi-Cyclic LDPC Codes
This paper is concerned with general analysis on the rank and row-redundancy
of an array of circulants whose null space defines a QC-LDPC code. Based on the
Fourier transform and the properties of conjugacy classes and Hadamard products
of matrices, we derive tight upper bounds on rank and row-redundancy for
general array of circulants, which make it possible to consider row-redundancy
in constructions of QC-LDPC codes to achieve better performance. We further
investigate the rank of two types of construction of QC-LDPC codes:
constructions based on Vandermonde Matrices and Latin Squares and give
combinatorial expression of the exact rank in some specific cases, which
demonstrates the tightness of the bound we derive. Moreover, several types of
new construction of QC-LDPC codes with large row-redundancy are presented and
analyzed.Comment: arXiv admin note: text overlap with arXiv:1004.118
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
Multiplicatively Repeated Non-Binary LDPC Codes
We propose non-binary LDPC codes concatenated with multiplicative repetition
codes. By multiplicatively repeating the (2,3)-regular non-binary LDPC mother
code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9,
1/12,... Surprisingly, such simple low-rate non-binary LDPC codes outperform
the best low-rate binary LDPC codes so far. Moreover, we propose the decoding
algorithm for the proposed codes, which can be decoded with almost the same
computational complexity as that of the mother code.Comment: To appear in IEEE Transactions on Information Theor
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