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August  2014, 8(3): 271-280. doi: 10.3934/amc.2014.8.271

On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric

Daniele Bartoli 1, and  Leo Storme 2,

1. 

Dipartimento di Matematica ed Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
2. 

Department of Mathematics, Ghent University, Krijgslaan 281 - S22, 9000 Ghent

Received  January 2013 Revised  January 2014 Published  August 2014

We discuss the functional codes $C_h(\mathcal{Q}_N)$, for small $h\geq 3$, $q>9$, and for $N\geq 6$. This continues the study of different classes of functional codes, performed on functional codes arising from quadrics and Hermitian varieties. Here, we consider the functional codes arising from the intersections of the algebraic hypersurfaces of small degree $h$ with a given non-singular quadric $\mathcal{Q}_N$ in PG$(N,q)$.
Keywords: intersections., small weight codewords, quadrics, Functional codes, algebraic varieties.
Mathematics Subject Classification: Primary: 51E20, 94B05; Secondary: 05B2.
Citation: Daniele Bartoli, Leo Storme. On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric. Advances in Mathematics of Communications, 2014, 8 (3) : 271-280. doi: 10.3934/amc.2014.8.271
References:
[1]

D. Bartoli, M. De Boeck, S. Fanali and L. Storme, On the functional codes defined by quadrics and Hermitian varieties, Des. Codes Cryptogr., 71 (2014), 21-46. doi: 10.1007/s10623-012-9712-4.  Google Scholar

[2]

A. Cafure and G. Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl., 12 (2006), 155-185. doi: 10.1016/j.ffa.2005.03.003.  Google Scholar

[3]

F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, A study of intersections of quadrics having applications on the small weight codewords of the functional codes $C_2(Q)$, $Q$ a non-singular quadric, J. Pure Appl. Algebra, 214 (2010), 1729-1739. doi: 10.1016/j.jpaa.2009.12.017.  Google Scholar

[4]

F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, On the small weight codewords of the functional codes $C_{herm}(X)$, $X$ a non-singular Hermitian variety, Des. Codes Cryptogr., 56 (2010), 219-233. doi: 10.1007/s10623-010-9401-0.  Google Scholar

[5]

A. Hallez and L. Storme, Functional codes arising from quadric intersections with Hermitian varieties, Finite Fields Appl., 16 (2010), 27-35. doi: 10.1016/j.ffa.2009.11.005.  Google Scholar

[6]

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

[7]

G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in Arithmetic, Geometry, and Coding Theory, Walter De Gruyter, Berlin, 1996, 77-104.  Google Scholar

show all references

References:
[1]

D. Bartoli, M. De Boeck, S. Fanali and L. Storme, On the functional codes defined by quadrics and Hermitian varieties, Des. Codes Cryptogr., 71 (2014), 21-46. doi: 10.1007/s10623-012-9712-4.  Google Scholar

[2]

A. Cafure and G. Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl., 12 (2006), 155-185. doi: 10.1016/j.ffa.2005.03.003.  Google Scholar

[3]

F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, A study of intersections of quadrics having applications on the small weight codewords of the functional codes $C_2(Q)$, $Q$ a non-singular quadric, J. Pure Appl. Algebra, 214 (2010), 1729-1739. doi: 10.1016/j.jpaa.2009.12.017.  Google Scholar

[4]

F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, On the small weight codewords of the functional codes $C_{herm}(X)$, $X$ a non-singular Hermitian variety, Des. Codes Cryptogr., 56 (2010), 219-233. doi: 10.1007/s10623-010-9401-0.  Google Scholar

[5]

A. Hallez and L. Storme, Functional codes arising from quadric intersections with Hermitian varieties, Finite Fields Appl., 16 (2010), 27-35. doi: 10.1016/j.ffa.2009.11.005.  Google Scholar

[6]

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

[7]

G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in Arithmetic, Geometry, and Coding Theory, Walter De Gruyter, Berlin, 1996, 77-104.  Google Scholar

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