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Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Christine A. Kelley
2013 IEEE Transactions on Communications  
An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices  ...  Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes.  ...  ACKNOWLEDGMENT The author would like to thank the three anonymous reviewers for their detailed comments that helped improve the quality of this paper.  ... 
doi:10.1109/tcomm.2013.012313.110513 fatcat:mjihtfnrkbdz3bnri5cpl6dtwy

Partially Quasi-Cyclic Protograph-Based LDPC Codes

R. Smarandache, D.G.M. Mitchell, Danniel J. Costello
2011 2011 IEEE International Conference on Communications (ICC)  
Despite their implementation advantages and algebraic properties that make them easy to analyze, protograph-based QC-LDPC codes have undesirable fixed upper limits on important code parameters.  ...  We perform both a minimum Hamming distance and girth analysis of these codes.  ...  ACKNOWLEDGEMENTS This work was partially supported by NSF Grants CCF-0830650, DMS-0708033, CCF-0830608, and NASA Grant NNX-09AI66G.  ... 
doi:10.1109/icc.2011.5963333 dblp:conf/icc/SmarandacheMC11 fatcat:gj63vlwzm5axxkqj3k5dseodq4

Algebraic constructions of graph-based nested codes from protographs

Christine A. Kelley, Jorg Kliewer
2010 2010 IEEE International Symposium on Information Theory  
on algebraic protograph LDPC codes.  ...  In this paper we illustrate how these methods can be used in the construction of nested codes from algebraic lifts of graphs.  ...  In this paper we focus on array-code type constructions [8] , [9] and propose an algebraic design of nested linear codes based on protograph LDPC codes [10] , [11] .  ... 
doi:10.1109/isit.2010.5513628 dblp:conf/isit/KelleyK10 fatcat:d4c6h7yibvd2xopihsnpn73n4a

Algebraic Constructions of Graph-Based Nested Codes from Protographs [article]

Christine A. Kelley, Joerg Kliewer
2010 arXiv   pre-print
on algebraic protograph LDPC codes.  ...  In this paper we illustrate how these methods can be used in the construction of nested codes from algebraic lifts of graphs.  ...  In this paper we focus on array-code type constructions [8] , [9] and propose an algebraic design of nested linear codes based on protograph LDPC codes [10] , [11] .  ... 
arXiv:1006.2977v1 fatcat:ukszspict5e7dphabivrqkdwqq

Quasi-Cyclic LDPC Codes Based on Pre-Lifted Protographs

David G. M. Mitchell, Roxana Smarandache, Daniel J. Costello
2014 IEEE Transactions on Information Theory  
Quasi-cyclic low-density parity-check (QC-LDPC) codes based on protographs are of great interest to code designers because of their implementation advantages and algebraic properties that make them easy  ...  In this paper, we show that the upper bound on the minimum Hamming distance of protograph-based QC codes can be improved by the careful application of a twostep lifting procedure applied to the protograph  ...  Algebraic conditions were derived for the permutation matrices describing the N -fold graph covers of a given protograph that ensure higher girth and minimum distance than the upper bounds for QC codes  ... 
doi:10.1109/tit.2014.2342735 fatcat:lbkqo7vtdjg3baj7rb2xm3ljoi

Quasi-cyclic LDPC codes based on pre-lifted protographs

David G. M. Mitchell, Roxana Smarandache, Daniel J. Costello
2011 2011 IEEE Information Theory Workshop  
Quasi-cyclic low-density parity-check (QC-LDPC) codes based on protographs are of great interest to code designers because of their implementation advantages and algebraic properties that make them easy  ...  In this paper, we show that the upper bound on the minimum Hamming distance of protograph-based QC codes can be improved by the careful application of a twostep lifting procedure applied to the protograph  ...  Algebraic conditions were derived for the permutation matrices describing the N -fold graph covers of a given protograph that ensure higher girth and minimum distance than the upper bounds for QC codes  ... 
doi:10.1109/itw.2011.6089477 dblp:conf/itw/MitchellSC11 fatcat:v4oc5al6sve3lemjqzomjlvq6e

On codes designed via algebraic lifts of graphs

Christine A. Kelley
2008 2008 46th Annual Allerton Conference on Communication, Control, and Computing  
In particular, we present a construction of LDPC codes by giving an algebraic method of choosing the permutation voltages and illustrate the usefulness of the proposed technique via simulation results.  ...  More recently, an algebraic analog of this approach was introduced using the theory of voltage graphs.  ...  For practical implementation, the design of short to moderate length codes with algebraic structure is desired.  ... 
doi:10.1109/allerton.2008.4797704 fatcat:gptjyj4savd3zpsiyzw4pvai5q

On distance properties of quasi-cyclic protograph-based LDPC codes

Brian K. Butler, Paul H. Siegel
2010 2010 IEEE International Symposium on Information Theory  
Recent work has shown that properly designed protograph-based LDPC codes may have minimum distance linearly increasing with block length.  ...  This notion rests on ensemble arguments over all possible expansions of the base protograph. When implementation complexity is considered, the expansion is typically chosen to be quite orderly.  ...  The bounds developed here and [3] are useful tools in validating future QC LDPC code designs both punctured and unpunctured. VIII.  ... 
doi:10.1109/isit.2010.5513638 dblp:conf/isit/ButlerS10 fatcat:bhfalywuonethcm3ekh4lpxb6u

The Development of Turbo and LDPC Codes for Deep-Space Applications

Kenneth S. Andrews, Dariush Divsalar, Sam Dolinar, Jon Hamkins, Christopher R. Jones, Fabrizio Pollara
2007 Proceedings of the IEEE  
A family of turbo codes was standardized and implemented and is currently in use by several spacecraft.  ...  JPL's LDPC codes are built from protographs and circulants, selected by analyses of decoding thresholds and methods to avoid loops in the code graph.  ...  permutation of size T for each edge in the protograph.  ... 
doi:10.1109/jproc.2007.905132 fatcat:qmmfxlxarnenhnz3ic2ph5luxu

Automorphism Ensemble Decoding of Quasi-Cyclic LDPC Codes by Breaking Graph Symmetries [article]

Marvin Geiselhart, Moustafa Ebada, Ahmed Elkelesh, Jannis Clausius, Stephan ten Brink
2022 arXiv   pre-print
The proposed method is demonstrated using LDPC codes from the CCSDS, 802.11n and 5G standards and produces gains of 0.2 to 0.3 dB over conventional BP decoding.  ...  We consider automorphism ensemble decoding (AED) of quasi-cyclic (QC) low-density parity-check (LDPC) codes.  ...  Therefore, AED using the standard H-matrices and QC permutations does not result in any performance gain for QC LDPC codes. IV.  ... 
arXiv:2202.00287v2 fatcat:2vzyv2im6nhazfbzxxy35xnsoq

A Unifying Framework to Construct QC-LDPC Tanner Graphs of Desired Girth [article]

Roxana Smarandache, David G. M. Mitchell
2021 arXiv   pre-print
We also give a new perspective on LDPC protograph-based parity-check matrices by viewing them as rows of a parity-check matrix equal to the sum of certain permutation matrices and obtain an important connection  ...  We also show that the results and methodology that we develop for the all-one protograph can be used and adapted to analyze the girth of the Tanner graph of any parity-check matrix and demonstrate how  ...  codes by constructing sums of permutation matrices with good girth, and either use them as the parity-check matrices (for a proper value of N), or take submatrices, like we did in the above example.  ... 
arXiv:2108.01637v1 fatcat:d5tk2lzvwfaplpsnm3u6yquidm

Coding Theory

Joachim Rosenthal, Mohammad Amin Shokrollahi
2007 Oberwolfach Reports  
Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete  ...  The workshop on coding theory covered many facets of the recent research advances.  ...  "protograph codes" [1] .  ... 
doi:10.4171/owr/2007/56 fatcat:ac2rttsrn5g2rn5la72nudiyum

Duality of Quantum and Classical Error Correction Codes: Design Principles and Examples

Zunaira Babar, Daryus Chandra, Hung Viet Nguyen, Panagiotis Botsinis, Dimitrios Alanis, Soon Xin Ng, Lajos Hanzo
2019 IEEE Communications Surveys and Tutorials  
Calderbank-Shor-Steane (CSS) codes, non-CSS codes and entanglement-assisted codes.  ...  Finally, we apply our discussions to two popular code families, namely to the family of Bose-Chaudhuri-Hocquenghem as well as of convolutional codes and provide detailed design examples for both their  ...  Another major milestone in algebraic coding was achieved with the development of non-binary Redundant Residue Number System (RRNS) codes [60] , [61] , which are also maximum minimum-distance codes and  ... 
doi:10.1109/comst.2018.2861361 fatcat:wj6fgl27srfwjegboaf54pmipi

Signal definition and resource management techniques for upcoming communication and broadcasting systems

Charbel Abdel Nour
2020 Zenodo  
The work concerns code design for binary and non-binary codes with simplified decoding algorithms, their association with high order modulations and multiple antennas, the study of high diversity schemes  ...  Beyond 5G systems are expected to support an everlasting increase in number of connected devices with limited spectral resources. Non Orthogonal Multiple Access (NOMA) can be used to this effect.  ...  In practical Turbo coded systems, algebraic permutations are preferred to random-based permutations.  ... 
doi:10.5281/zenodo.4968707 fatcat:d2zvqd5l5vhq7j6li5kwyc65je

Signal definition and resource management techniques for upcoming communication and broadcasting systems

Charbel Abdel Nour
2020 Zenodo  
The work concerns code design for binary and non-binary codes with simplified decoding algorithms, their association with high order modulations and multiple antennas, the study of high diversity schemes  ...  Beyond 5G systems are expected to support an everlasting increase in number of connected devices with limited spectral resources. Non Orthogonal Multiple Access (NOMA) can be used to this effect.  ...  In practical Turbo coded systems, algebraic permutations are preferred to random-based permutations.  ... 
doi:10.5281/zenodo.5020532 fatcat:nlqm7ld62nhpxiygmjt5sy5m54
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