Cite

[1] P. Beelen and M. Datta, Generalized Hamming weights of a ne Cartesian codes, Finite Fields Appl. 51 (2018), 130–145.10.1016/j.ffa.2018.01.006Search in Google Scholar

[2] C. Carvalho, On the second Hamming weight of some Reed-Muller type codes, Finite Fields Appl. 24 (2013), 88–94.10.1016/j.ffa.2013.06.004Search in Google Scholar

[3] D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1992.10.1007/978-1-4757-2181-2Search in Google Scholar

[4] M. Datta and S. Ghorpade, Number of solutions of systems of homogeneous polynomial equations over finite fields, Proc. Amer. Math. Soc. 145 (2017), no. 2, 525–541.Search in Google Scholar

[5] I. M. Duursma, C. Rentería and H. Tapia-Recillas, Reed-Muller codes on complete intersections, Appl. Algebra Engrg. Comm. Comput. 11 (2001), no. 6, 455–462.Search in Google Scholar

[6] M. González-Sarabia, E. Camps, E. Sarmiento and R. H. Villarreal, The second generalized Hamming weight of some evaluation codes arising from a projective torus, Finite Fields Appl. 52 (2018), 370–394.10.1016/j.ffa.2018.05.002Search in Google Scholar

[7] M. Gonzlez-Sarabia, J. Martnez-Bernal, R.H. Villarreal, and C.E. Vivares, Generalized minimum distance functions, J. Algebraic Combin. 50 (2019), no. 3, 317–346.10.1007/s10801-018-0855-xSearch in Google Scholar

[8] M. González–Sarabia and C. Rentería, The dual code of some Reed-Muller type codes, Appl. Algebra Engrg. Comm. Comput. 14 (2004), no. 5, 329–333.10.1007/s00200-003-0136-2Search in Google Scholar

[9] M. González–Sarabia and C. Rentería, Generalized Hamming weights and some parameterized codes, Discrete Math. 339 (2016), 813–821.10.1016/j.disc.2015.10.026Search in Google Scholar

[10] M. González–Sarabia, C. Rentería, and E. Sarmiento, Projective parameterized linear codes, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 23, 2 (2015) 223–240.10.1515/auom-2015-0039Search in Google Scholar

[11] M. González-Sarabia, C. Rentería and H. Tapia-Recillas, Reed-Muller-type codes over the Segre variety, Finite Fields Appl. 8 (2002), no. 4, 511–518.Search in Google Scholar

[12] D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.Search in Google Scholar

[13] P. Heijnen and R. Pellikaan, Generalized Hamming weights of q–ary Reed–Muller codes, IEEE Trans. Inform. Theory 44 (1998), no. 1, 181–196.Search in Google Scholar

[14] T. Helleseth, T. Kløve and J. Mykkelveit, The weight distribution of irreducible cyclic codes with block lengths n1((ql − 1) /N), Discrete Math. 18 (1977), 179–211.10.1016/0012-365X(77)90078-4Search in Google Scholar

[15] T. Johnsen and H. Verdure, Generalized Hamming weights for almost a ne codes, IEEE Trans. Inform. Theory 63 (2017), no. 4, 1941–1953.Search in Google Scholar

[16] T. Kløve, The weight distribution of linear codes over GF (ql) having generator matrix over GF (q), Discrete Math. 23 (1978), no. 2, 159–168.Search in Google Scholar

[17] H. H. López, C. Rentería and R. H. Villarreal, A ne cartesian codes, Des. Codes Cryptogr. 71 (2014), no. 1, 5–19.Search in Google Scholar

[18] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, North-Holland, 1977.Search in Google Scholar

[19] J. Martínez-Bernal, Y. Pitones and R. H. Villarreal, Minimum distance functions of graded ideals and Reed-Muller-type codes, J. Pure Appl. Algebra 221 (2017), 251–275.10.1016/j.jpaa.2016.06.006Search in Google Scholar

[20] J. Neves, M. Vaz Pinto and R. H. Villarreal, Vanishing ideals over graphs and even cycles, Comm. Algebra 43 3 (2015) 1050–1075.10.1080/00927872.2012.714025Search in Google Scholar

[21] W. Olaya–León and C. Granados–Pinzón, The second generalized Hamming weight of certain Castle codes, Des. Codes Cryptogr. 76 (2015), no. 1, 81–87.Search in Google Scholar

[22] C. Rentería, A. Simis and R. H. Villarreal, Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields, Finite Fields Appl. 17 (2011), no. 1, 81–104.Search in Google Scholar

[23] C. Rentería and H. Tapia-Recillas, Reed-Muller type codes on the Veronese variety over finite fields, Coding theory, cryptography and related areas (Guanajuato, 1998), 237–243, Springer, Berlin, 2000.10.1007/978-3-642-57189-3_21Search in Google Scholar

[24] E. Sarmiento, M. Vaz Pinto and R. H. Villarreal, The minimum distance of parameterized codes on projective tori, Appl. Algebra Engrg. Comm. Comput. 22 (2011), no. 4, 249–264.Search in Google Scholar

[25] H. G. Schaathun and W. Willems, A lower bound on the weight hierarchies of product codes, Discrete Appl. Math. 128 (2003), no. 1, 251–261.Search in Google Scholar

[26] A. Sørensen, Projective Reed-Muller codes, IEEE Trans. Inform. Theory 37 (1991), no. 6, 1567–1576.Search in Google Scholar

[27] M. Tsfasman, S. Vladut and D. Nogin, Algebraic geometric codes: basic notions, Mathematical Surveys and Monographs 139, American Mathematical Society, Providence, RI, 2007.10.1090/surv/139Search in Google Scholar

[28] R. H. Villarreal, Monomial Algebras, Second Edition, Monographs and Research Notes in Mathematics, Chapman and Hall/CRC, 2015.Search in Google Scholar

[29] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory 37 (1991), no. 5, 1412–1418.Search in Google Scholar

[30] V. K. Wei and K. Yang, On the generalized Hamming weights of product codes, IEEE Trans. Inform. Theory 39 (1993), no. 5, 1709–1713.Search in Google Scholar

[31] M. Yang, J. Lin, K. Feng and D. Lin, Generalized Hamming weights of irreducible cyclic codes, IEEE Trans. Inform. Theory 61 (2015), no. 9, 4905–4913.Search in Google Scholar

eISSN:
1844-0835
Language:
English
Publication timeframe:
Volume Open
Journal Subjects:
Mathematics, General Mathematics