doi: 10.3934/amc.2022051
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Constructions of symplectic LCD MDS codes from quasi-cyclic codes

1. 

School of Mathematics, Hefei University of Technology, Hefei 230601, China

2. 

Department of Information Management, Anhui Vocational College of Police Officers, Hefei 230031, China

*Corresponding author: Jin Li

Received  March 2022 Revised  May 2022 Early access July 2022

Fund Project: This paper was supported by the National Natural Science Foundation of China (No.62002093; No.12171134), the Fundamental Research Funds for the Central Universities of China (No.JZ2022HGTB0264) and the Key Projects of Natural Science Research of Universities in Anhui Province (No.KJ2021A1469)

Linear complementary dual (LCD) codes play an important role in data storage, communications systems and cryptography. In this paper, we construct some symplectic LCD codes from quasi-cyclic (QC) codes, and prove that these symplectic LCD codes with dimension two over $ \mathbb{F}_{2^a} $ are maximum-distance-separable (MDS) codes. By extending the generator matrices of symplectic LCD MDS codes with dimension two, we obtain a series of symplectic LCD codes over $ \mathbb{F}_{2^a} $. As an application, some new entanglement-assisted quantum error-correcting codes (EAQECCs) over $ \mathbb{F}_{2^a} $ are constructed by symplectic LCD codes.

Citation: Xia Huang, Jin Li, Shan Huang. Constructions of symplectic LCD MDS codes from quasi-cyclic codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2022051
References:
[1]

K. Boonniyoma and S. Jitman, Complementary dual subfield linear codes over finite fields, preprint, 2016, arXiv: 1605.06827.

[2]

S. T. DoughertyJ.-L. KimB. OzkayaL. Sok and P. Solé, The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices, Int. J. Inf. Coding Theory, 4 (2017), 116-128.  doi: 10.1504/IJICOT.2017.083827.

[3]

W. J. FangF. W. FuL. Q. Li and S. X. Zhu, Euclidean and Hermitian hulls of MDS codes and their applications to EAQECCs, IEEE Trans. Inf. Theory, 66 (2020), 3527-3537.  doi: 10.1109/tit.2019.2950245.

[4]

Q. FuR. H. LiF. W. Fu and Y. Rao, On the construction of binary optimal LCD codes with short length, Int. J. Found. Comput. S., 30 (2019), 1237-1245.  doi: 10.1142/S0129054119500242.

[5]

C. GalindoO. GeilF. Hernando and D. Ruano, New binary and ternary LCD codes, IEEE Trans. Inf. Theory, 65 (2019), 1008-1016.  doi: 10.1109/TIT.2018.2834500.

[6]

C. Galindo, F. Hernando, R. Matsumoto and D. Ruano, Entanglement-assisted quantum error-correcting codes over arbitrary finite fields, Quantum Inf. Process., 18 (2019), Paper No. 116, 18 pp. doi: 10.1007/s11128-019-2234-5.

[7]

K. GuendaS. Jitman and T. A. Gulliver, Constructions of good entanglement-assisted quantum error correcting codes, Des. Codes Cryptogr, 86 (2018), 121-136.  doi: 10.1007/s10623-017-0330-z.

[8]

C. GüneriB. Özkaya and P. Solé, Quasi-cyclic complementary dual codes, Finite Fields Appl., 42 (2016), 67-80.  doi: 10.1016/j.ffa.2016.07.005.

[9]

C. Y. LaiT. A. Brun and M. M. Wilde, Dualities and identities for entanglement-assisted quantum codes, Quantum Inf. Process., 13 (2014), 957-990. 

[10]

C. Y. LaiT. A. Brun and M. M. Wilde, Duality in entanglement assisted quantum error correction, IEEE Trans. Inf. Theory, 59 (2013), 4020-4024.  doi: 10.1109/TIT.2013.2246274.

[11]

E. R. J. Lina and E. G. Nocon, On the construction of some LCD codes over finite fields, Manila Journal of Science, 9 (2016), 67-82. 

[12]

X. S. LiuY. Fan and H. L. Liu, Galois LCD codes over finite fields, Finite Fields Appl., 49 (2018), 227-242.  doi: 10.1016/j.ffa.2017.10.001.

[13]

X. S. LiuL. Yu and P. Hu, New entanglement-assisted quantum codes from $k$-Galois dual codes, Finite Fields Appl., 55 (2019), 21-32.  doi: 10.1016/j.ffa.2018.09.001.

[14]

J. J. LvR. H. Li and J. L. Wang, New binary quantum codes derived from one-generator quasi-cyclic codes, IEEE Access, 7 (2019), 85782-85785. 

[15]

J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342.  doi: 10.1016/0012-365X(92)90563-U.

[16]

J. F. Qian and L. N. Zhang, On MDS linear complementary dual codes and entanglement-assisted quantum codes, Des. Codes Cryptogr, 86 (2018), 1565-1572.  doi: 10.1007/s10623-017-0413-x.

[17]

N. Sendrier, Linear codes with complementary duals meet the Gilbert-Varshamov bound, Discrete Math., 285 (2004), 345-347.  doi: 10.1016/j.disc.2004.05.005.

[18]

Q. Song, R. H. Li, Q. Fu and L. B. Guo, On the construction of LCD codes over $F_{5}$, ITM Web of Conferences, 12 (2017), 04006, 1–5.

[19]

H. Q. Xu and W. Du, Constructions of symplectic LCD MDS codes, Bull. Malays. Math. Sci. Soc., 44 (2021), 3377-3390.  doi: 10.1007/s40840-021-01114-x.

[20]

H. Q. Xu and W. Du, Hermitian LCD codes over $F_{q^{2}}+uF_{q^{2}}$ and their applications to maximal entanglement EAQECCs, Cryptogr. Commun., 14 (2022), 259-269.  doi: 10.1007/s12095-021-00510-1.

[21]

X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discrete Math., 126 (1994), 391-393.  doi: 10.1016/0012-365X(94)90283-6.

show all references

References:
[1]

K. Boonniyoma and S. Jitman, Complementary dual subfield linear codes over finite fields, preprint, 2016, arXiv: 1605.06827.

[2]

S. T. DoughertyJ.-L. KimB. OzkayaL. Sok and P. Solé, The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices, Int. J. Inf. Coding Theory, 4 (2017), 116-128.  doi: 10.1504/IJICOT.2017.083827.

[3]

W. J. FangF. W. FuL. Q. Li and S. X. Zhu, Euclidean and Hermitian hulls of MDS codes and their applications to EAQECCs, IEEE Trans. Inf. Theory, 66 (2020), 3527-3537.  doi: 10.1109/tit.2019.2950245.

[4]

Q. FuR. H. LiF. W. Fu and Y. Rao, On the construction of binary optimal LCD codes with short length, Int. J. Found. Comput. S., 30 (2019), 1237-1245.  doi: 10.1142/S0129054119500242.

[5]

C. GalindoO. GeilF. Hernando and D. Ruano, New binary and ternary LCD codes, IEEE Trans. Inf. Theory, 65 (2019), 1008-1016.  doi: 10.1109/TIT.2018.2834500.

[6]

C. Galindo, F. Hernando, R. Matsumoto and D. Ruano, Entanglement-assisted quantum error-correcting codes over arbitrary finite fields, Quantum Inf. Process., 18 (2019), Paper No. 116, 18 pp. doi: 10.1007/s11128-019-2234-5.

[7]

K. GuendaS. Jitman and T. A. Gulliver, Constructions of good entanglement-assisted quantum error correcting codes, Des. Codes Cryptogr, 86 (2018), 121-136.  doi: 10.1007/s10623-017-0330-z.

[8]

C. GüneriB. Özkaya and P. Solé, Quasi-cyclic complementary dual codes, Finite Fields Appl., 42 (2016), 67-80.  doi: 10.1016/j.ffa.2016.07.005.

[9]

C. Y. LaiT. A. Brun and M. M. Wilde, Dualities and identities for entanglement-assisted quantum codes, Quantum Inf. Process., 13 (2014), 957-990. 

[10]

C. Y. LaiT. A. Brun and M. M. Wilde, Duality in entanglement assisted quantum error correction, IEEE Trans. Inf. Theory, 59 (2013), 4020-4024.  doi: 10.1109/TIT.2013.2246274.

[11]

E. R. J. Lina and E. G. Nocon, On the construction of some LCD codes over finite fields, Manila Journal of Science, 9 (2016), 67-82. 

[12]

X. S. LiuY. Fan and H. L. Liu, Galois LCD codes over finite fields, Finite Fields Appl., 49 (2018), 227-242.  doi: 10.1016/j.ffa.2017.10.001.

[13]

X. S. LiuL. Yu and P. Hu, New entanglement-assisted quantum codes from $k$-Galois dual codes, Finite Fields Appl., 55 (2019), 21-32.  doi: 10.1016/j.ffa.2018.09.001.

[14]

J. J. LvR. H. Li and J. L. Wang, New binary quantum codes derived from one-generator quasi-cyclic codes, IEEE Access, 7 (2019), 85782-85785. 

[15]

J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342.  doi: 10.1016/0012-365X(92)90563-U.

[16]

J. F. Qian and L. N. Zhang, On MDS linear complementary dual codes and entanglement-assisted quantum codes, Des. Codes Cryptogr, 86 (2018), 1565-1572.  doi: 10.1007/s10623-017-0413-x.

[17]

N. Sendrier, Linear codes with complementary duals meet the Gilbert-Varshamov bound, Discrete Math., 285 (2004), 345-347.  doi: 10.1016/j.disc.2004.05.005.

[18]

Q. Song, R. H. Li, Q. Fu and L. B. Guo, On the construction of LCD codes over $F_{5}$, ITM Web of Conferences, 12 (2017), 04006, 1–5.

[19]

H. Q. Xu and W. Du, Constructions of symplectic LCD MDS codes, Bull. Malays. Math. Sci. Soc., 44 (2021), 3377-3390.  doi: 10.1007/s40840-021-01114-x.

[20]

H. Q. Xu and W. Du, Hermitian LCD codes over $F_{q^{2}}+uF_{q^{2}}$ and their applications to maximal entanglement EAQECCs, Cryptogr. Commun., 14 (2022), 259-269.  doi: 10.1007/s12095-021-00510-1.

[21]

X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discrete Math., 126 (1994), 391-393.  doi: 10.1016/0012-365X(94)90283-6.

Table 1.  symplectic LCD MDS codes
$\mathbb{F}_{q}$ $g^{\perp}(x)$ $[2n, k, d_{s}]$
$\mathbb{F}_{4}$ $\begin{array}{l} x^{2}+w^{i}x+1,\\ 1\leq i\leq 2 \end{array}$ $[10t, 2, 5t], \ i = 1, 2.$
$\mathbb{F}_{8}$ $\begin{array}{l} x^{2}+w^{i}x+1,\\ 1\leq i\leq 6 \end{array}$ $\begin{array}{l} \ [18t, 2, 9t], \ i = 1, 2, 4,\\ [14t, 2, 7t], \ i = 3, 5, 6. \end{array}$
$\mathbb{F}_{16}$ $\begin{array}{l} x^{2}+w^{i}x+1,\\ 1\leq i\leq 14 \end{array}$ $\begin{array}{l} \ [10t, 2, 5t], \ i = 5, 10,\\ [30t, 2, 15t], \ i = 7, 11, 13, 14,\\ [34t, 2, 17t], \ i = 1, 2, 3, 4, 6, 8, 9, 12. \end{array}$
$\mathbb{F}_{32}$ $\begin{array}{l} x^{2}+w^{i}x+1,\\ 1\leq i\leq 30 \end{array}$ $\begin{array}{l} \ [22t, 2, 11t], \ i = 5, 9, 10, 18, 20,\\ [62t, 2, 31t], \ i = 1, 2, 3, 4, 6, 8, 12, 15, 16, 17, 23, 24,\\ 27, 29, 30,\\ [66t, 2, 33t], \ i = 7, 11, 13, 14, 19, 21, 22, 25, 26, 28. \end{array}$
$\mathbb{F}_{64}$ $\begin{array}{l} x^{2}+w^{i}x+1,\\ 1\leq i\leq 62 \end{array}$ $\begin{array}{l} \ [10t, 2, 5t], \ i = 21, 42,\\ [14t, 2, 7t], \ i = 27, 45, 54,\\ [18t, 2, 9t], \ i = 9, 18, 36,\\ [26t, 2, 13t], \ i = 13, 19, 26, 38, 41, 52,\\ [42t, 2, 21t], \ i = 23, 29, 43, 46, 53, 58,\\ [126t, 2, 63t], \ i = 3, 6, 7, 12, 14, 24, 28, 31, 33, 35, 47,\\ 48, 49, 55, 56, 59, 61, 62,\\ [130t, 2, 65t], \ i = 1, 2, 4, 5, 8, 10, 11, 15, 16, 17, 20, 22, 25,\\ 30, 32, 34, 37, 39, 40, 44, 50, 51, 57, 60. \end{array}$
$\mathbb{F}_{q}$ $g^{\perp}(x)$ $[2n, k, d_{s}]$
$\mathbb{F}_{4}$ $\begin{array}{l} x^{2}+w^{i}x+1,\\ 1\leq i\leq 2 \end{array}$ $[10t, 2, 5t], \ i = 1, 2.$
$\mathbb{F}_{8}$ $\begin{array}{l} x^{2}+w^{i}x+1,\\ 1\leq i\leq 6 \end{array}$ $\begin{array}{l} \ [18t, 2, 9t], \ i = 1, 2, 4,\\ [14t, 2, 7t], \ i = 3, 5, 6. \end{array}$
$\mathbb{F}_{16}$ $\begin{array}{l} x^{2}+w^{i}x+1,\\ 1\leq i\leq 14 \end{array}$ $\begin{array}{l} \ [10t, 2, 5t], \ i = 5, 10,\\ [30t, 2, 15t], \ i = 7, 11, 13, 14,\\ [34t, 2, 17t], \ i = 1, 2, 3, 4, 6, 8, 9, 12. \end{array}$
$\mathbb{F}_{32}$ $\begin{array}{l} x^{2}+w^{i}x+1,\\ 1\leq i\leq 30 \end{array}$ $\begin{array}{l} \ [22t, 2, 11t], \ i = 5, 9, 10, 18, 20,\\ [62t, 2, 31t], \ i = 1, 2, 3, 4, 6, 8, 12, 15, 16, 17, 23, 24,\\ 27, 29, 30,\\ [66t, 2, 33t], \ i = 7, 11, 13, 14, 19, 21, 22, 25, 26, 28. \end{array}$
$\mathbb{F}_{64}$ $\begin{array}{l} x^{2}+w^{i}x+1,\\ 1\leq i\leq 62 \end{array}$ $\begin{array}{l} \ [10t, 2, 5t], \ i = 21, 42,\\ [14t, 2, 7t], \ i = 27, 45, 54,\\ [18t, 2, 9t], \ i = 9, 18, 36,\\ [26t, 2, 13t], \ i = 13, 19, 26, 38, 41, 52,\\ [42t, 2, 21t], \ i = 23, 29, 43, 46, 53, 58,\\ [126t, 2, 63t], \ i = 3, 6, 7, 12, 14, 24, 28, 31, 33, 35, 47,\\ 48, 49, 55, 56, 59, 61, 62,\\ [130t, 2, 65t], \ i = 1, 2, 4, 5, 8, 10, 11, 15, 16, 17, 20, 22, 25,\\ 30, 32, 34, 37, 39, 40, 44, 50, 51, 57, 60. \end{array}$
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