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

M. De Boeck; P. Vandendriessche

doi:10.3934/amc.2014.8.281

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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

Abstract
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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:

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[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. Google Scholar
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[9]	P. Vandendriessche, LDPC codes associated with linear representations of geometries,, Adv. Math. Commun., 4 (2010), 405. 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. 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, (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. 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). doi: 10.1103/PhysRevA.82.042338. Google Scholar
[4]	J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries,, Oxford University Press, (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. 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. 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, (2010), 185. 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. 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. 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. doi: 10.1007/s10623-009-9324-9. Google Scholar

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

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