-
Previous Article
Quasi-symmetric designs on $ 56 $ points
- AMC Home
- This Issue
-
Next Article
Character sums over a non-chain ring and their applications
Constructions of linear codes with small hulls from association schemes
School of Mathematical Sciences, Zhejiang University, Zhejiang, 310027, China |
The intersection of a linear code and its dual is called the hull of this code. The code is a linear complementary dual (LCD) code if the dimension of its hull is zero. In this paper, we develop a method to construct LCD codes and linear codes with one-dimensional hull by association schemes. One of constructions in this paper generalizes that of linear codes associated with Gauss periods given in [
References:
[1] |
E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, The Benjamin/Cummings, London, 1984. |
[2] |
L. D. Baumert, W. H. Mills and R. L. Ward,
Uniform cyclotomy, J. Number Theory, 14 (1982), 67-82.
doi: 10.1016/0022-314X(82)90058-0. |
[3] |
A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-regular Graphs, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-74341-2. |
[4] |
C. Carlet and S. Guilley,
Complementary dual codes for counter-measures to side-channel attacks, Adv. Math. Commun., 10 (2016), 131-150.
doi: 10.3934/amc.2016.10.131. |
[5] |
C. Carlet, C. Li and S. Mesnager,
Linear codes with small hulls in semi-primitive case, Des. Codes Cryptogr., 87 (2019), 3063-3075.
doi: 10.1007/s10623-019-00663-4. |
[6] |
C. Carlet, S. Mesnager, C. Tang and Y. Qi,
Euclidean and Hermitian LCD MDS codes, Des. Codes Cryptogr., 86 (2018), 2605-2618.
doi: 10.1007/s10623-018-0463-8. |
[7] |
C. Carlet, S. Mesnager, C. Tang and Y. Qi,
New characterization and parametrization of LCD codes, IEEE Trans. Inform. Theory, 65 (2019), 39-49.
doi: 10.1109/TIT.2018.2829873. |
[8] |
C. Carlet, S. Mesnager, C. Tang, Y. Qi and R. Pellikaan,
Linear codes over $\Bbb F_q$ are equivalent to LCD codes for $q>3$, IEEE Trans. Inform. Theory, 64 (2018), 3010-3017.
doi: 10.1109/TIT.2018.2789347. |
[9] |
C. Ding and J. Yang,
Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.
doi: 10.1016/j.disc.2012.11.009. |
[10] |
K. Feng, An Introduction to Algebraic Number Theory, Science Press, Beijing, 2000.
![]() |
[11] |
T. Feng and K. Momihara,
Three-class association schemes from cyclotomy, J. Combin. Theory Ser. A, 120 (2013), 1202-1215.
doi: 10.1016/j.jcta.2013.03.002. |
[12] |
K. Ireland and M. Rosen, A classical introduction to modern number theory, 2$^nd$ edition, Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4757-2103-4. |
[13] |
L. Jin and C. Xing,
Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound, IEEE Trans. Inform. Theory, 64 (2018), 6277-6282.
doi: 10.1109/TIT.2017.2773057. |
[14] |
H. Kharaghani and S. Suda,
Symmetric Bush-type generalized Hadamard matrices and association schemes, Finite Fields Appl., 37 (2016), 72-84.
doi: 10.1016/j.ffa.2015.09.003. |
[15] |
C. Li, C. Ding and S. Li,
LCD cyclic codes over finite fields, IEEE Trans. Inform. Theory, 63 (2017), 4344-4356.
doi: 10.1109/TIT.2017.2672961. |
[16] |
C. Li and P. Zeng,
Constructions of linear codes with one-dimensional hull, IEEE Trans. Inform. Theory, 65 (2019), 1668-1676.
doi: 10.1109/TIT.2018.2863693. |
[17] |
S. Li, C. Li, C. Ding and H. Liu,
Two families of LCD BCH codes, IEEE Trans. Inform. Theory, 63 (2017), 5699-5717.
|
[18] |
X. Liu and H. Liu, Matrix-product complementary dual codes, preprint, arXiv: 1604.03774. Google Scholar |
[19] |
J. L. Massey,
Linear codes with complementary duals, Discret. Math., 106 (1992), 337-342.
doi: 10.1016/0012-365X(92)90563-U. |
[20] |
S. Mesnager, C. Tang and Y. Qi,
Complementary dual algebraic geometry codes, IEEE Trans. Inform. Theory, 64 (2018), 2390-2397.
doi: 10.1109/TIT.2017.2766075. |
[21] |
B. Pang, S. Zhu and Z. Sun,
On LCD negacyclic codes over finite fields, J. Syst. Sci. Complex., 31 (2018), 1065-1077.
doi: 10.1007/s11424-017-6301-7. |
[22] |
N. Sendrier,
Finding the permutation between equivalent linear codes: The support splitting algorithm, IEEE Trans. Inform. Theory, 46 (2000), 1193-1203.
doi: 10.1109/18.850662. |
[23] |
N. Sendrier and G. Skersys, On the computation of the automorphism group of a linear code, in : Proceedings of IEEE ISIT2001, Washington, DC, 2001.
doi: 10.1109/ISIT.2001.935876. |
[24] |
X. Shi, Q. Yue and S. Yang, New LCD MDS codes constructed from generalized Reed-Solomon codes, J. Algebra Appl., 18 (2019), 1950150.
doi: 10.1142/S0219498819501500. |
[25] |
E. R. van Dam and M. Muzychuk,
Some implications on amorphic association schemes, J. Combin. Theory Ser. A, 117 (2010), 111-127.
doi: 10.1016/j.jcta.2009.03.018. |
show all references
References:
[1] |
E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, The Benjamin/Cummings, London, 1984. |
[2] |
L. D. Baumert, W. H. Mills and R. L. Ward,
Uniform cyclotomy, J. Number Theory, 14 (1982), 67-82.
doi: 10.1016/0022-314X(82)90058-0. |
[3] |
A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-regular Graphs, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-74341-2. |
[4] |
C. Carlet and S. Guilley,
Complementary dual codes for counter-measures to side-channel attacks, Adv. Math. Commun., 10 (2016), 131-150.
doi: 10.3934/amc.2016.10.131. |
[5] |
C. Carlet, C. Li and S. Mesnager,
Linear codes with small hulls in semi-primitive case, Des. Codes Cryptogr., 87 (2019), 3063-3075.
doi: 10.1007/s10623-019-00663-4. |
[6] |
C. Carlet, S. Mesnager, C. Tang and Y. Qi,
Euclidean and Hermitian LCD MDS codes, Des. Codes Cryptogr., 86 (2018), 2605-2618.
doi: 10.1007/s10623-018-0463-8. |
[7] |
C. Carlet, S. Mesnager, C. Tang and Y. Qi,
New characterization and parametrization of LCD codes, IEEE Trans. Inform. Theory, 65 (2019), 39-49.
doi: 10.1109/TIT.2018.2829873. |
[8] |
C. Carlet, S. Mesnager, C. Tang, Y. Qi and R. Pellikaan,
Linear codes over $\Bbb F_q$ are equivalent to LCD codes for $q>3$, IEEE Trans. Inform. Theory, 64 (2018), 3010-3017.
doi: 10.1109/TIT.2018.2789347. |
[9] |
C. Ding and J. Yang,
Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.
doi: 10.1016/j.disc.2012.11.009. |
[10] |
K. Feng, An Introduction to Algebraic Number Theory, Science Press, Beijing, 2000.
![]() |
[11] |
T. Feng and K. Momihara,
Three-class association schemes from cyclotomy, J. Combin. Theory Ser. A, 120 (2013), 1202-1215.
doi: 10.1016/j.jcta.2013.03.002. |
[12] |
K. Ireland and M. Rosen, A classical introduction to modern number theory, 2$^nd$ edition, Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4757-2103-4. |
[13] |
L. Jin and C. Xing,
Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound, IEEE Trans. Inform. Theory, 64 (2018), 6277-6282.
doi: 10.1109/TIT.2017.2773057. |
[14] |
H. Kharaghani and S. Suda,
Symmetric Bush-type generalized Hadamard matrices and association schemes, Finite Fields Appl., 37 (2016), 72-84.
doi: 10.1016/j.ffa.2015.09.003. |
[15] |
C. Li, C. Ding and S. Li,
LCD cyclic codes over finite fields, IEEE Trans. Inform. Theory, 63 (2017), 4344-4356.
doi: 10.1109/TIT.2017.2672961. |
[16] |
C. Li and P. Zeng,
Constructions of linear codes with one-dimensional hull, IEEE Trans. Inform. Theory, 65 (2019), 1668-1676.
doi: 10.1109/TIT.2018.2863693. |
[17] |
S. Li, C. Li, C. Ding and H. Liu,
Two families of LCD BCH codes, IEEE Trans. Inform. Theory, 63 (2017), 5699-5717.
|
[18] |
X. Liu and H. Liu, Matrix-product complementary dual codes, preprint, arXiv: 1604.03774. Google Scholar |
[19] |
J. L. Massey,
Linear codes with complementary duals, Discret. Math., 106 (1992), 337-342.
doi: 10.1016/0012-365X(92)90563-U. |
[20] |
S. Mesnager, C. Tang and Y. Qi,
Complementary dual algebraic geometry codes, IEEE Trans. Inform. Theory, 64 (2018), 2390-2397.
doi: 10.1109/TIT.2017.2766075. |
[21] |
B. Pang, S. Zhu and Z. Sun,
On LCD negacyclic codes over finite fields, J. Syst. Sci. Complex., 31 (2018), 1065-1077.
doi: 10.1007/s11424-017-6301-7. |
[22] |
N. Sendrier,
Finding the permutation between equivalent linear codes: The support splitting algorithm, IEEE Trans. Inform. Theory, 46 (2000), 1193-1203.
doi: 10.1109/18.850662. |
[23] |
N. Sendrier and G. Skersys, On the computation of the automorphism group of a linear code, in : Proceedings of IEEE ISIT2001, Washington, DC, 2001.
doi: 10.1109/ISIT.2001.935876. |
[24] |
X. Shi, Q. Yue and S. Yang, New LCD MDS codes constructed from generalized Reed-Solomon codes, J. Algebra Appl., 18 (2019), 1950150.
doi: 10.1142/S0219498819501500. |
[25] |
E. R. van Dam and M. Muzychuk,
Some implications on amorphic association schemes, J. Combin. Theory Ser. A, 117 (2010), 111-127.
doi: 10.1016/j.jcta.2009.03.018. |
[1] |
Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044 |
[2] |
Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 |
[3] |
Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021006 |
[4] |
Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011 |
[5] |
Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 |
[6] |
Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 |
[7] |
Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009 |
[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] |
Chih-Chiang Fang. Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020127 |
[10] |
Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182 |
2019 Impact Factor: 0.734
Tools
Article outline
[Back to Top]