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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

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