• 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 $\mathcal{H}(2n+1,q^{2})$

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.
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).   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

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

[1]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[2]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[3]

Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167

[4]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[5]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]