-
Previous Article
Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$
- AMC Home
- This Issue
-
Next Article
An extension of binary threshold sequences from Fermat quotients
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 |
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. |
[1] |
Ghislain Fourier, Gabriele Nebe. Degenerate flag varieties in network coding. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021027 |
[2] |
J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413 |
[3] |
Fabrizio Catanese, Luca Cesarano. Canonical maps of general hypersurfaces in Abelian varieties. Electronic Research Archive, 2021, 29 (6) : 4315-4325. doi: 10.3934/era.2021087 |
[4] |
Alexander Gorodnik, Frédéric Paulin. Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows. Journal of Modern Dynamics, 2014, 8 (1) : 25-59. doi: 10.3934/jmd.2014.8.25 |
[5] |
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 |
[6] |
Carla Mascia, Giancarlo Rinaldo, Massimiliano Sala. Hilbert quasi-polynomial for order domains and application to coding theory. Advances in Mathematics of Communications, 2018, 12 (2) : 287-301. doi: 10.3934/amc.2018018 |
[7] |
Simone Fiori, Italo Cervigni, Mattia Ippoliti, Claudio Menotta. Synchronization of dynamical systems on Riemannian manifolds by an extended PID-type control theory: Numerical evaluation. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022047 |
[8] |
María Chara, Ricardo A. Podestá, Ricardo Toledano. The conorm code of an AG-code. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021018 |
[9] |
Roland Martin. On simple Igusa local zeta functions. Electronic Research Announcements, 1995, 1: 108-111. |
[10] |
V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511 |
[11] |
Lawrence Ein, Wenbo Niu, Jinhyung Park. On blowup of secant varieties of curves. Electronic Research Archive, 2021, 29 (6) : 3649-3654. doi: 10.3934/era.2021055 |
[12] |
Simon Scott. Relative zeta determinants and the geometry of the determinant line bundle. Electronic Research Announcements, 2001, 7: 8-16. |
[13] |
Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1 |
[14] |
Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003 |
[15] |
Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45 |
[16] |
D. Warren, K Najarian. Learning theory applied to Sigmoid network classification of protein biological function using primary protein structure. Conference Publications, 2003, 2003 (Special) : 898-904. doi: 10.3934/proc.2003.2003.898 |
[17] |
Anton Izosimov. Pentagrams, inscribed polygons, and Prym varieties. Electronic Research Announcements, 2016, 23: 25-40. doi: 10.3934/era.2016.23.004 |
[18] |
G. Mashevitzky, B. Plotkin and E. Plotkin. Automorphisms of categories of free algebras of varieties. Electronic Research Announcements, 2002, 8: 1-10. |
[19] |
A. Giambruno and M. Zaicev. Minimal varieties of algebras of exponential growth. Electronic Research Announcements, 2000, 6: 40-44. |
[20] |
Laurenţiu Maxim, Jörg Schürmann. Characteristic classes of singular toric varieties. Electronic Research Announcements, 2013, 20: 109-120. doi: 10.3934/era.2013.20.109 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]