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), 57-60.

[2]

E. Ballico and A. Cossidente, Finite field nullstellensatz and Grassmannians, Austr. J. Combin., 24 (2001), 313-315.

[3]

R. Bernt, An Introduction to Symplectic Geometry, Amer. Math. Soc., Providence, 1998.

[4]

J. Buczynski, Properties of Legendrian subvarieties of projective space, Geometria Dedicata, 118 (2006), 87-103. doi: 10.1007/s10711-005-9027-y.

[5]

J. Carrillo-Pacheco and F. Zaldivar, On Lagrangian-Grassmannian codes, Des. Codes Crypt., 60 (2011), 291-298. doi: 10.1007/s10623-010-9434-4.

[6]

H. Chen, On the minimum distance of Schubert codes, IEEE Trans. Inf. Theory, 46 (2000), 1535-1538. doi: 10.1109/18.850689.

[7]

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

[8]

S. R. Ghorpade and G. Lachaud, Higher weights of Grassmann codes, in Coding Theory, Cryptography and Related Areas, Springer-Verlag, Berlin, 2000, 122-131.

[9]

S. R. Ghorpade and G. Lachaud, Hyperplane sections of Grassmannians and the number of MDS linear codes, Finite Fields Appl., 7 (2001), 468-506. doi: 10.1006/ffta.2000.0299.

[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), 54-68. doi: 10.1016/j.ffa.2008.08.001.

[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, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2009, 87-99. doi: 10.1090/conm/487/09526.

[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), Linkping Univ., 2003, 75-84.

[13]

S. R. Ghorpade and M. A Tsfasman, Schubert varieties, linear codes and enumerative combinatorics, Finite Fields Appl., 11 (2005), 684-699. doi: 10.1016/j.ffa.2004.09.002.

[14]

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

[15]

L. Guerra and R. Vincenti, On the linear codes arising from Schubert varieties, Des. Codes Crypt., 33 (2004), 173-180. doi: 10.1023/B:DESI.0000035470.05639.2b.

[16]

G.-M. Hana, Schubert unions and codes from $l$-step flag varieties, in Arithmetic, Geometry, and Coding Theory, Soc. Math. France, Paris, 2009, 43-61.

[17]

G. M. Hana and T. Johnsen, Scroll codes, Des. Codes Crypt., 45 (2007), 365-377. doi: 10.1007/s10623-007-9131-0.

[18]

J. P. Hansen, T. Johnsen and K. Ranestad, Schubert unions in Grassmann varieties, Finite Fields Appl., 13 (2007), 738-750. doi: 10.1016/j.ffa.2007.06.003.

[19]

J. P. Hansen, T. Johnsen and K. Ranestad, Grassman codes and Schubert unions, in Arithmetic, Geometry, and Coding Theory, Soc. Math. France, Paris, 2009, 103-121.

[20]

R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford, 1986.

[21]

T. Ikeda, Schubert classes in the equivariant cohomology of the Lagrangian-Grassmannian, Adv. Math., 215 (2007), 1-23. doi: 10.1016/j.aim.2007.04.008.

[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), 383-417. doi: 10.1307/mmj/1123090775.

[23]

A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian-Grassmannian, J. Algebraic Geometry, 12 (2003), 777-810. doi: 10.1090/S1056-3911-03-00347-3.

[24]

D. Yu Nogin, Codes associated to Grassmannians, in Arithmetic, Geometry and Coding Theory, Walter de Gruyter, Berlin, 1996, 145-154.

[25]

D. Yu Nogin, Generalized Hamming weights of codes on multidimensional quadrics, Prob. Inf. Trans. Theory, 29 (1993), 21-30.

[26]

V. Pless, Power moment identities on weight distribution in error-correcting codes, Inf. Contr., 6 (1962), 147-152.

[27]

F. Rodier, Codes from flag varieties over a finite field, J. Pure Apppl. Algebra, 178 (2003), 203-214. doi: 10.1016/S0022-4049(02)00188-3.

[28]

C. T. Ryan, An application of Grassmannian varieties to coding theory, Congr. Numer, 57 (1987), 257-271.

[29]

C. T. Ryan, Projective codes based on Grassmann varieties, Congr. Num, 57 (1987), 273-279.

[30]

C. T. Ryan and K. M. Ryan, The minimum weight of Grassmannian codes $C(k,n)$, Disc. Appl. Math, 28 (1990), 149-156. doi: 10.1016/0166-218X(90)90112-P.

[31]

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

[32]

M. A. Tsfasman and S. G. Vladut, Geometric approach to higher weights, IEEE Trans. Inf. Theory, 41 (1995), 1564-1588. doi: 10.1109/18.476213.

[33]

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

[34]

V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inf. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259.

[35]

X. Xiang, On the minimum distance conjecture for Schubert codes, IEEE Trans. Inf. Theory 54 (2008), 486-488. doi: 10.1109/TIT.2007.911283.

show all references

References:
[1]

E. Ballico and A. Cossidente, On the finite field nullstellensatz, Austr. J. Combin., 21 (2000), 57-60.

[2]

E. Ballico and A. Cossidente, Finite field nullstellensatz and Grassmannians, Austr. J. Combin., 24 (2001), 313-315.

[3]

R. Bernt, An Introduction to Symplectic Geometry, Amer. Math. Soc., Providence, 1998.

[4]

J. Buczynski, Properties of Legendrian subvarieties of projective space, Geometria Dedicata, 118 (2006), 87-103. doi: 10.1007/s10711-005-9027-y.

[5]

J. Carrillo-Pacheco and F. Zaldivar, On Lagrangian-Grassmannian codes, Des. Codes Crypt., 60 (2011), 291-298. doi: 10.1007/s10623-010-9434-4.

[6]

H. Chen, On the minimum distance of Schubert codes, IEEE Trans. Inf. Theory, 46 (2000), 1535-1538. doi: 10.1109/18.850689.

[7]

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

[8]

S. R. Ghorpade and G. Lachaud, Higher weights of Grassmann codes, in Coding Theory, Cryptography and Related Areas, Springer-Verlag, Berlin, 2000, 122-131.

[9]

S. R. Ghorpade and G. Lachaud, Hyperplane sections of Grassmannians and the number of MDS linear codes, Finite Fields Appl., 7 (2001), 468-506. doi: 10.1006/ffta.2000.0299.

[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), 54-68. doi: 10.1016/j.ffa.2008.08.001.

[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, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2009, 87-99. doi: 10.1090/conm/487/09526.

[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), Linkping Univ., 2003, 75-84.

[13]

S. R. Ghorpade and M. A Tsfasman, Schubert varieties, linear codes and enumerative combinatorics, Finite Fields Appl., 11 (2005), 684-699. doi: 10.1016/j.ffa.2004.09.002.

[14]

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

[15]

L. Guerra and R. Vincenti, On the linear codes arising from Schubert varieties, Des. Codes Crypt., 33 (2004), 173-180. doi: 10.1023/B:DESI.0000035470.05639.2b.

[16]

G.-M. Hana, Schubert unions and codes from $l$-step flag varieties, in Arithmetic, Geometry, and Coding Theory, Soc. Math. France, Paris, 2009, 43-61.

[17]

G. M. Hana and T. Johnsen, Scroll codes, Des. Codes Crypt., 45 (2007), 365-377. doi: 10.1007/s10623-007-9131-0.

[18]

J. P. Hansen, T. Johnsen and K. Ranestad, Schubert unions in Grassmann varieties, Finite Fields Appl., 13 (2007), 738-750. doi: 10.1016/j.ffa.2007.06.003.

[19]

J. P. Hansen, T. Johnsen and K. Ranestad, Grassman codes and Schubert unions, in Arithmetic, Geometry, and Coding Theory, Soc. Math. France, Paris, 2009, 103-121.

[20]

R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford, 1986.

[21]

T. Ikeda, Schubert classes in the equivariant cohomology of the Lagrangian-Grassmannian, Adv. Math., 215 (2007), 1-23. doi: 10.1016/j.aim.2007.04.008.

[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), 383-417. doi: 10.1307/mmj/1123090775.

[23]

A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian-Grassmannian, J. Algebraic Geometry, 12 (2003), 777-810. doi: 10.1090/S1056-3911-03-00347-3.

[24]

D. Yu Nogin, Codes associated to Grassmannians, in Arithmetic, Geometry and Coding Theory, Walter de Gruyter, Berlin, 1996, 145-154.

[25]

D. Yu Nogin, Generalized Hamming weights of codes on multidimensional quadrics, Prob. Inf. Trans. Theory, 29 (1993), 21-30.

[26]

V. Pless, Power moment identities on weight distribution in error-correcting codes, Inf. Contr., 6 (1962), 147-152.

[27]

F. Rodier, Codes from flag varieties over a finite field, J. Pure Apppl. Algebra, 178 (2003), 203-214. doi: 10.1016/S0022-4049(02)00188-3.

[28]

C. T. Ryan, An application of Grassmannian varieties to coding theory, Congr. Numer, 57 (1987), 257-271.

[29]

C. T. Ryan, Projective codes based on Grassmann varieties, Congr. Num, 57 (1987), 273-279.

[30]

C. T. Ryan and K. M. Ryan, The minimum weight of Grassmannian codes $C(k,n)$, Disc. Appl. Math, 28 (1990), 149-156. doi: 10.1016/0166-218X(90)90112-P.

[31]

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

[32]

M. A. Tsfasman and S. G. Vladut, Geometric approach to higher weights, IEEE Trans. Inf. Theory, 41 (1995), 1564-1588. doi: 10.1109/18.476213.

[33]

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

[34]

V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inf. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259.

[35]

X. Xiang, On the minimum distance conjecture for Schubert codes, IEEE Trans. Inf. Theory 54 (2008), 486-488. doi: 10.1109/TIT.2007.911283.

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