# American Institute of Mathematical Sciences

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

## A note on diagonal and Hermitian hypersurfaces

 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

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.
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, (1992), 4.  doi: 10.1007/BFb0087988.  Google Scholar [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.  doi: 10.1007/s10623-012-9712-4.  Google Scholar [3] R. C. Bose, On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements,, Calcutta Math. Soc., (1959), 341.   Google Scholar [4] R. C. Bose and I. M. Chakravarti, Hermitian varieties in a finite projective space $PG(N,q)$,, Canad. J. Math., 18 (1966), 1161.   Google Scholar [5] R. Calderbank and W. Kantor, The geometry of two weight codes,, Bull. London Math. Soc., 18 (1986), 97.  doi: 10.1112/blms/18.2.97.  Google Scholar [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.  doi: 10.1006/ffta.1996.0014.  Google Scholar [7] F. Edoukou, Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen's conjecture,, Finite Fields Appl., 13 (2008), 616.  doi: 10.1016/j.ffa.2006.07.001.  Google Scholar [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.  doi: 10.1007/s10623-010-9401-0.  Google Scholar [9] F. Edoukou, S. Ling and C. Xing, Intersection of two quadrics with no common hyperplane in $\mathbbP^n (\mathbbF_q )$,, preprint, ().   Google Scholar [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.  doi: 10.1016/j.jcta.2011.05.006.  Google Scholar [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, (2002), 269.   Google Scholar [12] A. Hallez and L. Storme, Functional codes arising from quadric intersections with Hermitian varieties,, Finite Fields Appl., 16 (2010), 27.  doi: 10.1016/j.ffa.2009.11.005.  Google Scholar [13] S. H. Hansen, Error-correcting codes from higher dimensional varieties,, Finite Fields Appl., 7 (2001), 530.  doi: 10.1006/ffta.2001.0313.  Google Scholar [14] K. Ireland and M. Rosen, A Cclassical Introduction to Modern Number Theory,, Springer-Verlag, (1990).  doi: 10.1007/978-1-4757-2103-4.  Google Scholar [15] N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta Functions,, Springer-Verlag, (1977).   Google Scholar [16] G. Lachaud, The parameters of projective Reed-Müller codes,, Discrete Math., 81 (1990), 217.  doi: 10.1016/0012-365X(90)90155-B.  Google Scholar [17] G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties,, in Arithmetic, (1993).   Google Scholar [18] J. B. Little, Algebraic geometry codes from higher dimensional varieties,, in Advances in Algebraic Geometry Codes, (2008), 257.  doi: 10.1142/9789812794017_0007.  Google Scholar [19] A. Sboui, Second highest number of points of hypersurfaces in $\F_q^n$,, Finite fields and their applications, 13 (2007), 444.  doi: 10.1016/j.ffa.2005.11.002.  Google Scholar [20] A. B. Sørensen, Projective Reed-Müller codes,, IEEE Trans. Inform. Theory, 17 (1991), 1567.  doi: 10.1109/18.104317.  Google Scholar [21] A. B. Sørensen, Rational Points on Hypersurfaces, Reed-Muller Codes and Algebraic Geometric Codes,, Ph.D. thesis, (1991).   Google Scholar [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.  doi: 10.1016/0012-365X(93)E0009-S.  Google Scholar [23] H. Stichtenoth, Algebraic Function Fields and Codes,, Springer-Verlag, (1991).   Google Scholar [24] M. Tsfasman, S. Vladut and D. Nogin, Algebraic Geometric Codes: Basic Notions,, AMS, (2007).  doi: 10.1090/surv/139.  Google Scholar [25] A. Weil, Numbers of solutions of equations in finite fields,, Bull. Amer. Math. Soc., 55 (1949), 497.   Google Scholar [26] J. Wolfmann, The number of solutions of certain diagonal equations over finite fields,, J. Number Theory, 42 (1992), 247.  doi: 10.1016/0022-314X(92)90091-3.  Google Scholar [27] K. Yang and V. Kumar, On the true minimum distance of Hermitian codes,, Coding Theory and Algebraic Geometry, (1992), 99.  doi: 10.1007/BFb0087995.  Google Scholar [28] M. Zarzar, Error-correcting codes on low rank surfaces,, Finite Fields Appl., 13 (2007), 727.  doi: 10.1016/j.ffa.2007.05.001.  Google Scholar

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##### References:
 [1] Y. Aubry, Reed-Muller codes associated to projective algebraic varieties,, in Coding Theory and Algebraic Geometry, (1992), 4.  doi: 10.1007/BFb0087988.  Google Scholar [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.  doi: 10.1007/s10623-012-9712-4.  Google Scholar [3] R. C. Bose, On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements,, Calcutta Math. Soc., (1959), 341.   Google Scholar [4] R. C. Bose and I. M. Chakravarti, Hermitian varieties in a finite projective space $PG(N,q)$,, Canad. J. Math., 18 (1966), 1161.   Google Scholar [5] R. Calderbank and W. Kantor, The geometry of two weight codes,, Bull. London Math. Soc., 18 (1986), 97.  doi: 10.1112/blms/18.2.97.  Google Scholar [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.  doi: 10.1006/ffta.1996.0014.  Google Scholar [7] F. Edoukou, Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen's conjecture,, Finite Fields Appl., 13 (2008), 616.  doi: 10.1016/j.ffa.2006.07.001.  Google Scholar [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.  doi: 10.1007/s10623-010-9401-0.  Google Scholar [9] F. Edoukou, S. Ling and C. Xing, Intersection of two quadrics with no common hyperplane in $\mathbbP^n (\mathbbF_q )$,, preprint, ().   Google Scholar [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.  doi: 10.1016/j.jcta.2011.05.006.  Google Scholar [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, (2002), 269.   Google Scholar [12] A. Hallez and L. Storme, Functional codes arising from quadric intersections with Hermitian varieties,, Finite Fields Appl., 16 (2010), 27.  doi: 10.1016/j.ffa.2009.11.005.  Google Scholar [13] S. H. Hansen, Error-correcting codes from higher dimensional varieties,, Finite Fields Appl., 7 (2001), 530.  doi: 10.1006/ffta.2001.0313.  Google Scholar [14] K. Ireland and M. Rosen, A Cclassical Introduction to Modern Number Theory,, Springer-Verlag, (1990).  doi: 10.1007/978-1-4757-2103-4.  Google Scholar [15] N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta Functions,, Springer-Verlag, (1977).   Google Scholar [16] G. Lachaud, The parameters of projective Reed-Müller codes,, Discrete Math., 81 (1990), 217.  doi: 10.1016/0012-365X(90)90155-B.  Google Scholar [17] G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties,, in Arithmetic, (1993).   Google Scholar [18] J. B. Little, Algebraic geometry codes from higher dimensional varieties,, in Advances in Algebraic Geometry Codes, (2008), 257.  doi: 10.1142/9789812794017_0007.  Google Scholar [19] A. Sboui, Second highest number of points of hypersurfaces in $\F_q^n$,, Finite fields and their applications, 13 (2007), 444.  doi: 10.1016/j.ffa.2005.11.002.  Google Scholar [20] A. B. Sørensen, Projective Reed-Müller codes,, IEEE Trans. Inform. Theory, 17 (1991), 1567.  doi: 10.1109/18.104317.  Google Scholar [21] A. B. Sørensen, Rational Points on Hypersurfaces, Reed-Muller Codes and Algebraic Geometric Codes,, Ph.D. thesis, (1991).   Google Scholar [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.  doi: 10.1016/0012-365X(93)E0009-S.  Google Scholar [23] H. Stichtenoth, Algebraic Function Fields and Codes,, Springer-Verlag, (1991).   Google Scholar [24] M. Tsfasman, S. Vladut and D. Nogin, Algebraic Geometric Codes: Basic Notions,, AMS, (2007).  doi: 10.1090/surv/139.  Google Scholar [25] A. Weil, Numbers of solutions of equations in finite fields,, Bull. Amer. Math. Soc., 55 (1949), 497.   Google Scholar [26] J. Wolfmann, The number of solutions of certain diagonal equations over finite fields,, J. Number Theory, 42 (1992), 247.  doi: 10.1016/0022-314X(92)90091-3.  Google Scholar [27] K. Yang and V. Kumar, On the true minimum distance of Hermitian codes,, Coding Theory and Algebraic Geometry, (1992), 99.  doi: 10.1007/BFb0087995.  Google Scholar [28] M. Zarzar, Error-correcting codes on low rank surfaces,, Finite Fields Appl., 13 (2007), 727.  doi: 10.1016/j.ffa.2007.05.001.  Google Scholar
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