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

Previous Article
On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric
AMC Home
This Issue
Next Article
Linear complexity of cyclotomic sequences of order six and BCH codes over GF(3)

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

Abstract
Related Papers

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, (1992).
[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. doi: 10.1007/s10623-006-0022-6.
[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). doi: 10.1103/PhysRevA.82.042338.
[4]	J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries,, Oxford University Press, (1991).
[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. doi: 10.1007/s10623-006-9017-6.
[6]	A. Klein, K. Metsch and L. Storme, Small maximal partial spreads in classical finite polar spaces,, Adv. Geom., 10 (2010), 379. doi: 10.1515/ADVGEOM.2010.007.
[7]	M. Lavrauw, L. Storme and G. Van de Voorde, Linear codes from projective spaces,, in Error-Correcting Codes, (2010), 185. doi: 10.1090/conm/523/10326.
[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. doi: 10.1016/j.disc.2009.06.010.
[9]	P. Vandendriessche, LDPC codes associated with linear representations of geometries,, Adv. Math. Commun., 4 (2010), 405. doi: 10.3934/amc.2010.4.405.
[10]	P. Vandendriessche, Some low-density parity-check codes derived from finite geometries,, Des. Codes. Cryptogr., 54 (2010), 287. doi: 10.1007/s10623-009-9324-9.

show all references

References:

[1]	E. F. Assmus and J. D. Key, Designs and their Codes,, Cambridge University Press, (1992).
[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. doi: 10.1007/s10623-006-0022-6.
[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). doi: 10.1103/PhysRevA.82.042338.
[4]	J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries,, Oxford University Press, (1991).
[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. doi: 10.1007/s10623-006-9017-6.
[6]	A. Klein, K. Metsch and L. Storme, Small maximal partial spreads in classical finite polar spaces,, Adv. Geom., 10 (2010), 379. doi: 10.1515/ADVGEOM.2010.007.
[7]	M. Lavrauw, L. Storme and G. Van de Voorde, Linear codes from projective spaces,, in Error-Correcting Codes, (2010), 185. doi: 10.1090/conm/523/10326.
[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. doi: 10.1016/j.disc.2009.06.010.
[9]	P. Vandendriessche, LDPC codes associated with linear representations of geometries,, Adv. Math. Commun., 4 (2010), 405. doi: 10.3934/amc.2010.4.405.
[10]	P. Vandendriessche, Some low-density parity-check codes derived from finite geometries,, Des. Codes. Cryptogr., 54 (2010), 287. doi: 10.1007/s10623-009-9324-9.

[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]	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
[10]	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
[11]	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
[12]	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
[13]	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
[14]	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
[15]	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
[16]	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
[17]	Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393
[18]	Ekkasit Sangwisut, Somphong Jitman, Patanee Udomkavanich. Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields. Advances in Mathematics of Communications, 2017, 11 (3) : 595-613. doi: 10.3934/amc.2017045
[19]	Daniel Heinlein, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. A subspace code of size $ \bf{333} $ in the setting of a binary $ \bf{q} $-analog of the Fano plane. Advances in Mathematics of Communications, 2019, 13 (3) : 457-475. doi: 10.3934/amc.2019029
[20]	V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511

2017 Impact Factor: 0.564

American Institute of Mathematical Sciences