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Designing LDPC Codes for ECOC Classification Systems [chapter]

Claudio Marrocco, Francesco Tortorella
2014 Lecture Notes in Computer Science  
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

M. Amin Khajehnejad, Alexandros G. Dimakis, Babak Hassibi
2010 2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers  
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

M. Amin Khajehnejad, Alexandros G. Dimakis, Babak Hassibi, William Bradley
2010 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)  
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]

Lior Eldar, Aram W. Harrow
2016 arXiv   pre-print
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

J. Conrad, C. Chamberland, N. P. Breuckmann, B. M. Terhal
2018 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
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

Vida Ravanmehr, Mehrdad Khatami, David Declercq, Bane Vasic
2016 Transactions on Emerging Telecommunications Technologies  
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]

Vida Ravanmehr, Mehrdad Khatami, David Declercq, Bane Vasic
2016 arXiv   pre-print
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]

Amin Khajehnejad, Alexandros G. Dimakis, Babak Hassibi, Benjamin Vigoda, William Bradley
2011 arXiv   pre-print
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

Lior Eldar, Aram W. Harrow
2017 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)  
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

Amin Khajehnejad, Alexandros G. Dimakis, Babak Hassibi, Benjamin Vigoda, William Bradley
2012 IEEE Transactions on Information Theory  
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]

Salah A. Aly
2008 arXiv   pre-print
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]

Anthony Leverrier, Gilles Zémor
2022 arXiv   pre-print
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

Nicholas Bonello, Sheng Chen, Lajos Hanzo
2011 IEEE Communications Surveys and Tutorials  
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

Nikolas P. Breuckmann, Jens Niklas Eberhardt
2021 PRX Quantum  
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]

Nikolas P. Breuckmann, Jens Niklas Eberhardt
2021 arXiv   pre-print
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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