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Designing LDPC Codes for ECOC Classification Systems
[chapter]
2014
Lecture Notes in Computer Science
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In this paper we analyze a framework for an ECOC classification system founded on the use of LPDC codes, a class of codes well-known in Coding Theory. ...
Some experiments have been accomplished with the focus on the parity-check matrix used to define the codewords of the LDPC code, so as to determine how the code parameters influence the performance of ...
This is an important issue since the sparseness of the paritycheck matrix guarantees a minimum Hamming distance linearly increasing with the code length while the decoding complexity linearly increases ...
doi:10.1007/978-3-662-44415-3_46
fatcat:a3uziynpibcpncmwiakq7zkt3y
Iterative weighted ℓ1 Optimization for compressed sensing and coding
2010
2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers
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We introduce a novel algorithm for decoding binary linear codes by linear programming. ...
Our analysis shows that for some LDPC ensembles we can improve the provable threshold guarantees compared to standard LP decoding. ...
Using the fact that the LP relaxation with P relaxes the entries of x from x i ∈ {0, 1} to x i ∈ [0, 1], we may replace the Hamming distance with the 1 distance x − x (r) 1 . ...
doi:10.1109/acssc.2010.5757668
fatcat:wpz7ipptojhf7pxygue6vqgssi
Reweighted LP decoding for LDPC codes
2010
2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)
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We introduce a novel algorithm for decoding binary linear codes by linear programming. ...
Our analysis shows that for some LDPC ensembles we can improve the provable threshold guarantees compared to standard LP decoding. ...
Using the fact that the LP relaxation with P relaxes the entries of x from x i ∈ {0, 1} to x i ∈ [0, 1], we may replace the Hamming distance with the 1 distance x − x (r) 1 . ...
doi:10.1109/allerton.2010.5706905
fatcat:igtsogw4orhivm3l27gkshxfay
Local Hamiltonians Whose Ground States are Hard to Approximate
[article]
2016
arXiv
pre-print
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Our Hamiltonian is based on applying the hypergraph product by Tillich and Zemor to a classical locally testable code. ...
Formally, it provides positive evidence for two wide-open conjectures in condensed-matter physics and quantum complexity theory which are the qLDPC conjecture, positing the existence of "good" quantum LDPC codes ...
We note that while the minimal distance of C is exactly n, as it is the repetition code, the minimal distance of C T is much smaller, i.e. ...
arXiv:1510.02082v3
fatcat:4la4js5ctzf5hokemvt6fsft3a
The small stellated dodecahedron code and friends
2018
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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We explore a distance-3 homological CSS quantum code, namely the small stellated dodecahedron code, for dense storage of quantum information and we compare its performance with the distance-3 surface code ...
code. ...
LPDC codes or more specifically hyperbolic surface codes. ...
doi:10.1098/rsta.2017.0323
pmid:29807900
pmcid:PMC5990658
fatcat:4scahegyxbfopbv3mi74ldkkyi
Check-hybrid GLDPC codes: Systematic elimination of trapping sets and guaranteed error correction capability
2016
Transactions on Emerging Telecommunications Technologies
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Moreover, we provide a fixed set for a class of constructed check-hybrid codes. The guaranteed error correction capability of the CH-GLDPC codes is also studied. ...
The second purpose is to minimize the rate lose caused by replacing the super checks through finding the minimum number of such critical checks. ...
[2] , low-rate GLDPC codes are constructed by doping quasi-cyclic (QC)-LDPC codes with Hamming codes. ...
doi:10.1002/ett.3122
fatcat:r6mn24hibfgolifaqkahk6qmwi
Check-hybrid GLDPC Codes: Systematic Elimination of Trapping Sets and Guaranteed Error Correction Capability
[article]
2016
arXiv
pre-print
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Moreover, we provide a fixed set for a class of constructed check-hybrid codes. The guaranteed error correction capability of the CH-GLDPC codes is also studied. ...
The second purpose is to minimize the rate loss caused by replacing the super checks through finding the minimum number of such critical checks. ...
[2] , low-rate GLDPC codes are constructed by doping quasi-cyclic (QC)-LDPC codes with Hamming codes. ...
arXiv:1604.05256v1
fatcat:4sn57s7zlzarbjv4cyr3thioci
Reweighted LP Decoding for LDPC Codes
[article]
2011
arXiv
pre-print
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We introduce a novel algorithm for decoding binary linear codes by linear programming. ...
Our analysis shows that for some LDPC ensembles we can improve the provable threshold guarantees compared to standard LP decoding. ...
Using the fact that the LP relaxation with P relaxes the entries of x from x i ∈ {0, 1} to x i ∈ [0, 1], we may replace the Hamming distance with the 1 distance x − x (r) 1 . ...
arXiv:1103.2837v1
fatcat:koswmom43zdvbddb5aqjwlsupa
Local Hamiltonians Whose Ground States Are Hard to Approximate
2017
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
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We not only show that d-trivial states cannot be ε-impostors, but we show that these sets of quantum states are separated by a trace distance of n −Ω(1) . 5. ...
Our Hamiltonian is based on applying the hypergraph product by Tillich-Zémor [43] to a classical locally testable code. ...
We note that while the minimal distance of C is exactly n, as it is the repetition code, the minimal distance of C T is much smaller, i.e. ...
doi:10.1109/focs.2017.46
dblp:conf/focs/EldarH17
fatcat:4vjnm2n7vrgcdl362amxgojtni
Reweighted LP Decoding for LDPC Codes
2012
IEEE Transactions on Information Theory
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In contrast to message-passing decoding, however, the * LP decoder either concedes failure on a problem, or returns a codeword along with a guarantee that it is the ML codeword, thereby eliminating any ...
This concept was inspired by a similar reweighted iterative 1 minimization idea for compressive sensing [21, 20] . ...
Using the fact that the LP relaxation with P relaxes the entries of x from x i ∈ {0, 1} to x i ∈ [0, 1], we may replace the Hamming distance with the 1 distance x − x (r) 1 . ...
doi:10.1109/tit.2012.2202211
fatcat:xergtyez5zakxjifa7n4rkuquq
On Quantum and Classical Error Control Codes: Constructions and Applications
[article]
2008
arXiv
pre-print
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However, it is a formidable task to build a quantum computer, since the quantum mechanical systems storing the information unavoidably interact with their environment. ...
codes over symmetric and asymmetric channels. ...
The constructed codes have cycles with length 4 to guarantee self-orthogonality as we will show in section 15.2. ...
arXiv:0812.5104v1
fatcat:oqjbenemizczjmfuivp4vt6jny
Quantum Tanner codes
[article]
2022
arXiv
pre-print
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This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. ...
Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. ...
We would like to thank Benjamin Audoux, Alain Couvreur, Shai Evra, Omar Fawzi, Tali Kaufman, Jean-Pierre Tillich, and Christophe Vuillot for many fruitful discussions on quantum codes over the years. ...
arXiv:2202.13641v2
fatcat:262hwtpgrngmdgzocbagudhqf4
Low-Density Parity-Check Codes and Their Rateless Relatives
2011
IEEE Communications Surveys and Tutorials
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This survey guides the reader through the extensive open literature that is covering the family of low-density paritycheck (LDPC) codes and their rateless relatives. ...
This was only verified for regular LPDC codes. ...
. , K
Hamming codes are considered to be a very efficient class of short codes having a minimum distance equal to 3. ...
doi:10.1109/surv.2011.040410.00042
fatcat:l6hrbfv3zvf6tpe37h7lhpeuoi
Quantum Low-Density Parity-Check Codes
2021
PRX Quantum
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We introduce the zoo of quantum LDPC codes and discuss their potential for making quantum computers robust with regard to noise. ...
In this Perspective we discuss a particular class of quantum codes called "quantum low-density parity-check (LDPC) codes." ...
The Z distance d Z and the X distance d X are the minimum Hamming weight of all nontrivial homology and cohomology classes, respectively. The distance d is the minimum of d X and d Z . ...
doi:10.1103/prxquantum.2.040101
fatcat:37ut3gxfwbdkhmnaydp2mqixya
LDPC Quantum Codes
[article]
2021
arXiv
pre-print
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2021 accepted arXiv:2103.06309v2 fatcat:hbn26ab6cfaf3krlj4cl4ten3u
In this Perspective we discuss a particular class of quantum codes called low-density parity-check (LDPC) quantum codes. ...
The codes we discuss are alternatives to the surface code, which is the currently leading candidate to implement quantum fault-tolerance. ...
The Z-distance d Z and X-distance d X is the minimum Hamming weight of all nontrivial homology and cohomology classes, respectively. The distance d is the minimum of d X and d Z . ...
arXiv:2103.06309v1
fatcat:nrzxvirsdfhhvhqtbj7rqzghrm
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