# American Institute of Mathematical Sciences

May  2022, 16(2): 349-364. doi: 10.3934/amc.2020114

## Constructions of linear codes with small hulls from association schemes

 School of Mathematical Sciences, Zhejiang University, Zhejiang, 310027, China

* Corresponding author: Ran Tao

Received  April 2020 Revised  August 2020 Published  May 2022 Early access  October 2020

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 [5]. In addition, we present a generalized construction of linear codes, which can provide more LCD codes and linear codes with one-dimensional hull. We also present some examples of LCD MDS, LCD almost MDS codes, and MDS, almost MDS codes with one-dimensional hull from our constructions.

Citation: Ye Wang, Ran Tao. Constructions of linear codes with small hulls from association schemes. Advances in Mathematics of Communications, 2022, 16 (2) : 349-364. doi: 10.3934/amc.2020114
##### 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. [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.

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##### 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. [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.
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