doi: 10.3934/amc.2021013
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Generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes

Indian Institute of Science Education and Research, Bhopal, India

* Corresponding author: Nupur Patanker

Received  December 2020 Revised  March 2021 Early access May 2021

Let $ \mathbb{F}_{q} $ be a finite field with $ q $ elements, where $ q $ is a power of a prime $ p $. A polynomial over $ \mathbb{F}_{q} $ is monomially squarefree if all its monomials are squarefree. In this paper, we determine an upper bound on the number of common zeroes of any set of $ r $ linearly independent monomially squarefree polynomials of $ \mathbb{F}_{q}[t_{1}, t_{2}, \dots, t_{s}] $ in the affine torus $ T = (\mathbb{F}_{q}^{*})^{s} $ under certain conditions on $ r $, $ s $ and the degree of these polynomials. Applying the results, we obtain the generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes, as defined in [14].

Citation: Nupur Patanker, Sanjay Kumar Singh. Generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021013
References:
[1]

P. Beelen and M. Datta, Generalized Hamming weights of affine Cartesian codes, Finite Fields Appl., 51 (2018), 130-145.  doi: 10.1016/j.ffa.2018.01.006.

[2]

M. Bras-Amorós and M. E. O'Sullivan, Duality for some families of correction capability optimized evaluation codes, Adv. Math. Commun., 2 (2008), 15-33.  doi: 10.3934/amc.2008.2.15.

[3]

D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3$^{rd}$ edition, Undergraduate Texts in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-35651-8.

[4]

C. Galindo, O. Geil, F. Hernando and D. Ruano, On the distance of stabilizer quantum codes from J-affine variety codes, Quantum Inf. Process., 16 (2017), Paper No. 111, 32 pp. doi: 10.1007/s11128-017-1559-1.

[5]

C. Galindo and F. Hernando, Quantum codes from affine variety codes and their subfield-subcodes, Des. Codes Cryptogr., 76 (2015), 89-100.  doi: 10.1007/s10623-014-0016-8.

[6]

C. GalindoF. Hernando and D. Ruano, Stabilizer quantum codes from J-affine variety codes and a new Steane-like enlargement, Quantum Inf. Process., 14 (2015), 3211-3231.  doi: 10.1007/s11128-015-1057-2.

[7]

O. Geil, Evaluation codes from an affine variety code perspective, in Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol., 5, World Sci. Publ., Hackensack, NJ, 2008,153–180. doi: 10.1142/9789812794017_0004.

[8]

M. González-SarabiaE. CampsE. Sarmiento and R. H. Villarreal, The second generalized Hamming weight of some evaluation codes arising from a projective torus, Finite Fields Appl., 52 (2018), 370-394.  doi: 10.1016/j.ffa.2018.05.002.

[9]

M. González-SarabiaJ. Martínez-BernalR. H. Villarreal and C. E. Vivares, Generalized minimum distance functions, J. Algebraic Combin., 50 (2019), 317-346.  doi: 10.1007/s10801-018-0855-x.

[10]

J. P. Hansen, Toric surfaces and error-correcting codes, in Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), Springer, Berlin, 2000,132–142.

[11]

J. P. Hansen, Toric varieties Hirzebruch surfaces and error-correcting codes, Appl. Algebra Engrg. Comm. Comput., 13 (2002), 289-300.  doi: 10.1007/s00200-002-0106-0.

[12]

P. Heijnen and R. Pellikaan, Generalized Hamming weights of $q$-ary Reed-Muller codes, IEEE Trans. Inform. Theory, 44 (1998), 181-196.  doi: 10.1109/18.651015.

[13]

T. HellesethT. Kløve and J. Mykkeltveit, The weight distribution of irreducible cyclic codes with block length $n_{1}((q^l-1)/N)$, Discrete Math., 18 (1977), 179-211.  doi: 10.1016/0012-365X(77)90078-4.

[14]

D. JaramilloM. V. Pinto and R. H. Villarreal, Evaluation codes and their basic parameters, Des. Codes Cryptogr., 89 (2021), 269-300.  doi: 10.1007/s10623-020-00818-8.

[15]

D. Joyner, Toric codes over finite fields, Appl. Algebra Engrg. Comm. Comput., 15 (2004), 63-79.  doi: 10.1007/s00200-004-0152-x.

[16]

T. Kløve, The weight distribution of linear codes over $GF(q^l)$ having generator matrix over $GF(q)$, Discrete Math., 23 (1978), 159-168.  doi: 10.1016/0012-365X(78)90114-0.

[17]

J. Little and H. Schenck, Toric surface codes and Minkowski sums, SIAM J. Discrete Math., 20 (2006), 999-1014.  doi: 10.1137/050637054.

[18]

J. Little and R. Schwarz, On toric codes and multivariate Vandermonde matrices, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 349-367.  doi: 10.1007/s00200-007-0041-1.

[19]

Z. Nie and A. Y. Wang, Hilbert functions and the finite degree Zariski closure in finite field combinatorial geometry, J. Combin. Theory Ser. A, 134 (2015), 196-220.  doi: 10.1016/j.jcta.2015.03.011.

[20]

C. Rentería-MárquezA. Simis and R. H. Villarreal, Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields, Finite Fields Appl., 17 (2011), 81-104.  doi: 10.1016/j.ffa.2010.09.007.

[21]

D. Ruano, On the parameters of r-dimensional toric codes, Finite Fields Appl., 13 (2007), 962-976.  doi: 10.1016/j.ffa.2007.02.002.

[22]

D. Ruano, On the structure of generalized toric codes, J. Symbolic Comput., 44 (2009), 499-506.  doi: 10.1016/j.jsc.2007.07.018.

[23]

E. SarmientoM. V. Pinto and R. H. Villarreal, The minimum distance of parameterized codes on projective tori, Appl. Algebra Engrg. Comm. Comput., 22 (2011), 249-264.  doi: 10.1007/s00200-011-0148-2.

[24]

I. Soprunov and J. Soprunova, Toric surface codes and Minkowski length of polygons, SIAM J. Discrete Math., 23 (2008/09), 384-400.  doi: 10.1137/080716554.

[25]

I. Soprunov and J. Soprunova, Bringing toric codes to the next dimension, SIAM J. Discrete Math., 24 (2010), 655-665.  doi: 10.1137/090762592.

[26]

V. G. Umaña and M. Velasco, Dual toric codes and polytopes of degree one, SIAM J. Discrete Math., 29 (2015), 683-692.  doi: 10.1137/140966228.

[27]

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

show all references

References:
[1]

P. Beelen and M. Datta, Generalized Hamming weights of affine Cartesian codes, Finite Fields Appl., 51 (2018), 130-145.  doi: 10.1016/j.ffa.2018.01.006.

[2]

M. Bras-Amorós and M. E. O'Sullivan, Duality for some families of correction capability optimized evaluation codes, Adv. Math. Commun., 2 (2008), 15-33.  doi: 10.3934/amc.2008.2.15.

[3]

D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3$^{rd}$ edition, Undergraduate Texts in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-35651-8.

[4]

C. Galindo, O. Geil, F. Hernando and D. Ruano, On the distance of stabilizer quantum codes from J-affine variety codes, Quantum Inf. Process., 16 (2017), Paper No. 111, 32 pp. doi: 10.1007/s11128-017-1559-1.

[5]

C. Galindo and F. Hernando, Quantum codes from affine variety codes and their subfield-subcodes, Des. Codes Cryptogr., 76 (2015), 89-100.  doi: 10.1007/s10623-014-0016-8.

[6]

C. GalindoF. Hernando and D. Ruano, Stabilizer quantum codes from J-affine variety codes and a new Steane-like enlargement, Quantum Inf. Process., 14 (2015), 3211-3231.  doi: 10.1007/s11128-015-1057-2.

[7]

O. Geil, Evaluation codes from an affine variety code perspective, in Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol., 5, World Sci. Publ., Hackensack, NJ, 2008,153–180. doi: 10.1142/9789812794017_0004.

[8]

M. González-SarabiaE. CampsE. Sarmiento and R. H. Villarreal, The second generalized Hamming weight of some evaluation codes arising from a projective torus, Finite Fields Appl., 52 (2018), 370-394.  doi: 10.1016/j.ffa.2018.05.002.

[9]

M. González-SarabiaJ. Martínez-BernalR. H. Villarreal and C. E. Vivares, Generalized minimum distance functions, J. Algebraic Combin., 50 (2019), 317-346.  doi: 10.1007/s10801-018-0855-x.

[10]

J. P. Hansen, Toric surfaces and error-correcting codes, in Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), Springer, Berlin, 2000,132–142.

[11]

J. P. Hansen, Toric varieties Hirzebruch surfaces and error-correcting codes, Appl. Algebra Engrg. Comm. Comput., 13 (2002), 289-300.  doi: 10.1007/s00200-002-0106-0.

[12]

P. Heijnen and R. Pellikaan, Generalized Hamming weights of $q$-ary Reed-Muller codes, IEEE Trans. Inform. Theory, 44 (1998), 181-196.  doi: 10.1109/18.651015.

[13]

T. HellesethT. Kløve and J. Mykkeltveit, The weight distribution of irreducible cyclic codes with block length $n_{1}((q^l-1)/N)$, Discrete Math., 18 (1977), 179-211.  doi: 10.1016/0012-365X(77)90078-4.

[14]

D. JaramilloM. V. Pinto and R. H. Villarreal, Evaluation codes and their basic parameters, Des. Codes Cryptogr., 89 (2021), 269-300.  doi: 10.1007/s10623-020-00818-8.

[15]

D. Joyner, Toric codes over finite fields, Appl. Algebra Engrg. Comm. Comput., 15 (2004), 63-79.  doi: 10.1007/s00200-004-0152-x.

[16]

T. Kløve, The weight distribution of linear codes over $GF(q^l)$ having generator matrix over $GF(q)$, Discrete Math., 23 (1978), 159-168.  doi: 10.1016/0012-365X(78)90114-0.

[17]

J. Little and H. Schenck, Toric surface codes and Minkowski sums, SIAM J. Discrete Math., 20 (2006), 999-1014.  doi: 10.1137/050637054.

[18]

J. Little and R. Schwarz, On toric codes and multivariate Vandermonde matrices, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 349-367.  doi: 10.1007/s00200-007-0041-1.

[19]

Z. Nie and A. Y. Wang, Hilbert functions and the finite degree Zariski closure in finite field combinatorial geometry, J. Combin. Theory Ser. A, 134 (2015), 196-220.  doi: 10.1016/j.jcta.2015.03.011.

[20]

C. Rentería-MárquezA. Simis and R. H. Villarreal, Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields, Finite Fields Appl., 17 (2011), 81-104.  doi: 10.1016/j.ffa.2010.09.007.

[21]

D. Ruano, On the parameters of r-dimensional toric codes, Finite Fields Appl., 13 (2007), 962-976.  doi: 10.1016/j.ffa.2007.02.002.

[22]

D. Ruano, On the structure of generalized toric codes, J. Symbolic Comput., 44 (2009), 499-506.  doi: 10.1016/j.jsc.2007.07.018.

[23]

E. SarmientoM. V. Pinto and R. H. Villarreal, The minimum distance of parameterized codes on projective tori, Appl. Algebra Engrg. Comm. Comput., 22 (2011), 249-264.  doi: 10.1007/s00200-011-0148-2.

[24]

I. Soprunov and J. Soprunova, Toric surface codes and Minkowski length of polygons, SIAM J. Discrete Math., 23 (2008/09), 384-400.  doi: 10.1137/080716554.

[25]

I. Soprunov and J. Soprunova, Bringing toric codes to the next dimension, SIAM J. Discrete Math., 24 (2010), 655-665.  doi: 10.1137/090762592.

[26]

V. G. Umaña and M. Velasco, Dual toric codes and polytopes of degree one, SIAM J. Discrete Math., 29 (2015), 683-692.  doi: 10.1137/140966228.

[27]

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

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