May  2016, 10(2): 209-220. doi: 10.3934/amc.2016001

On codes over FFN$(1,q)$-projective varieties

1. 

Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, 09790, México, D. F., Mexico

2. 

Departamento de Matemáticas, Universidad Autónoma Metropolitana-I, 09340, México, D. F., Mexico

Received  March 2013 Revised  April 2015 Published  April 2016

For projective varieties defined over a finite field ${\mathbb F}_q$ we show that they contain a unique subvariety that satisfies the Finite Field Nullstellensatz property [1,2], for homogeneous linear polynomials over ${\mathbb F}_q$. Using these subvarieties we construct linear codes and estimate some of their parameters.
Citation: Jesús Carrillo-Pacheco, Felipe Zaldivar. On codes over FFN$(1,q)$-projective varieties. Advances in Mathematics of Communications, 2016, 10 (2) : 209-220. doi: 10.3934/amc.2016001
References:
[1]

E. Ballico and A. Cossidente, On the finite field nullstellensatz,, Austr. J. Combin., 21 (2000), 57.   Google Scholar

[2]

E. Ballico and A. Cossidente, Finite field nullstellensatz and Grassmannians,, Austr. J. Combin., 24 (2001), 313.   Google Scholar

[3]

R. Bernt, An Introduction to Symplectic Geometry,, Amer. Math. Soc., (1998).   Google Scholar

[4]

J. Buczynski, Properties of Legendrian subvarieties of projective space,, Geometria Dedicata, 118 (2006), 87.  doi: 10.1007/s10711-005-9027-y.  Google Scholar

[5]

J. Carrillo-Pacheco and F. Zaldivar, On Lagrangian-Grassmannian codes,, Des. Codes Crypt., 60 (2011), 291.  doi: 10.1007/s10623-010-9434-4.  Google Scholar

[6]

H. Chen, On the minimum distance of Schubert codes,, IEEE Trans. Inf. Theory, 46 (2000), 1535.  doi: 10.1109/18.850689.  Google Scholar

[7]

W. Fulton, Young Tableaux, with Applications to Representation Theory and Geometry,, Cambridge Univ. Press, (1997).   Google Scholar

[8]

S. R. Ghorpade and G. Lachaud, Higher weights of Grassmann codes,, in Coding Theory, (2000), 122.   Google Scholar

[9]

S. R. Ghorpade and G. Lachaud, Hyperplane sections of Grassmannians and the number of MDS linear codes,, Finite Fields Appl., 7 (2001), 468.  doi: 10.1006/ffta.2000.0299.  Google Scholar

[10]

S. R. Ghorpade, A. R. Patil and H. K. Pillai, Decomposable subspaces, linear sections of Grassmann varieties, and higher weights of Grassmann codes,, Finite Fields Appl., 15 (2009), 54.  doi: 10.1016/j.ffa.2008.08.001.  Google Scholar

[11]

S. R. Ghorpade, A. R. Patil and H. K. Pillai, Subclose families, threshold graphs, and the weight hierarchy of Grassmann and Schubert codes,, in Arithmetic, (2009), 87.  doi: 10.1090/conm/487/09526.  Google Scholar

[12]

S. R. Ghorpade and M. A. Tsfasman, Classical varieties, codes and combinatorics,, in Formal Power Series and Algebraic Combinatorics (eds. K. Eriksson and S. Linusson), (2003), 75.   Google Scholar

[13]

S. R. Ghorpade and M. A Tsfasman, Schubert varieties, linear codes and enumerative combinatorics,, Finite Fields Appl., 11 (2005), 684.  doi: 10.1016/j.ffa.2004.09.002.  Google Scholar

[14]

M. Grass, http://codetables.de, ., ().   Google Scholar

[15]

L. Guerra and R. Vincenti, On the linear codes arising from Schubert varieties,, Des. Codes Crypt., 33 (2004), 173.  doi: 10.1023/B:DESI.0000035470.05639.2b.  Google Scholar

[16]

G.-M. Hana, Schubert unions and codes from $l$-step flag varieties,, in Arithmetic, (2009), 43.   Google Scholar

[17]

G. M. Hana and T. Johnsen, Scroll codes,, Des. Codes Crypt., 45 (2007), 365.  doi: 10.1007/s10623-007-9131-0.  Google Scholar

[18]

J. P. Hansen, T. Johnsen and K. Ranestad, Schubert unions in Grassmann varieties,, Finite Fields Appl., 13 (2007), 738.  doi: 10.1016/j.ffa.2007.06.003.  Google Scholar

[19]

J. P. Hansen, T. Johnsen and K. Ranestad, Grassman codes and Schubert unions,, in Arithmetic, (2009), 103.   Google Scholar

[20]

R. Hill, A First Course in Coding Theory,, Clarendon Press, (1986).   Google Scholar

[21]

T. Ikeda, Schubert classes in the equivariant cohomology of the Lagrangian-Grassmannian,, Adv. Math., 215 (2007), 1.  doi: 10.1016/j.aim.2007.04.008.  Google Scholar

[22]

A. Iliek and K. Ranestad, Geometry of the Lagrangian-Grassmannian $LG(3,6)$ with applications to Brill-Noether Loci,, Michigan Math. J., 53 (2005), 383.  doi: 10.1307/mmj/1123090775.  Google Scholar

[23]

A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian-Grassmannian,, J. Algebraic Geometry, 12 (2003), 777.  doi: 10.1090/S1056-3911-03-00347-3.  Google Scholar

[24]

D. Yu Nogin, Codes associated to Grassmannians,, in Arithmetic, (1996), 145.   Google Scholar

[25]

D. Yu Nogin, Generalized Hamming weights of codes on multidimensional quadrics,, Prob. Inf. Trans. Theory, 29 (1993), 21.   Google Scholar

[26]

V. Pless, Power moment identities on weight distribution in error-correcting codes,, Inf. Contr., 6 (1962), 147.   Google Scholar

[27]

F. Rodier, Codes from flag varieties over a finite field,, J. Pure Apppl. Algebra, 178 (2003), 203.  doi: 10.1016/S0022-4049(02)00188-3.  Google Scholar

[28]

C. T. Ryan, An application of Grassmannian varieties to coding theory,, Congr. Numer, 57 (1987), 257.   Google Scholar

[29]

C. T. Ryan, Projective codes based on Grassmann varieties,, Congr. Num, 57 (1987), 273.   Google Scholar

[30]

C. T. Ryan and K. M. Ryan, The minimum weight of Grassmannian codes $C(k,n)$,, Disc. Appl. Math, 28 (1990), 149.  doi: 10.1016/0166-218X(90)90112-P.  Google Scholar

[31]

M. A. Tsfasman and S. G. Vladut, Algebraic Geometric Codes,, Kluwer, (1991).  doi: 10.1007/978-94-011-3810-9.  Google Scholar

[32]

M. A. Tsfasman and S. G. Vladut, Geometric approach to higher weights,, IEEE Trans. Inf. Theory, 41 (1995), 1564.  doi: 10.1109/18.476213.  Google Scholar

[33]

M. A. Tsfasman, S. G. Vladut and D. Nogin, Algebraic Geometric Codes: Basic Notions,, Amer. Math. Soc., (2007).  doi: 10.1090/surv/139.  Google Scholar

[34]

V. K. Wei, Generalized Hamming weights for linear codes,, IEEE Trans. Inf. Theory, 37 (1991), 1412.  doi: 10.1109/18.133259.  Google Scholar

[35]

X. Xiang, On the minimum distance conjecture for Schubert codes,, IEEE Trans. Inf. Theory 54 (2008), 54 (2008), 486.  doi: 10.1109/TIT.2007.911283.  Google Scholar

show all references

References:
[1]

E. Ballico and A. Cossidente, On the finite field nullstellensatz,, Austr. J. Combin., 21 (2000), 57.   Google Scholar

[2]

E. Ballico and A. Cossidente, Finite field nullstellensatz and Grassmannians,, Austr. J. Combin., 24 (2001), 313.   Google Scholar

[3]

R. Bernt, An Introduction to Symplectic Geometry,, Amer. Math. Soc., (1998).   Google Scholar

[4]

J. Buczynski, Properties of Legendrian subvarieties of projective space,, Geometria Dedicata, 118 (2006), 87.  doi: 10.1007/s10711-005-9027-y.  Google Scholar

[5]

J. Carrillo-Pacheco and F. Zaldivar, On Lagrangian-Grassmannian codes,, Des. Codes Crypt., 60 (2011), 291.  doi: 10.1007/s10623-010-9434-4.  Google Scholar

[6]

H. Chen, On the minimum distance of Schubert codes,, IEEE Trans. Inf. Theory, 46 (2000), 1535.  doi: 10.1109/18.850689.  Google Scholar

[7]

W. Fulton, Young Tableaux, with Applications to Representation Theory and Geometry,, Cambridge Univ. Press, (1997).   Google Scholar

[8]

S. R. Ghorpade and G. Lachaud, Higher weights of Grassmann codes,, in Coding Theory, (2000), 122.   Google Scholar

[9]

S. R. Ghorpade and G. Lachaud, Hyperplane sections of Grassmannians and the number of MDS linear codes,, Finite Fields Appl., 7 (2001), 468.  doi: 10.1006/ffta.2000.0299.  Google Scholar

[10]

S. R. Ghorpade, A. R. Patil and H. K. Pillai, Decomposable subspaces, linear sections of Grassmann varieties, and higher weights of Grassmann codes,, Finite Fields Appl., 15 (2009), 54.  doi: 10.1016/j.ffa.2008.08.001.  Google Scholar

[11]

S. R. Ghorpade, A. R. Patil and H. K. Pillai, Subclose families, threshold graphs, and the weight hierarchy of Grassmann and Schubert codes,, in Arithmetic, (2009), 87.  doi: 10.1090/conm/487/09526.  Google Scholar

[12]

S. R. Ghorpade and M. A. Tsfasman, Classical varieties, codes and combinatorics,, in Formal Power Series and Algebraic Combinatorics (eds. K. Eriksson and S. Linusson), (2003), 75.   Google Scholar

[13]

S. R. Ghorpade and M. A Tsfasman, Schubert varieties, linear codes and enumerative combinatorics,, Finite Fields Appl., 11 (2005), 684.  doi: 10.1016/j.ffa.2004.09.002.  Google Scholar

[14]

M. Grass, http://codetables.de, ., ().   Google Scholar

[15]

L. Guerra and R. Vincenti, On the linear codes arising from Schubert varieties,, Des. Codes Crypt., 33 (2004), 173.  doi: 10.1023/B:DESI.0000035470.05639.2b.  Google Scholar

[16]

G.-M. Hana, Schubert unions and codes from $l$-step flag varieties,, in Arithmetic, (2009), 43.   Google Scholar

[17]

G. M. Hana and T. Johnsen, Scroll codes,, Des. Codes Crypt., 45 (2007), 365.  doi: 10.1007/s10623-007-9131-0.  Google Scholar

[18]

J. P. Hansen, T. Johnsen and K. Ranestad, Schubert unions in Grassmann varieties,, Finite Fields Appl., 13 (2007), 738.  doi: 10.1016/j.ffa.2007.06.003.  Google Scholar

[19]

J. P. Hansen, T. Johnsen and K. Ranestad, Grassman codes and Schubert unions,, in Arithmetic, (2009), 103.   Google Scholar

[20]

R. Hill, A First Course in Coding Theory,, Clarendon Press, (1986).   Google Scholar

[21]

T. Ikeda, Schubert classes in the equivariant cohomology of the Lagrangian-Grassmannian,, Adv. Math., 215 (2007), 1.  doi: 10.1016/j.aim.2007.04.008.  Google Scholar

[22]

A. Iliek and K. Ranestad, Geometry of the Lagrangian-Grassmannian $LG(3,6)$ with applications to Brill-Noether Loci,, Michigan Math. J., 53 (2005), 383.  doi: 10.1307/mmj/1123090775.  Google Scholar

[23]

A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian-Grassmannian,, J. Algebraic Geometry, 12 (2003), 777.  doi: 10.1090/S1056-3911-03-00347-3.  Google Scholar

[24]

D. Yu Nogin, Codes associated to Grassmannians,, in Arithmetic, (1996), 145.   Google Scholar

[25]

D. Yu Nogin, Generalized Hamming weights of codes on multidimensional quadrics,, Prob. Inf. Trans. Theory, 29 (1993), 21.   Google Scholar

[26]

V. Pless, Power moment identities on weight distribution in error-correcting codes,, Inf. Contr., 6 (1962), 147.   Google Scholar

[27]

F. Rodier, Codes from flag varieties over a finite field,, J. Pure Apppl. Algebra, 178 (2003), 203.  doi: 10.1016/S0022-4049(02)00188-3.  Google Scholar

[28]

C. T. Ryan, An application of Grassmannian varieties to coding theory,, Congr. Numer, 57 (1987), 257.   Google Scholar

[29]

C. T. Ryan, Projective codes based on Grassmann varieties,, Congr. Num, 57 (1987), 273.   Google Scholar

[30]

C. T. Ryan and K. M. Ryan, The minimum weight of Grassmannian codes $C(k,n)$,, Disc. Appl. Math, 28 (1990), 149.  doi: 10.1016/0166-218X(90)90112-P.  Google Scholar

[31]

M. A. Tsfasman and S. G. Vladut, Algebraic Geometric Codes,, Kluwer, (1991).  doi: 10.1007/978-94-011-3810-9.  Google Scholar

[32]

M. A. Tsfasman and S. G. Vladut, Geometric approach to higher weights,, IEEE Trans. Inf. Theory, 41 (1995), 1564.  doi: 10.1109/18.476213.  Google Scholar

[33]

M. A. Tsfasman, S. G. Vladut and D. Nogin, Algebraic Geometric Codes: Basic Notions,, Amer. Math. Soc., (2007).  doi: 10.1090/surv/139.  Google Scholar

[34]

V. K. Wei, Generalized Hamming weights for linear codes,, IEEE Trans. Inf. Theory, 37 (1991), 1412.  doi: 10.1109/18.133259.  Google Scholar

[35]

X. Xiang, On the minimum distance conjecture for Schubert codes,, IEEE Trans. Inf. Theory 54 (2008), 54 (2008), 486.  doi: 10.1109/TIT.2007.911283.  Google Scholar

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