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.  Google Scholar

[2]

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

[3]

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

[4]

J. Buczynski, Properties of Legendrian subvarieties of projective space, Geometria Dedicata, 118 (2006), 87-103. 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), 291-298. 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), 1535-1538. 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, Cryptography and Related Areas, Springer-Verlag, Berlin, 2000, 122-131.  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), 468-506. 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), 54-68. 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, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2009, 87-99. 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), Linkping Univ., 2003, 75-84. Google Scholar

[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.  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), 173-180. 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, Geometry, and Coding Theory, Soc. Math. France, Paris, 2009, 43-61.  Google Scholar

[17]

G. M. Hana and T. Johnsen, Scroll codes, Des. Codes Crypt., 45 (2007), 365-377. 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), 738-750. 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, Geometry, and Coding Theory, Soc. Math. France, Paris, 2009, 103-121.  Google Scholar

[20]

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

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

[23]

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

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

M. A. Tsfasman and S. G. Vladut, Algebraic Geometric Codes, Kluwer, Amsterdam, 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), 1564-1588. 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., Providence, 2007. doi: 10.1090/surv/139.  Google Scholar

[34]

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

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

J. Buczynski, Properties of Legendrian subvarieties of projective space, Geometria Dedicata, 118 (2006), 87-103. 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), 291-298. 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), 1535-1538. 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, Cryptography and Related Areas, Springer-Verlag, Berlin, 2000, 122-131.  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), 468-506. 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), 54-68. 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, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2009, 87-99. 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), Linkping Univ., 2003, 75-84. Google Scholar

[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.  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), 173-180. 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, Geometry, and Coding Theory, Soc. Math. France, Paris, 2009, 43-61.  Google Scholar

[17]

G. M. Hana and T. Johnsen, Scroll codes, Des. Codes Crypt., 45 (2007), 365-377. 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), 738-750. 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, Geometry, and Coding Theory, Soc. Math. France, Paris, 2009, 103-121.  Google Scholar

[20]

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

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

[23]

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

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

M. A. Tsfasman and S. G. Vladut, Algebraic Geometric Codes, Kluwer, Amsterdam, 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), 1564-1588. 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., Providence, 2007. doi: 10.1090/surv/139.  Google Scholar

[34]

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

[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.  Google Scholar

[1]

Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543

[2]

Peter Beelen, Kristian Brander. Efficient list decoding of a class of algebraic-geometry codes. Advances in Mathematics of Communications, 2010, 4 (4) : 485-518. doi: 10.3934/amc.2010.4.485

[3]

Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021

[4]

Elisa Gorla, Felice Manganiello, Joachim Rosenthal. An algebraic approach for decoding spread codes. Advances in Mathematics of Communications, 2012, 6 (4) : 443-466. doi: 10.3934/amc.2012.6.443

[5]

Angela Aguglia, Antonio Cossidente, Giuseppe Marino, Francesco Pavese, Alessandro Siciliano. Orbit codes from forms on vector spaces over a finite field. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020105

[6]

Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83

[7]

Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355-365. doi: 10.3934/amc.2016010

[8]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265

[9]

Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034

[10]

Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39

[11]

Alexandre Fotue-Tabue, Edgar Martínez-Moro, J. Thomas Blackford. On polycyclic codes over a finite chain ring. Advances in Mathematics of Communications, 2020, 14 (3) : 455-466. doi: 10.3934/amc.2020028

[12]

Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395

[13]

Irene Márquez-Corbella, Edgar Martínez-Moro. Algebraic structure of the minimal support codewords set of some linear codes. Advances in Mathematics of Communications, 2011, 5 (2) : 233-244. doi: 10.3934/amc.2011.5.233

[14]

Carlos Munuera, Wanderson Tenório, Fernando Torres. Locally recoverable codes from algebraic curves with separated variables. Advances in Mathematics of Communications, 2020, 14 (2) : 265-278. doi: 10.3934/amc.2020019

[15]

Grégory Berhuy. Algebraic space-time codes based on division algebras with a unitary involution. Advances in Mathematics of Communications, 2014, 8 (2) : 167-189. doi: 10.3934/amc.2014.8.167

[16]

Seungkook Park. Coherence of sensing matrices coming from algebraic-geometric codes. Advances in Mathematics of Communications, 2016, 10 (2) : 429-436. doi: 10.3934/amc.2016016

[17]

Ettore Fornasini, Telma Pinho, Raquel Pinto, Paula Rocha. Composition codes. Advances in Mathematics of Communications, 2016, 10 (1) : 163-177. doi: 10.3934/amc.2016.10.163

[18]

Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039

[19]

Michael Braun. On lattices, binary codes, and network codes. Advances in Mathematics of Communications, 2011, 5 (2) : 225-232. doi: 10.3934/amc.2011.5.225

[20]

David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (111)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]