A note on diagonal and Hermitian hypersurfaces

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November 2016, 10(4): 753-764. doi: 10.3934/amc.2016039

A note on diagonal and Hermitian hypersurfaces

Ian Blake ^1, , V. Kumar Murty ^2, and Hamid Usefi ^3,

1.	Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada
2.	Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Canada M5S 2E4
3.	Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, A1C 5S7, Canada

Received October 2014 Revised August 2015 Published November 2016

Abstract
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Aspects of the properties, enumeration and construction of points on diagonal and Hermitian hypersurfaces have been considered extensively in the literature and are further considered here. The zeta function of diagonal hypersurfaces is given as a direct result of the work of Wolfmann. Recursive construction techniques for the set of rational points of Hermitian hypersurfaces are of interest. The relationship of these techniques here to the construction of codes on hypersurfaces is briefly noted.

Keywords: coding theory, diagonal hypersurfaces, evaluation code., zeta function, Hermitian varieties.

Mathematics Subject Classification: Primary: 11T71, 94B27; Secondary: 14G10, 14G5.

Citation: Ian Blake, V. Kumar Murty, Hamid Usefi. A note on diagonal and Hermitian hypersurfaces. Advances in Mathematics of Communications, 2016, 10 (4) : 753-764. doi: 10.3934/amc.2016039

References:

[1]	Y. Aubry, Reed-Muller codes associated to projective algebraic varieties, in Coding Theory and Algebraic Geometry, Springer, Berlin, 1992, 4-17. doi: 10.1007/BFb0087988.
[2]	D. Bartoli, M. De Boeck, S. Fanali and L. Storme, On the functional codes defined by quadrics and Hermitian varieties, Des. Codes Crypt., 71 (2014), 21-46. doi: 10.1007/s10623-012-9712-4.
[3]	R. C. Bose, On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements, Calcutta Math. Soc., Golden Jubilee Commemoration Volume, Part II, 1959, 341-356.
[4]	R. C. Bose and I. M. Chakravarti, Hermitian varieties in a finite projective space $PG(N,q)$, Canad. J. Math., 18 (1966), 1161-1182.
[5]	R. Calderbank and W. Kantor, The geometry of two weight codes, Bull. London Math. Soc., 18 (1986), 97-122. doi: 10.1112/blms/18.2.97.
[6]	J. P. Cherdieu and R. Rolland, On the number of points of some hypersurfaces in $\F_q^n$, Finite Fields Appl., 2 (1996), 214-224. doi: 10.1006/ffta.1996.0014.
[7]	F. Edoukou, Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen's conjecture, Finite Fields Appl., 13 (2008), 616-627. doi: 10.1016/j.ffa.2006.07.001.
[8]	F. Edoukou, A. Hallex, F. Rodier and L. Storme, The small weight codewords of the functional codes associated to non-singular Hermitian varieties, Des. Codes Crypt., 56 (2010), 219-233. doi: 10.1007/s10623-010-9401-0.
[9]	F. Edoukou, S. Ling and C. Xing, Intersection of two quadrics with no common hyperplane in $\mathbbP^n (\mathbbF_q )$,, preprint, ().
[10]	F. Edoukou, S. Ling and C. Xing, Structure of functional codes defined on non-degenerate Hermitian varieties, J. Combin. Theory Ser. A, 118 (2011), 2436-2444. doi: 10.1016/j.jcta.2011.05.006.
[11]	S. R. Ghorpade and G. Lachaud, Number of solutions of equations over finite fields and a conjecture of Lang and Weil, in Number Theory and Discrete Mathematics, Birkhäuser, Basel, 2002, 269-291.
[12]	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.
[13]	S. H. Hansen, Error-correcting codes from higher dimensional varieties, Finite Fields Appl., 7 (2001), 530-552. doi: 10.1006/ffta.2001.0313.
[14]	K. Ireland and M. Rosen, A Cclassical Introduction to Modern Number Theory, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2103-4.
[15]	N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta Functions, Springer-Verlag, New York, 1977.
[16]	G. Lachaud, The parameters of projective Reed-Müller codes, Discrete Math., 81 (1990), 217-221. doi: 10.1016/0012-365X(90)90155-B.
[17]	G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in Arithmetic, Geometry and Coding Theory, Walter de Gruyter, 1993.
[18]	J. B. Little, Algebraic geometry codes from higher dimensional varieties, in Advances in Algebraic Geometry Codes, World Sci. Publ., Hackensack, NJ, 2008, 257-293. doi: 10.1142/9789812794017_0007.
[19]	A. Sboui, Second highest number of points of hypersurfaces in $\F_q^n$, Finite fields and their applications, 13 (2007), 444-449. doi: 10.1016/j.ffa.2005.11.002.
[20]	A. B. Sørensen, Projective Reed-Müller codes, IEEE Trans. Inform. Theory, 17 (1991), 1567-1576. doi: 10.1109/18.104317.
[21]	A. B. Sørensen, Rational Points on Hypersurfaces, Reed-Muller Codes and Algebraic Geometric Codes, Ph.D. thesis, Aarhus, 1991.
[22]	A. B. Sørensen, On the number of rational points on codimension-1 algebraic sets in $P^n (F_q)$, Discrete Math., 135 (1994), 321-324. doi: 10.1016/0012-365X(93)E0009-S.
[23]	H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, New York, 1991.
[24]	M. Tsfasman, S. Vladut and D. Nogin, Algebraic Geometric Codes: Basic Notions, AMS, 2007. doi: 10.1090/surv/139.
[25]	A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc., 55 (1949), 497-508.
[26]	J. Wolfmann, The number of solutions of certain diagonal equations over finite fields, J. Number Theory, 42 (1992), 247-257. doi: 10.1016/0022-314X(92)90091-3.
[27]	K. Yang and V. Kumar, On the true minimum distance of Hermitian codes, Coding Theory and Algebraic Geometry, Springer, Berlin, 1992, 99-107. doi: 10.1007/BFb0087995.
[28]	M. Zarzar, Error-correcting codes on low rank surfaces, Finite Fields Appl., 13 (2007), 727-737. doi: 10.1016/j.ffa.2007.05.001.

show all references

References:

[1]	Y. Aubry, Reed-Muller codes associated to projective algebraic varieties, in Coding Theory and Algebraic Geometry, Springer, Berlin, 1992, 4-17. doi: 10.1007/BFb0087988.
[2]	D. Bartoli, M. De Boeck, S. Fanali and L. Storme, On the functional codes defined by quadrics and Hermitian varieties, Des. Codes Crypt., 71 (2014), 21-46. doi: 10.1007/s10623-012-9712-4.
[3]	R. C. Bose, On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements, Calcutta Math. Soc., Golden Jubilee Commemoration Volume, Part II, 1959, 341-356.
[4]	R. C. Bose and I. M. Chakravarti, Hermitian varieties in a finite projective space $PG(N,q)$, Canad. J. Math., 18 (1966), 1161-1182.
[5]	R. Calderbank and W. Kantor, The geometry of two weight codes, Bull. London Math. Soc., 18 (1986), 97-122. doi: 10.1112/blms/18.2.97.
[6]	J. P. Cherdieu and R. Rolland, On the number of points of some hypersurfaces in $\F_q^n$, Finite Fields Appl., 2 (1996), 214-224. doi: 10.1006/ffta.1996.0014.
[7]	F. Edoukou, Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen's conjecture, Finite Fields Appl., 13 (2008), 616-627. doi: 10.1016/j.ffa.2006.07.001.
[8]	F. Edoukou, A. Hallex, F. Rodier and L. Storme, The small weight codewords of the functional codes associated to non-singular Hermitian varieties, Des. Codes Crypt., 56 (2010), 219-233. doi: 10.1007/s10623-010-9401-0.
[9]	F. Edoukou, S. Ling and C. Xing, Intersection of two quadrics with no common hyperplane in $\mathbbP^n (\mathbbF_q )$,, preprint, ().
[10]	F. Edoukou, S. Ling and C. Xing, Structure of functional codes defined on non-degenerate Hermitian varieties, J. Combin. Theory Ser. A, 118 (2011), 2436-2444. doi: 10.1016/j.jcta.2011.05.006.
[11]	S. R. Ghorpade and G. Lachaud, Number of solutions of equations over finite fields and a conjecture of Lang and Weil, in Number Theory and Discrete Mathematics, Birkhäuser, Basel, 2002, 269-291.
[12]	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.
[13]	S. H. Hansen, Error-correcting codes from higher dimensional varieties, Finite Fields Appl., 7 (2001), 530-552. doi: 10.1006/ffta.2001.0313.
[14]	K. Ireland and M. Rosen, A Cclassical Introduction to Modern Number Theory, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2103-4.
[15]	N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta Functions, Springer-Verlag, New York, 1977.
[16]	G. Lachaud, The parameters of projective Reed-Müller codes, Discrete Math., 81 (1990), 217-221. doi: 10.1016/0012-365X(90)90155-B.
[17]	G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in Arithmetic, Geometry and Coding Theory, Walter de Gruyter, 1993.
[18]	J. B. Little, Algebraic geometry codes from higher dimensional varieties, in Advances in Algebraic Geometry Codes, World Sci. Publ., Hackensack, NJ, 2008, 257-293. doi: 10.1142/9789812794017_0007.
[19]	A. Sboui, Second highest number of points of hypersurfaces in $\F_q^n$, Finite fields and their applications, 13 (2007), 444-449. doi: 10.1016/j.ffa.2005.11.002.
[20]	A. B. Sørensen, Projective Reed-Müller codes, IEEE Trans. Inform. Theory, 17 (1991), 1567-1576. doi: 10.1109/18.104317.
[21]	A. B. Sørensen, Rational Points on Hypersurfaces, Reed-Muller Codes and Algebraic Geometric Codes, Ph.D. thesis, Aarhus, 1991.
[22]	A. B. Sørensen, On the number of rational points on codimension-1 algebraic sets in $P^n (F_q)$, Discrete Math., 135 (1994), 321-324. doi: 10.1016/0012-365X(93)E0009-S.
[23]	H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, New York, 1991.
[24]	M. Tsfasman, S. Vladut and D. Nogin, Algebraic Geometric Codes: Basic Notions, AMS, 2007. doi: 10.1090/surv/139.
[25]	A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc., 55 (1949), 497-508.
[26]	J. Wolfmann, The number of solutions of certain diagonal equations over finite fields, J. Number Theory, 42 (1992), 247-257. doi: 10.1016/0022-314X(92)90091-3.
[27]	K. Yang and V. Kumar, On the true minimum distance of Hermitian codes, Coding Theory and Algebraic Geometry, Springer, Berlin, 1992, 99-107. doi: 10.1007/BFb0087995.
[28]	M. Zarzar, Error-correcting codes on low rank surfaces, Finite Fields Appl., 13 (2007), 727-737. doi: 10.1016/j.ffa.2007.05.001.

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