American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Linear complexity of cyclotomic sequences of order six and BCH codes over GF(3)
  • AMC Home
  • This Issue
  • Next Article
    On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric
August  2014, 8(3): 281-296. doi: 10.3934/amc.2014.8.281

On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$

M. De Boeck 1, and  P. Vandendriessche 1,

1. 

UGent, Department of Mathematics, Krijgslaan 281-S22, 9000 Gent, Flanders, Belgium, Belgium

Received  March 2013 Published  August 2014

We study the dual linear code of points and generators on a non-singular Hermitian variety $\mathcal{H}(2n+1,q^2)$. We improve the earlier results for $n=2$, we solve the minimum distance problem for general $n$, we classify the $n$ smallest types of code words and we characterize the small weight code words as being a linear combination of these $n$ types.
Keywords: small weight code words., Dual code, Hermitian varieties.
Mathematics Subject Classification: Primary: 51E20, 51E22; Secondary: 05B25, 94B0.
Citation: M. De Boeck, P. Vandendriessche. On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$. Advances in Mathematics of Communications, 2014, 8 (3) : 281-296. doi: 10.3934/amc.2014.8.281
References:
[1]

E. F. Assmus and J. D. Key, Designs and their Codes, Cambridge University Press, Cambridge, 1992.  Google Scholar

[2]

S. V. Droms, K. E. Mellinger and C. Meyer, LDPC codes generated by conics in the classical projective plane, Des. Codes Cryptogr., 40 (2006), 343-356. doi: 10.1007/s10623-006-0022-6.  Google Scholar

[3]

Y. Fujiwara, D. Clark, P. Vandendriessche, M. De Boeck and V. D. Tonchev, Entanglement-assisted quantum low-density parity-check codes, Phys. Rev. A, 82 (2010), 042338. doi: 10.1103/PhysRevA.82.042338.  Google Scholar

[4]

J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford University Press, Oxford, 1991.  Google Scholar

[5]

J.-L. Kim, K. Mellinger and L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des. Codes Cryptogr., 42 (2007), 73-92. doi: 10.1007/s10623-006-9017-6.  Google Scholar

[6]

A. Klein, K. Metsch and L. Storme, Small maximal partial spreads in classical finite polar spaces, Adv. Geom., 10 (2010), 379-402. doi: 10.1515/ADVGEOM.2010.007.  Google Scholar

[7]

M. Lavrauw, L. Storme and G. Van de Voorde, Linear codes from projective spaces, in Error-Correcting Codes, Finite Geometries, and Cryptography (eds. A.A. Bruen and D.L. Wehlau), 2010, 185-202. doi: 10.1090/conm/523/10326.  Google Scholar

[8]

V. Pepe, L. Storme and G. Van de Voorde, On codewords in the dual code of classical generalised quadrangles and classical polar spaces, Discrete Math., 310 (2010), 3132-3148. doi: 10.1016/j.disc.2009.06.010.  Google Scholar

[9]

P. Vandendriessche, LDPC codes associated with linear representations of geometries, Adv. Math. Commun., 4 (2010), 405-417. doi: 10.3934/amc.2010.4.405.  Google Scholar

[10]

P. Vandendriessche, Some low-density parity-check codes derived from finite geometries, Des. Codes. Cryptogr., 54 (2010), 287-297. doi: 10.1007/s10623-009-9324-9.  Google Scholar

show all references

References:
[1]

E. F. Assmus and J. D. Key, Designs and their Codes, Cambridge University Press, Cambridge, 1992.  Google Scholar

[2]

S. V. Droms, K. E. Mellinger and C. Meyer, LDPC codes generated by conics in the classical projective plane, Des. Codes Cryptogr., 40 (2006), 343-356. doi: 10.1007/s10623-006-0022-6.  Google Scholar

[3]

Y. Fujiwara, D. Clark, P. Vandendriessche, M. De Boeck and V. D. Tonchev, Entanglement-assisted quantum low-density parity-check codes, Phys. Rev. A, 82 (2010), 042338. doi: 10.1103/PhysRevA.82.042338.  Google Scholar

[4]

J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford University Press, Oxford, 1991.  Google Scholar

[5]

J.-L. Kim, K. Mellinger and L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des. Codes Cryptogr., 42 (2007), 73-92. doi: 10.1007/s10623-006-9017-6.  Google Scholar

[6]

A. Klein, K. Metsch and L. Storme, Small maximal partial spreads in classical finite polar spaces, Adv. Geom., 10 (2010), 379-402. doi: 10.1515/ADVGEOM.2010.007.  Google Scholar

[7]

M. Lavrauw, L. Storme and G. Van de Voorde, Linear codes from projective spaces, in Error-Correcting Codes, Finite Geometries, and Cryptography (eds. A.A. Bruen and D.L. Wehlau), 2010, 185-202. doi: 10.1090/conm/523/10326.  Google Scholar

[8]

V. Pepe, L. Storme and G. Van de Voorde, On codewords in the dual code of classical generalised quadrangles and classical polar spaces, Discrete Math., 310 (2010), 3132-3148. doi: 10.1016/j.disc.2009.06.010.  Google Scholar

[9]

P. Vandendriessche, LDPC codes associated with linear representations of geometries, Adv. Math. Commun., 4 (2010), 405-417. doi: 10.3934/amc.2010.4.405.  Google Scholar

[10]

P. Vandendriessche, Some low-density parity-check codes derived from finite geometries, Des. Codes. Cryptogr., 54 (2010), 287-297. doi: 10.1007/s10623-009-9324-9.  Google Scholar

[1]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[2]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261
[3]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[4]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1
[5]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003
[6]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
[7]

Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027
[8]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399
[9]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2021, 15 (3) : 415-422. doi: 10.3934/amc.2020074
[10]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
[11]

Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503
[12]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407
[13]

Jorge P. Arpasi. On the non-Abelian group code capacity of memoryless channels. Advances in Mathematics of Communications, 2020, 14 (3) : 423-436. doi: 10.3934/amc.2020058
[14]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83
[15]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Information--bit error rate and false positives in an MDS code. Advances in Mathematics of Communications, 2015, 9 (2) : 149-168. doi: 10.3934/amc.2015.9.149
[16]

Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275
[17]

Anna-Lena Horlemann-Trautmann, Kyle Marshall. New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 533-548. doi: 10.3934/amc.2017042
[18]

Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2021, 15 (1) : 113-130. doi: 10.3934/amc.2020046
[19]

Terry Shue Chien Lau, Chik How Tan. Polynomial-time plaintext recovery attacks on the IKKR code-based cryptosystems. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020132
[20]

Jianying Fang. 5-SEEDs from the lifted Golay code of length 24 over Z4. Advances in Mathematics of Communications, 2017, 11 (1) : 259-266. doi: 10.3934/amc.2017017

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences