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Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles

Jon-Lark Kim, Keith E. Mellinger, Leo Storme
2006 Designs, Codes and Cryptography  
We find lower bounds on the minimum distance and characterize codewords of small weight in low-density parity check codes defined by (dual) classical generalized quadrangles.  ...  Similar results are also presented for the LDPC codes LU(3, q) given in [10] .  ...  We contribute in this article to the problem of finding the minimum distance of LDPC codes defined by the incidence structure of (dual) classical finite generalized quadrangles.  ... 
doi:10.1007/s10623-006-9017-6 fatcat:crvnnioblzh7lpay6pru6xv6vq

On codewords in the dual code of classical generalised quadrangles and classical polar spaces [article]

Valentina Pepe, Leo Storme, Geertrui Van de Voorde
2012 arXiv   pre-print
Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q+(5, q) and H(5, q2), and we present lower  ...  In [9], the codewords of small weight in the dual code of the code of points and lines of Q(4, q) are characterised.  ...  In [9] , the codewords of small weight in the dual code of the code of points and lines of classical generalised quadrangles are characterised, and in [20] , the authors characterise codewords of small  ... 
arXiv:1201.3294v1 fatcat:bl6rlrplcbaxbmnmz2gmfv6zjy

On codewords in the dual code of classical generalised quadrangles and classical polar spaces

Valentina Pepe, Leo Storme, Geertrui Van de Voorde
2010 Discrete Mathematics  
In [J.L. Kim, K. Mellinger, L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des.  ...  Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q + (5, q) and H(5, q 2 ), and we present lower  ...  The research of the first author was supported by the Scientific Research Network FWO-Flanders: Fundamental Methods and Techniques in Mathematics.  ... 
doi:10.1016/j.disc.2009.06.010 fatcat:bnyedbo35za2bffsgjjwenssjy

Small weight codewords in the LDPC codes arising from linear representations of geometries

V. Pepe, L. Storme, G. Van de Voorde
2009 Journal of combinatorial designs (Print)  
In this paper, we investigate the minimum distance and small weight codewords of the LDPC codes of linear representations, using only geometrical methods.  ...  When investigating the small weight codewords of T * 2 (Θ) D , we deal with the case of Θ a regular hyperoval, i.e. a conic and its nucleus, separately, since in this case, we have a larger upper bound  ...  Furthermore, Liu and Pados proved in [15] that the binary LDPC codes defined by generalized quadrangles only have codewords of even weight.  ... 
doi:10.1002/jcd.20179 fatcat:h7bc3r2p7ben7fgz4qu7sj7rwq

Locally repairable codes with high availability based on generalised quadrangles [article]

Michel Lavrauw, Geertrui Van de Voorde
2019 arXiv   pre-print
In order to do so, we determine the minimum weight of the codes of translation generalised quadrangles and characterise the codewords of minimum weight.  ...  Locally Repairable Codes (LRC's) based on generalised quadrangles were introduced by Pamies-Juarez, Hollmann and Oggier in , and bounds on the repairability and availability were derived.  ...  This research was made possible by an Erskine Fellowship from the University of Canterbury, Christchurch, New Zealand.  ... 
arXiv:1912.06372v1 fatcat:c2kcldhvuzexjlskrl62kyu36q

Locally repairable codes with high availability based on generalised quadrangles

Michel Lavrauw, ,Faculty of Engineering and Natural Sciences, Sabancı University, 34956 Orhanlı, Tuzla, Istanbul, Turkey, Geertrui Van de Voorde, ,School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, 8140 Christchurch, New Zealand
2019 Advances in Mathematics of Communications  
In order to do so, we determine the minimum weight of the codes of translation generalised quadrangles and characterise the codewords of minimum weight. 2020 Mathematics Subject Classification: 51E12,51E20,94B05  ...  Locally Repairable Codes (LRC's) based on generalised quadrangles were introduced by Pamies-Juarez, Hollmann and Oggier in [3] , and bounds on the repairability and availability were derived.  ...  In [3] , the authors define a pg − BLRC-code as the dual code of the binary code generated by N .  ... 
doi:10.3934/amc.2020099 fatcat:xha44quxdzbavcynrzxxbw6ibu

Quantum Tanner codes [article]

Anthony Leverrier, Gilles Zémor
2022 arXiv   pre-print
By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code.  ...  Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes.  ...  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

On the pseudocodeword weight and parity-check matrix redundancy of linear codes

Christine A. Kelley, Deepak Sridhara
2007 2007 IEEE Information Theory Workshop  
The minimum pseudocodeword weight wmin of a linear graph-based code is more influential in determining decoding performance when decoded via iterative and linear programming decoding algorithms than the  ...  We provide some upper bounds on the BSCpseudoweight redundancy and illustrate the concept with some results for Hamming codes, tree-based and finite geometry LDPC codes, Reed-Muller codes and Hadamard  ...  The Type II, ℓ = 4 tree-based LDPC codes are equivalent to those obtained from finite generalized quadrangles.  ... 
doi:10.1109/itw.2007.4313040 fatcat:xlduof3m7rg6nnhv5bqjfdjjfm

Moderate Density Parity-Check Codes from Projective Bundles [article]

Jessica Bariffi, Sam Mattheus, Alessandro Neri, Joachim Rosenthal
2021 arXiv   pre-print
In this setting, our codes have the best possible error-correction performance for this range of parameters.  ...  A projective bundle is a special collection of ovals which pairwise meet in a unique point.  ...  Neri was supported by the Swiss National Science Foundation through grant no. 187711. The work of J. Rosenthal was supported by the Swiss National Science Foundation through grant no. 188430.  ... 
arXiv:2103.09722v1 fatcat:tsvuul4l3zedfggz5klvs6pj2m

Minimal-Entanglement Entanglement-Assisted Quantum Error Correction Codes from Modified Circulant Matrices

Duc Manh Nguyen, Sunghwan Kim
2017 Symmetry  
In this paper, new construction methods of entanglement-assisted quantum error correction code (EAQECC) from circulant matrices are proposed.  ...  The proposed EAQECC with small lengths are presented to explain the practicality of this construction of EAQECC.  ...  Acknowledgments: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF-2016R1D1A1B03934653).  ... 
doi:10.3390/sym9070122 fatcat:augbjgcms5ekngrngmdtu4pvgi

Classes of codes from quadratic surfaces of P G(3, q)

Keith Mellinger
2006 unpublished
In particular, we use geometric techniques to find small weight codewords, and hence, bound the minimum distance.  ...  We examine classes of binary linear error correcting codes constructed from certain sets of lines defined relative to one of the two classical quadratic surfaces in P G(3, q).  ...  In 2001, Fossorier, et al [7] examined classes of LDPC codes generated by incidence structures in finite geometries.  ... 
fatcat:qrv5v6cdkvcwvjxysuzdbqpjcm

Classes of codes from quadratic surfaces of PG(3,q)

Keith E Mellinger
2007
In particular, we use geometric techniques to find small weight codewords, and hence, bound the minimum distance.  ...  We examine classes of binary linear error correcting codes constructed from certain sets of lines defined relative to one of the two classical quadratic surfaces in $PG(3,q)$.  ...  Quite simply, LDPC codes are defined by a sparse parity-check matrix rather than a generator matrix which is perhaps more common. In 2001, Fossorier et al.  ... 
doi:10.11575/cdm.v2i1.61875 fatcat:qemvmlwrpfcq7po4bvgu2w2464