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doi: 10.3934/amc.2021070
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Construction for both self-dual codes and LCD codes

Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan

*Corresponding author: Keita Ishizuka

Received  August 2021 Revised  November 2021 Early access January 2022

From a given [n, k] code C, we give a method for constructing many [n, k] codes C' such that the hull dimensions of C and C' are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even [56, 28, 12] codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length 26 ≤ n ≤ 40.

Citation: Keita Ishizuka, Ken Saito. Construction for both self-dual codes and LCD codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021070
References:
[1]

M. Araya and M. Harada, On the minimum weights of binary linear complementary dual codes, Cryptogr. Commun., 12 (2020), 285-300.  doi: 10.1007/s12095-019-00402-5.

[2]

M. Araya and M. Harada, On the classification of quaternary optimal Hermitian LCD codes, arXiv: 2011.04139.

[3]

M. ArayaM. Harada and K. Saito, Characterization and classification of optimal LCD codes, Des. Codes Cryptogr., 89 (2021), 617-640.  doi: 10.1007/s10623-020-00834-8.

[4]

M. ArayaM. Harada and K. Saito, On the minimum weights of binary LCD codes and ternary LCD codes, Finite Fields Appl., 76 (2021), 101925.  doi: 10.1016/j.ffa.2021.101925.

[5]

V. K. BhargavaG. Young and A. K. Bhargava, A characterization of a (56, 28) extremal self-dual code, IEEE Trans. Inform. Theory, 27 (1981), 258-260.  doi: 10.1109/TIT.1981.1056308.

[6]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system. Ⅰ. The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.

[7]

S. Bouyuklieva, Optimal binary LCD codes, Des. Codes Cryptogr., 89 (2021), 2445-2461.  doi: 10.1007/s10623-021-00929-w.

[8]

F. C. Bussemarker and V. D. Tonchev, New extremal doubly-even codes of length $56$ derived from Hadamard matrices of order 28, Discrete Math., 76 (1989), 45-49.  doi: 10.1016/0012-365X(89)90286-0.

[9]

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.

[10]

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.

[11]

L. GalvezJ.-L. KimN. LeeY. G. Roe and B.-S. Won, Some bounds on binary LCD codes, Cryptogr. Commun., 10 (2018), 719-728.  doi: 10.1007/s12095-017-0258-1.

[12]

M. Grassl, Code tables: Bounds on the parameters of various types of codes, Available online at http://www.codetables.de/, Accessed on 5 Aug 2021.

[13]

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.

[14]

M. Harada, Existence of new extremal doubly-even codes and extremal singly-even codes, Des. Codes Cryptogr., 8 (1996), 273-283.  doi: 10.1023/A:1027303722125.

[15]

M. Harada, Self-orthogonal $3$-$(56, 12, 65)$ designs and extremal doubly-even self-dual codes of length $56$, Des. Codes Cryptogr., 38 (2006), 5-16.  doi: 10.1007/s10623-004-5657-6.

[16]

M. Harada, Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes, Des. Codes Cryptogr., 89 (2021), 2295-2312.  doi: 10.1007/s10623-021-00916-1.

[17]

M. HaradaT. A. Gulliver and H. Kaneta, Classification of extremal double-circulant self-dual codes of length up to $62$, Discrete Math., 188 (1998), 127-136.  doi: 10.1016/S0012-365X(97)00250-1.

[18]

M. Harada and H. Kimura, New extremal doubly-even $[64, 32, 12]$ codes, Des. Codes Cryptogr., 6 (1995), 91-96.  doi: 10.1007/BF01398007.

[19]

M. Harada and K. Saito, Binary linear complementary dual codes, Cryptogr. Commun., 11 (2019), 677-696.  doi: 10.1007/s12095-018-0319-0.

[20] W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes, Cambridge University Press, 2003.  doi: 10.1017/CBO9780511807077.
[21]

H. Kimura, Extremal doubly even (56, 28, 12) codes and Hadamard matrices of order 28, Australas. J. Combin., 10 (1994), 171-180. 

[22]

C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control., 22 (1973), 188-200.  doi: 10.1016/S0019-9958(73)90273-8.

[23]

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

[24]

N. Yankov and M. H. Lee, New binary self-dual codes of length 50–60, Des. Codes Cryptogr., 73 (2014), 983-996.  doi: 10.1007/s10623-013-9839-y.

[25]

N. Yankov and R. Russeva, Binary self-dual codes of length 52 to 60 with an automorphism of order 7 or 13, IEEE Trans. Inform. Theory, 57 (2011), 7498-7506.  doi: 10.1109/TIT.2011.2155619.

[26]

V. Yorgov, A method for constructing inequivalent self-dual codes with application to length 56, IEEE Trans. Inform. Theory, 33 (1987), 72-82.  doi: 10.1109/TIT.1987.1057273.

show all references

References:
[1]

M. Araya and M. Harada, On the minimum weights of binary linear complementary dual codes, Cryptogr. Commun., 12 (2020), 285-300.  doi: 10.1007/s12095-019-00402-5.

[2]

M. Araya and M. Harada, On the classification of quaternary optimal Hermitian LCD codes, arXiv: 2011.04139.

[3]

M. ArayaM. Harada and K. Saito, Characterization and classification of optimal LCD codes, Des. Codes Cryptogr., 89 (2021), 617-640.  doi: 10.1007/s10623-020-00834-8.

[4]

M. ArayaM. Harada and K. Saito, On the minimum weights of binary LCD codes and ternary LCD codes, Finite Fields Appl., 76 (2021), 101925.  doi: 10.1016/j.ffa.2021.101925.

[5]

V. K. BhargavaG. Young and A. K. Bhargava, A characterization of a (56, 28) extremal self-dual code, IEEE Trans. Inform. Theory, 27 (1981), 258-260.  doi: 10.1109/TIT.1981.1056308.

[6]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system. Ⅰ. The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.

[7]

S. Bouyuklieva, Optimal binary LCD codes, Des. Codes Cryptogr., 89 (2021), 2445-2461.  doi: 10.1007/s10623-021-00929-w.

[8]

F. C. Bussemarker and V. D. Tonchev, New extremal doubly-even codes of length $56$ derived from Hadamard matrices of order 28, Discrete Math., 76 (1989), 45-49.  doi: 10.1016/0012-365X(89)90286-0.

[9]

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.

[10]

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.

[11]

L. GalvezJ.-L. KimN. LeeY. G. Roe and B.-S. Won, Some bounds on binary LCD codes, Cryptogr. Commun., 10 (2018), 719-728.  doi: 10.1007/s12095-017-0258-1.

[12]

M. Grassl, Code tables: Bounds on the parameters of various types of codes, Available online at http://www.codetables.de/, Accessed on 5 Aug 2021.

[13]

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.

[14]

M. Harada, Existence of new extremal doubly-even codes and extremal singly-even codes, Des. Codes Cryptogr., 8 (1996), 273-283.  doi: 10.1023/A:1027303722125.

[15]

M. Harada, Self-orthogonal $3$-$(56, 12, 65)$ designs and extremal doubly-even self-dual codes of length $56$, Des. Codes Cryptogr., 38 (2006), 5-16.  doi: 10.1007/s10623-004-5657-6.

[16]

M. Harada, Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes, Des. Codes Cryptogr., 89 (2021), 2295-2312.  doi: 10.1007/s10623-021-00916-1.

[17]

M. HaradaT. A. Gulliver and H. Kaneta, Classification of extremal double-circulant self-dual codes of length up to $62$, Discrete Math., 188 (1998), 127-136.  doi: 10.1016/S0012-365X(97)00250-1.

[18]

M. Harada and H. Kimura, New extremal doubly-even $[64, 32, 12]$ codes, Des. Codes Cryptogr., 6 (1995), 91-96.  doi: 10.1007/BF01398007.

[19]

M. Harada and K. Saito, Binary linear complementary dual codes, Cryptogr. Commun., 11 (2019), 677-696.  doi: 10.1007/s12095-018-0319-0.

[20] W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes, Cambridge University Press, 2003.  doi: 10.1017/CBO9780511807077.
[21]

H. Kimura, Extremal doubly even (56, 28, 12) codes and Hadamard matrices of order 28, Australas. J. Combin., 10 (1994), 171-180. 

[22]

C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control., 22 (1973), 188-200.  doi: 10.1016/S0019-9958(73)90273-8.

[23]

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

[24]

N. Yankov and M. H. Lee, New binary self-dual codes of length 50–60, Des. Codes Cryptogr., 73 (2014), 983-996.  doi: 10.1007/s10623-013-9839-y.

[25]

N. Yankov and R. Russeva, Binary self-dual codes of length 52 to 60 with an automorphism of order 7 or 13, IEEE Trans. Inform. Theory, 57 (2011), 7498-7506.  doi: 10.1109/TIT.2011.2155619.

[26]

V. Yorgov, A method for constructing inequivalent self-dual codes with application to length 56, IEEE Trans. Inform. Theory, 33 (1987), 72-82.  doi: 10.1109/TIT.1987.1057273.

Figure 1.  Matrices $ A_{37,22,5},A_{38,13,10} $
Table 1.  Inequivalent extremal doubly even $ [56,28,12] $ codes
y4 y8 y12 y16 y20 y24 total
D11 45 19 15 2 33 4 118
C56,1 16 3 1 0 10 26 56
C56,2 34 27 26 1 3 14 105
C56,3 10 0 23 2 0 24 59
C56,4 10 109 17 58 2 16 212
C56,5 17 53 25 11 5 4 115
y4 y8 y12 y16 y20 y24 total
D11 45 19 15 2 33 4 118
C56,1 16 3 1 0 10 26 56
C56,2 34 27 26 1 3 14 105
C56,3 10 0 23 2 0 24 59
C56,4 10 109 17 58 2 16 212
C56,5 17 53 25 11 5 4 115
Table 2.  $ P_i,S_i $ for $ i = 1,2,3 $
i Pi Si
1 {2,5} {1,2,3,4,5,6,7,8,9,11}
2 {1,2,4} {3}
i Pi Si
1 {2,5} {1,2,3,4,5,6,7,8,9,11}
2 {1,2,4} {3}
Table 3.  $ C_{n,k,d} $ with $ x,y $
Cn,k,d x y
C37,22,6 (010110011011111) (110010110000001)
C38,13,11 (0010011100110001011000100) (1110100001110101110001101)
Cn,k,d x y
C37,22,6 (010110011011111) (110010110000001)
C38,13,11 (0010011100110001011000100) (1110100001110101110001101)
Table 4.  $ d_{LCD}(n,k) $, where $ 26 \le n \le 40, 9 \le k \le 17 $
n\k 9 10 11 12 13 14 15 16 17
26 9 8 8 8 7 6 5-6 5 4
27 9 9 8 8 7 6 6 6 5
28 10 10 8 8 8 7 6 6 5-6
29 10 10 9 8 8 8 6 6 6
30 11 10 9-10 9 8 8 6-7 6 6
31 11 10 10 10 9 8 7-8 6-7 6
32 12 11 10 10 9-10 8-9 7-8 7-8 6-7
33 12 12 10-11 10 10 9-10 8-9 8 6-8
34 13 12 11-12 10-12 10 10 9-10 8-9 7-8
35 13-14 12-13 12 10-12 10-11 10 9-10 9-10 7-8
36 14 12-14 12-13 11-12 10-12 10-11 10 10 8-9
37 14 12-14 12-14 12-13 10-12 10-12 10-11 10 9-10
38 14-15 13-14 12-14 12-14 11*-12 10-12 10-12 10-11 9-10
39 14-16 14-15 13-14 12-14 11-13 11-12 10-12 10-12 10-11
40 15-16 14-16 13-15 13-14 12-14 11-13 10-12 10-12 10-12
n\k 9 10 11 12 13 14 15 16 17
26 9 8 8 8 7 6 5-6 5 4
27 9 9 8 8 7 6 6 6 5
28 10 10 8 8 8 7 6 6 5-6
29 10 10 9 8 8 8 6 6 6
30 11 10 9-10 9 8 8 6-7 6 6
31 11 10 10 10 9 8 7-8 6-7 6
32 12 11 10 10 9-10 8-9 7-8 7-8 6-7
33 12 12 10-11 10 10 9-10 8-9 8 6-8
34 13 12 11-12 10-12 10 10 9-10 8-9 7-8
35 13-14 12-13 12 10-12 10-11 10 9-10 9-10 7-8
36 14 12-14 12-13 11-12 10-12 10-11 10 10 8-9
37 14 12-14 12-14 12-13 10-12 10-12 10-11 10 9-10
38 14-15 13-14 12-14 12-14 11*-12 10-12 10-12 10-11 9-10
39 14-16 14-15 13-14 12-14 11-13 11-12 10-12 10-12 10-11
40 15-16 14-16 13-15 13-14 12-14 11-13 10-12 10-12 10-12
Table 5.  dLCD(n, k), where 26 ≤ n ≤ 40, 18 ≤ k ≤ 26
n\k 18 19 20 21 22 23 24 25 26
26 4 4 4
27 4 4 4 3
28 5 4 4 4 3
29 6 5 4 4 4 3
30 6 5 5 4 4 4 3
31 6 6 6 5 4 4 4 3
32 6 6 6 5-6 5 4 4 3-4 3
33 6-7 6 6 6 6 5 4 4 4
34 6-8 6-7 6 6 6 5-6 4 4 4
35 7-8 6-8 6-7 5-6 6 6 5 4 4
36 8 7-8 6-8 6-7 6 6 6 5 4
37 8-9 7-8 7-8 6-8 6*-7 6 6 5-6 5
38 9-10 8-9 8 7-8 6-8 6-7 6 6 6
39 10 9-10 8-9 7-8 7-8 6-8 6-7 6 6
40 10-11 9-10 9-10 7-9 8 7-8 6-8 6-7 6
n\k 18 19 20 21 22 23 24 25 26
26 4 4 4
27 4 4 4 3
28 5 4 4 4 3
29 6 5 4 4 4 3
30 6 5 5 4 4 4 3
31 6 6 6 5 4 4 4 3
32 6 6 6 5-6 5 4 4 3-4 3
33 6-7 6 6 6 6 5 4 4 4
34 6-8 6-7 6 6 6 5-6 4 4 4
35 7-8 6-8 6-7 5-6 6 6 5 4 4
36 8 7-8 6-8 6-7 6 6 6 5 4
37 8-9 7-8 7-8 6-8 6*-7 6 6 5-6 5
38 9-10 8-9 8 7-8 6-8 6-7 6 6 6
39 10 9-10 8-9 7-8 7-8 6-8 6-7 6 6
40 10-11 9-10 9-10 7-9 8 7-8 6-8 6-7 6
Table 6.  dLCD(n, k), where 33 ≤ n ≤ 40, 27 ≤ k ≤ 34
n\k 27 28 29 30 31 32 33 34
33 3
34 3-4 3
35 4 4 3
36 4 4 3-4 3
37 4 4 4 4 3
38 5 4 4 4 3-4 3
39 5-6 5 4 4 4 4 2-3
40 6 6 5 4 4 4 3-4 2-3
n\k 27 28 29 30 31 32 33 34
33 3
34 3-4 3
35 4 4 3
36 4 4 3-4 3
37 4 4 4 4 3
38 5 4 4 4 3-4 3
39 5-6 5 4 4 4 4 2-3
40 6 6 5 4 4 4 3-4 2-3
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