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

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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

Abstract
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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.
[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.
[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.
[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.
[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.
[6]	J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, The Clarendon Press, Oxford University Press, 1991.
[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.

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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.
[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.
[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.
[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.
[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.
[6]	J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, The Clarendon Press, Oxford University Press, 1991.
[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.

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