American Institute of Mathematical Sciences

August  2014, 8(3): 271-280. doi: 10.3934/amc.2014.8.271

On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric

 1 Dipartimento di Matematica ed Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy 2 Department of Mathematics, Ghent University, Krijgslaan 281 - S22, 9000 Ghent

Received  January 2013 Revised  January 2014 Published  August 2014

We discuss the functional codes $C_h(\mathcal{Q}_N)$, for small $h\geq 3$, $q>9$, and for $N\geq 6$. This continues the study of different classes of functional codes, performed on functional codes arising from quadrics and Hermitian varieties. Here, we consider the functional codes arising from the intersections of the algebraic hypersurfaces of small degree $h$ with a given non-singular quadric $\mathcal{Q}_N$ in PG$(N,q)$.
Citation: Daniele Bartoli, Leo Storme. On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric. Advances in Mathematics of Communications, 2014, 8 (3) : 271-280. doi: 10.3934/amc.2014.8.271
References:
 [1] D. Bartoli, M. De Boeck, S. Fanali and L. Storme, On the functional codes defined by quadrics and Hermitian varieties, Des. Codes Cryptogr., 71 (2014), 21-46. doi: 10.1007/s10623-012-9712-4. [2] A. Cafure and G. Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl., 12 (2006), 155-185. doi: 10.1016/j.ffa.2005.03.003. [3] F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, A study of intersections of quadrics having applications on the small weight codewords of the functional codes $C_2(Q)$, $Q$ a non-singular quadric, J. Pure Appl. Algebra, 214 (2010), 1729-1739. doi: 10.1016/j.jpaa.2009.12.017. [4] F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, On the small weight codewords of the functional codes $C_{herm}(X)$, $X$ a non-singular Hermitian variety, Des. Codes Cryptogr., 56 (2010), 219-233. doi: 10.1007/s10623-010-9401-0. [5] A. Hallez and L. Storme, Functional codes arising from quadric intersections with Hermitian varieties, Finite Fields Appl., 16 (2010), 27-35. doi: 10.1016/j.ffa.2009.11.005. [6] J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, The Clarendon Press, Oxford University Press, 1991. [7] G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in Arithmetic, Geometry, and Coding Theory, Walter De Gruyter, Berlin, 1996, 77-104.

show all references

References:
 [1] D. Bartoli, M. De Boeck, S. Fanali and L. Storme, On the functional codes defined by quadrics and Hermitian varieties, Des. Codes Cryptogr., 71 (2014), 21-46. doi: 10.1007/s10623-012-9712-4. [2] A. Cafure and G. Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl., 12 (2006), 155-185. doi: 10.1016/j.ffa.2005.03.003. [3] F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, A study of intersections of quadrics having applications on the small weight codewords of the functional codes $C_2(Q)$, $Q$ a non-singular quadric, J. Pure Appl. Algebra, 214 (2010), 1729-1739. doi: 10.1016/j.jpaa.2009.12.017. [4] F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, On the small weight codewords of the functional codes $C_{herm}(X)$, $X$ a non-singular Hermitian variety, Des. Codes Cryptogr., 56 (2010), 219-233. doi: 10.1007/s10623-010-9401-0. [5] A. Hallez and L. Storme, Functional codes arising from quadric intersections with Hermitian varieties, Finite Fields Appl., 16 (2010), 27-35. doi: 10.1016/j.ffa.2009.11.005. [6] J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, The Clarendon Press, Oxford University Press, 1991. [7] G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in Arithmetic, Geometry, and Coding Theory, Walter De Gruyter, Berlin, 1996, 77-104.
 [1] 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 [2] Andreas Klein, Leo Storme. On the non-minimality of the largest weight codewords in the binary Reed-Muller codes. Advances in Mathematics of Communications, 2011, 5 (2) : 333-337. doi: 10.3934/amc.2011.5.333 [3] Claude Carlet. Expressing the minimum distance, weight distribution and covering radius of codes by means of the algebraic and numerical normal forms of their indicators. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022047 [4] Sylvain E. Cappell, Anatoly Libgober, Laurentiu Maxim and Julius L. Shaneson. Hodge genera and characteristic classes of complex algebraic varieties. Electronic Research Announcements, 2008, 15: 1-7. doi: 10.3934/era.2008.15.1 [5] 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 [6] Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039 [7] Tim Alderson, Alessandro Neri. Maximum weight spectrum codes. Advances in Mathematics of Communications, 2019, 13 (1) : 101-119. doi: 10.3934/amc.2019006 [8] 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 [9] László Mérai, Igor E. Shparlinski. Unlikely intersections over finite fields: Polynomial orbits in small subgroups. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1065-1073. doi: 10.3934/dcds.2020070 [10] 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 [11] Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195 [12] Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433 [13] Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. On the weight distribution of the cosets of MDS codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021042 [14] Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2022, 16 (3) : 571-596. doi: 10.3934/amc.2020124 [15] Chengju Li, Sunghan Bae, Shudi Yang. Some two-weight and three-weight linear codes. Advances in Mathematics of Communications, 2019, 13 (1) : 195-211. doi: 10.3934/amc.2019013 [16] 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 [17] Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017 [18] Zihui Liu, Xiangyong Zeng. The geometric structure of relative one-weight codes. Advances in Mathematics of Communications, 2016, 10 (2) : 367-377. doi: 10.3934/amc.2016011 [19] Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044 [20] Nigel Boston, Jing Hao. The weight distribution of quasi-quadratic residue codes. Advances in Mathematics of Communications, 2018, 12 (2) : 363-385. doi: 10.3934/amc.2018023

2021 Impact Factor: 1.015