doi: 10.3934/amc.2022021
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Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours

1. 

Department of Computing Science and Mathematics, School of Informatics and Creative Arts, Dundalk Institute of Technology, Dundalk, County Louth, A91 K584, Ireland

2. 

Department of Physical, Mathematical and Engineering Sciences, University of Chester, Exton Park, Chester, CH1 4AR, United Kingdom

3. 

Ferenc Rákóczi II Transcarpathian Hungarian College of Higher Education, Berehove, Zakarpattia Oblast, 90201, Ukraine

*Corresponding author: Adam M. Roberts

Received  November 2021 Revised  February 2022 Early access March 2022

In this work, we define a modification of a bordered construction for self-dual codes which utilises $ \lambda $-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist.

Citation: Joe Gildea, Adrian Korban, Adam M. Roberts, Alexander Tylyshchak. Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours. Advances in Mathematics of Communications, doi: 10.3934/amc.2022021
References:
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M. Bortos, J. Gildea, A. Kaya, A. Korban and A. Tylyshchak, New self-dual codes of length 68 from a $2\times2$ block matrix construction and group rings, Adv. Math. Commun., 16 (2022), 269-. doi: 10.3934/amc.2020111.

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J. Gildea, H. Hamilton, A. Kaya and B. Yildiz, Modified quadratic residue constructions and new extremal binary self-dual codes of lengths 64, 66 and 68, Inform. Process. Lett., 157 (2020), 105927, 8 pp. doi: 10.1016/j.ipl.2020.105927.

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show all references

References:
[1]

K. BetsumiyaS. GeorgiouT. A. GulliverM. Harada and C. Koukouvinos, On self-dual codes over some prime fields, Discrete Math., 262 (2003), 37-58.  doi: 10.1016/S0012-365X(02)00520-4.

[2]

M. Bortos, J. Gildea, A. Kaya, A. Korban and A. Tylyshchak, New self-dual codes of length 68 from a $2\times2$ block matrix construction and group rings, Adv. Math. Commun., 16 (2022), 269-. doi: 10.3934/amc.2020111.

[3]

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.

[4]

I. Boukliev and S. Buyuklieva, Some New Extremal Self-Dual Codes with Lengths $44, 50, 54$, and $58$, IEEE Trans. Inform. Theory, 44 (1998), 809-812.  doi: 10.1109/18.661526.

[5]

I. G. Bouyukliev, What is Q-extension?, Serdica J. Comput., 1 (2007), 115-130. 

[6]

S. Buyuklieva and I. Boukliev, Extremal self-dual codes with an automorphism of order $2$, IEEE Trans. Inform. Theory, 44 (1998), 323-328.  doi: 10.1109/18.651059.

[7]

S. Bouyuklieva and P. R. J. Östergøard, New constructions of optimal self-dual binary codes of length 54, Des. Codes Cryptogr., 41 (2006), 101-109.  doi: 10.1007/s10623-006-0018-2.

[8]

S. BouyuklievaR. Russeva and N. Yankov, On the structure of binary self-dual codes having an automorphism of order a square of an odd prime, IEEE Trans. Inform. Theory, 51 (2005), 3678-3686.  doi: 10.1109/TIT.2005.855616.

[9]

P. ÇomakJ.-L. Kim and F. Özbudak, New cubic self-dual codes of length 54, 60 and 66, Appl. Algebra Engrg. Comm. Comput., 29 (2018), 303-312.  doi: 10.1007/s00200-017-0343-x.

[10]

J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.  doi: 10.1109/18.59931.

[11]

S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.

[12]

S. T. DoughertyP. GaboritM. Harada and P. Solé, Type Ⅱ Codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45.  doi: 10.1109/18.746770.

[13]

S. T. Dougherty, J. Gildea and A. Kaya, $2^n$ Bordered constructions of self-dual codes from group rings, Finite Fields Appl., 67 (2020), 101692, 17 pp. doi: 10.1016/j.ffa.2020.101692.

[14]

S. T. DoughertyJ. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings, Cryptogr. Commun., 12 (2020), 127-146.  doi: 10.1007/s12095-019-00380-8.

[15]

S. T. DoughertyJ. GildeaA. Kaya and B. Yildiz, New self-dual and formally self-dual codes from group ring constructions, Adv. Math. Commun., 14 (2020), 11-22.  doi: 10.3934/amc.2020002.

[16]

S. T. DoughertyJ. GildeaA. Korban and A. Kaya, Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68, Adv. Math. Commun., 14 (2020), 677-702.  doi: 10.3934/amc.2020037.

[17]

S. T. DoughertyJ. GildeaA. Korban and A. Kaya, New extremal self-dual binary codes of length 68 via composite construction, $\mathbb{F}_2+u\mathbb{F}_2$ lifts, extensions and neighbours, Int. J. Inf. Coding Theory, 5 (2018), 211-226. 

[18]

S. T. DoughertyJ. GildeaA. Korban and A. Kaya, Composite matrices from group rings, composite $G$-codes and constructions of self-dual codes, Des. Codes Cryptogr., 89 (2021), 1615-1638.  doi: 10.1007/s10623-021-00882-8.

[19]

S. T. DoughertyJ. GildeaA. KorbanA. KayaA. Tylyshchak and B. Yildiz, Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes, Finite Fields Appl., 57 (2019), 108-127.  doi: 10.1016/j.ffa.2019.02.004.

[20]

S. T. DoughertyT. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.  doi: 10.1109/18.641574.

[21]

S. T. Dougherty and M. Harada, New extremal self-dual codes of length $68$, IEEE Trans. Inform. Theory, 45 (1999), 2133-2136.  doi: 10.1109/18.782158.

[22]

J. Gildea, H. Hamilton, A. Kaya and B. Yildiz, Modified quadratic residue constructions and new extremal binary self-dual codes of lengths 64, 66 and 68, Inform. Process. Lett., 157 (2020), 105927, 8 pp. doi: 10.1016/j.ipl.2020.105927.

[23]

J. Gildea, A. Kaya, A. Korban and A. Tylyshchak, Self-dual codes using bisymmetric matrices and group rings, Discrete Math., 343 (2020), 112085, 10 pp. doi: 10.1016/j.disc.2020.112085.

[24]

J. Gildea, A. Kaya, A. Korban and B. Yildiz, New extremal binary self-dual codes of length 68 from generalized neighbors, Finite Fields Appl., 67 (2020), 101727, 12 pp. doi: 10.1016/j.ffa.2020.101727.

[25]

J. Gildea, A. Kaya, A. M. Roberts, R. Taylor and A. Tylyshchak, New self-dual codes from $2\times 2$ block circulant matrices, group rings and neighbours of neighbours, Adv. Math. Commun., (2021). doi: 10.3934/amc.2021039.

[26]

J. Gildea, A. Kaya, R. Taylor, A. Tylyshchak and B. Yildiz, New extremal binary self-dual codes from block circulant matrices and block quadratic residue circulant matrices, Discrete Math., 344 (2021), 112590, 11 pp. doi: 10.1016/j.disc.2021.112590.

[27]

J. GildeaA. KayaR. Taylor and B. Yildiz, Constructions for self-dual codes induced from group rings, Finite Fields Appl., 51 (2018), 71-92.  doi: 10.1016/j.ffa.2018.01.002.

[28]

J. Gildea, A. Kaya, A. Tylyshchak and B. Yildiz, A group induced four-circulant construction for self-dual codes and new extremal binary self-dual codes, Discrete Math., 342 (2019), 111620, 8 pp, arXiv: 1912.11758. doi: 10.1016/j.disc.2019.111620.

[29]

J. GildeaA. KayaA. Tylyshchak and B. Yildiz, A modified bordered construction for self-dual codes from group rings, J. Algebra Comb. Discrete Struct. Appl., 7 (2020), 103-119.  doi: 10.13069/jacodesmath.729402.

[30]

J. Gildea, A. Kaya and B. Yildiz, An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes. I, Discrete Math., 342 (2019), 112620, 8 pp. doi: 10.1016/j.disc.2019.111620.

[31]

J. GildeaA. Kaya and B. Yildiz, New binary self-dual codes via a variation of the four-circulant construction, Math. Commun., 25 (2020), 213-226. 

[32]

J. GildeaA. KorbanA. Kaya and B. Yildiz, Constructing self-dual codes from group rings and reverse circulant matrices, Adv. Math. Commun., 15 (2021), 471-485.  doi: 10.3934/amc.2020077.

[33]

J. Gildea, A. Korban and A. M. Roberts, New binary self-dual codes of lengths 56, 58, 64, 80 and 92 from a modification of the four circulant construction, Finite Fields Appl., 75 (2021), 101876, 21 pp. doi: 10.1016/j.ffa.2021.101876.

[34]

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Table 1.  Quaternary notation system for elements of $ \mathbb{F}_2+u \mathbb{F}_2 $
$ \mathbb{F}_2+u \mathbb{F}_2 $ Symbol
$ 0 $ $ \texttt{0} $
$ 1 $ $ \texttt{1} $
$ u $ $ \texttt{2} $
$ 1+u $ $ \texttt{3} $
$ \mathbb{F}_2+u \mathbb{F}_2 $ Symbol
$ 0 $ $ \texttt{0} $
$ 1 $ $ \texttt{1} $
$ u $ $ \texttt{2} $
$ 1+u $ $ \texttt{3} $
Table 2.  Code of length 54 over $ \mathbb{F}_2 $ from Theorem 3.1 to which we apply Remark 4.1 to obtain the code in Table 3, where $ \boldsymbol{{\xi}} = (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) $
$ \mathcal{C}_{54,i}^* $ $ {\bf{{a}}} $ $ {\bf{{b}}} $ $ {\bf{{c}}} $ $ \boldsymbol{{\xi}} $
1 $ \texttt{(0111000101101)} $ $ \texttt{(1101110000100)} $ $ \texttt{(0101111110011)} $ $ \texttt{(001101)} $
$ \mathcal{C}_{54,i}^* $ $ {\bf{{a}}} $ $ {\bf{{b}}} $ $ {\bf{{c}}} $ $ \boldsymbol{{\xi}} $
1 $ \texttt{(0111000101101)} $ $ \texttt{(1101110000100)} $ $ \texttt{(0101111110011)} $ $ \texttt{(001101)} $
Table 3.  New binary self-dual $ [54,27,10] $ code from searching for neighbours of $ \mathcal{C}_{54,j}^* $ as given in Table 2 using Remark 4.1 with $ {\bf{{x}}} = ({\bf{{0}}},{\bf{{x}}}_0) $
$ \mathcal{C}_{54,i} $ $ \mathcal{C}_{54,j}^* $ $ {\bf{{x}}}_0 $ $ W_{54,k} $ $ \alpha $ $ |\text{Aut}({\mathcal{C}_{54,i}})| $
1 1 $ \texttt{(000001100101001000111101101)} $ 1 $ 23 $ $ 3 $
$ \mathcal{C}_{54,i} $ $ \mathcal{C}_{54,j}^* $ $ {\bf{{x}}}_0 $ $ W_{54,k} $ $ \alpha $ $ |\text{Aut}({\mathcal{C}_{54,i}})| $
1 1 $ \texttt{(000001100101001000111101101)} $ 1 $ 23 $ $ 3 $
Table 4.  New binary self-dual $ [68,34,12] $ codes from Theorem 3.1 over $ \mathbb{F}_2+u \mathbb{F}_2 $, where $ \boldsymbol{{\xi}} = (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) $
$ \mathcal{C}_{68,i} $ $ \lambda $ $ \mu $ $ {\bf{{a}}} $ $ {\bf{{b}}} $ $ {\bf{{c}}} $ $ \boldsymbol{{\xi}} $ $ W_{68,j} $ $ \alpha $ $ \beta $ $ |\text{Aut}({\mathcal{C}_{68,i}})| $
1 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(22120031)} $ $ \texttt{(02331100)} $ $ \texttt{(33331213)} $ $ \texttt{(101132)} $ 1 $ 110 $ $ - $ $ 2 $
2 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(10021300)} $ $ \texttt{(31232012)} $ $ \texttt{(30313131)} $ $ \texttt{(120023)} $ 1 $ 124 $ $ - $ $ 2 $
3 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(01323103)} $ $ \texttt{(20022123)} $ $ \texttt{(00300222)} $ $ \texttt{(013332)} $ 2 $ 20 $ $ 1 $ $ 2 $
4 $ \texttt{1} $ $ \texttt{3} $ $ \texttt{(01230200)} $ $ \texttt{(13010312)} $ $ \texttt{(22003002)} $ $ \texttt{(102232)} $ 2 $ 28 $ $ 1 $ $ 2 $
5 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(31221023)} $ $ \texttt{(30003111)} $ $ \texttt{(13012103)} $ $ \texttt{(233310)} $ 2 $ 32 $ $ 1 $ $ 2 $
6 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(03210210)} $ $ \texttt{(32221121)} $ $ \texttt{(13331101)} $ $ \texttt{(122201)} $ 2 $ 34 $ $ 1 $ $ 2 $
7 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(00030320)} $ $ \texttt{(21031233)} $ $ \texttt{(32100012)} $ $ \texttt{(122201)} $ 2 $ 36 $ $ 1 $ $ 2 $
$ \mathcal{C}_{68,i} $ $ \lambda $ $ \mu $ $ {\bf{{a}}} $ $ {\bf{{b}}} $ $ {\bf{{c}}} $ $ \boldsymbol{{\xi}} $ $ W_{68,j} $ $ \alpha $ $ \beta $ $ |\text{Aut}({\mathcal{C}_{68,i}})| $
1 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(22120031)} $ $ \texttt{(02331100)} $ $ \texttt{(33331213)} $ $ \texttt{(101132)} $ 1 $ 110 $ $ - $ $ 2 $
2 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(10021300)} $ $ \texttt{(31232012)} $ $ \texttt{(30313131)} $ $ \texttt{(120023)} $ 1 $ 124 $ $ - $ $ 2 $
3 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(01323103)} $ $ \texttt{(20022123)} $ $ \texttt{(00300222)} $ $ \texttt{(013332)} $ 2 $ 20 $ $ 1 $ $ 2 $
4 $ \texttt{1} $ $ \texttt{3} $ $ \texttt{(01230200)} $ $ \texttt{(13010312)} $ $ \texttt{(22003002)} $ $ \texttt{(102232)} $ 2 $ 28 $ $ 1 $ $ 2 $
5 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(31221023)} $ $ \texttt{(30003111)} $ $ \texttt{(13012103)} $ $ \texttt{(233310)} $ 2 $ 32 $ $ 1 $ $ 2 $
6 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(03210210)} $ $ \texttt{(32221121)} $ $ \texttt{(13331101)} $ $ \texttt{(122201)} $ 2 $ 34 $ $ 1 $ $ 2 $
7 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(00030320)} $ $ \texttt{(21031233)} $ $ \texttt{(32100012)} $ $ \texttt{(122201)} $ 2 $ 36 $ $ 1 $ $ 2 $
Table 5.  Code of length 34 over $ \mathbb{F}_2+u \mathbb{F}_2 $ from Theorem 3.1 to the image of which under $ \varphi_{ \mathbb{F}_2+u \mathbb{F}_2} $ we then apply Remark 4.1 to obtain the codes in Table 6, where $ \boldsymbol{{\xi}} = (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) $
$ \mathcal{C}_{34,i}^* $ $ \lambda $ $ \mu $ $ {\bf{{a}}} $ $ {\bf{{b}}} $ $ {\bf{{c}}} $ $ \boldsymbol{{\xi}} $
1 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(01323103)} $ $ \texttt{(20022123)} $ $ \texttt{(00300222)} $ $ \texttt{(013332)} $
$ \mathcal{C}_{34,i}^* $ $ \lambda $ $ \mu $ $ {\bf{{a}}} $ $ {\bf{{b}}} $ $ {\bf{{c}}} $ $ \boldsymbol{{\xi}} $
1 $ \texttt{1} $ $ \texttt{1} $ $ \texttt{(01323103)} $ $ \texttt{(20022123)} $ $ \texttt{(00300222)} $ $ \texttt{(013332)} $
Table 6.  New binary self-dual $ [68,34,12] $ codes from searching for neighbours of $ \varphi_{ \mathbb{F}_2+u \mathbb{F}_2}(\mathcal{C}_{34,j}^*) $ using Remark 4.1 with $ {\bf{{x}}} = ({\bf{{0}}},{\bf{{x}}}_0) $, where $ \mathcal{C}_{34,j}^* $ are as given in Table 5
$ \mathcal{C}_{68,i} $ $ \mathcal{C}_{34,j}^* $ $ {\bf{{x}}}_0 $ $ W_{68,k} $ $ \alpha $ $ \beta $ $ |\text{Aut}({\mathcal{C}_{68,i}})| $
8 1 $ \texttt{(0101010011111010001101100011011100)} $ 1 $ 113 $ $ - $ $ 1 $
9 1 $ \texttt{(1110010011100001110010110111100100)} $ 1 $ 114 $ $ - $ $ 1 $
10 1 $ \texttt{(1010100100010111000000100111010111)} $ 1 $ 116 $ $ - $ $ 1 $
11 1 $ \texttt{(0011000011011101010101010100010000)} $ 1 $ 118 $ $ - $ $ 1 $
12 1 $ \texttt{(0101010001111010000101100011011111)} $ 1 $ 121 $ $ - $ $ 1 $
13 1 $ \texttt{(0011001001011000000110010111110101)} $ 1 $ 123 $ $ - $ $ 1 $
14 1 $ \texttt{(0101110101111010001101100011011101)} $ 2 $ 37 $ $ 1 $ $ 1 $
$ \mathcal{C}_{68,i} $ $ \mathcal{C}_{34,j}^* $ $ {\bf{{x}}}_0 $ $ W_{68,k} $ $ \alpha $ $ \beta $ $ |\text{Aut}({\mathcal{C}_{68,i}})| $
8 1 $ \texttt{(0101010011111010001101100011011100)} $ 1 $ 113 $ $ - $ $ 1 $
9 1 $ \texttt{(1110010011100001110010110111100100)} $ 1 $ 114 $ $ - $ $ 1 $
10 1 $ \texttt{(1010100100010111000000100111010111)} $ 1 $ 116 $ $ - $ $ 1 $
11 1 $ \texttt{(0011000011011101010101010100010000)} $ 1 $ 118 $ $ - $ $ 1 $
12 1 $ \texttt{(0101010001111010000101100011011111)} $ 1 $ 121 $ $ - $ $ 1 $
13 1 $ \texttt{(0011001001011000000110010111110101)} $ 1 $ 123 $ $ - $ $ 1 $
14 1 $ \texttt{(0101110101111010001101100011011101)} $ 2 $ 37 $ $ 1 $ $ 1 $
Table 7.  New binary self-dual $ [82,41,14] $ codes from Theorem 3.1 over $ \mathbb{F}_2 $, where $ \boldsymbol{{\xi}} = (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) $
$ \mathcal{C}_{82,i} $ $ {\bf{{a}}} $ $ {\bf{{b}}} $ $ {\bf{{c}}} $ $ \boldsymbol{{\xi}} $
1 $ \texttt{(00110011100000000110)} $ $ \texttt{(00100110011101010011)} $ $ \texttt{(00010010010001000001)} $ $ \texttt{(101010)} $
2 $ \texttt{(11001011011010110101)} $ $ \texttt{(10011010011011010000)} $ $ \texttt{(01010011100101001010)} $ $ \texttt{(101010)} $
3 $ \texttt{(00011110011001011110)} $ $ \texttt{(01010101010011110100)} $ $ \texttt{(10101110111000111011)} $ $ \texttt{(101010)} $
4 $ \texttt{(00000110100111111111)} $ $ \texttt{(00110110000111101000)} $ $ \texttt{(11111011010111011000)} $ $ \texttt{(101001)} $
5 $ \texttt{(11100011011110101011)} $ $ \texttt{(11110001101100110011)} $ $ \texttt{(00100010100000001010)} $ $ \texttt{(101010)} $
6 $ \texttt{(11111110010110010010)} $ $ \texttt{(10001001101001001110)} $ $ \texttt{(01111010111110011001)} $ $ \texttt{(101001)} $
7 $ \texttt{(00111010001011010100)} $ $ \texttt{(11001010111101110001)} $ $ \texttt{(10001100011010110001)} $ $ \texttt{(101010)} $
8 $ \texttt{(00110011011011110001)} $ $ \texttt{(00101110100101000100)} $ $ \texttt{(10110001110000000001)} $ $ \texttt{(101110)} $
9 $ \texttt{(10000011001000100011)} $ $ \texttt{(00110001010001110100)} $ $ \texttt{(00010001110001000101)} $ $ \texttt{(101101)} $
10 $ \texttt{(11101110100101100010)} $ $ \texttt{(01110011001100110001)} $ $ \texttt{(00010100000110011010)} $ $ \texttt{(101101)} $
11 $ \texttt{(00011011111101000011)} $ $ \texttt{(11000000001100111001)} $ $ \texttt{(10100000101010010010)} $ $ \texttt{(101110)} $
12 $ \texttt{(00011110101110000110)} $ $ \texttt{(11000011010011000101)} $ $ \texttt{(01001010001111101110)} $ $ \texttt{(101110)} $
13 $ \texttt{(00100000101100010000)} $ $ \texttt{(11010101010010100011)} $ $ \texttt{(01011101110000111001)} $ $ \texttt{(101101)} $
14 $ \texttt{(10001111010001011100)} $ $ \texttt{(00000001010010011000)} $ $ \texttt{(01101011111010000110)} $ $ \texttt{(101101)} $
15 $ \texttt{(10011111001010110001)} $ $ \texttt{(11000010101110010110)} $ $ \texttt{(01000011001011110111)} $ $ \texttt{(101110)} $
16 $ \texttt{(11100100001011100001)} $ $ \texttt{(00101100110000110100)} $ $ \texttt{(00011111001001111100)} $ $ \texttt{(101101)} $
17 $ \texttt{(10001110110000101100)} $ $ \texttt{(00111010000111110010)} $ $ \texttt{(01110111101001100001)} $ $ \texttt{(101110)} $
18 $ \texttt{(00001101111100100101)} $ $ \texttt{(00011001110100011111)} $ $ \texttt{(01001100001011101111)} $ $ \texttt{(101110)} $
$\mathcal{C}_{82,i}$ $W_{82,j}$ $\alpha$ $\beta$ $|\text{Aut}({\mathcal{C}_{82,i}})|$
1 2 $-738$ $18$ $1$
2 2 $-736$ $18$ $1$
3 2 $-734$ $18$ $1$
4 2 $-714$ $18$ $1$
5 2 $-706$ $18$ $1$
6 2 $-688$ $18$ $1$
7 2 $-662$ $18$ $1$
8 3 $-828$ $0$ $1$
9 3 $-816$ $0$ $1$
10 3 $-812$ $0$ $1$
11 3 $-798$ $0$ $1$
12 3 $-786$ $0$ $1$
13 3 $-778$ $0$ $1$
14 3 $-776$ $0$ $1$
15 3 $-818$ $1$ $1$
16 3 $-838$ $2$ $1$
17 3 $-818$ $2$ $1$
18 3 $-854$ $5$ $1$
$ \mathcal{C}_{82,i} $ $ {\bf{{a}}} $ $ {\bf{{b}}} $ $ {\bf{{c}}} $ $ \boldsymbol{{\xi}} $
1 $ \texttt{(00110011100000000110)} $ $ \texttt{(00100110011101010011)} $ $ \texttt{(00010010010001000001)} $ $ \texttt{(101010)} $
2 $ \texttt{(11001011011010110101)} $ $ \texttt{(10011010011011010000)} $ $ \texttt{(01010011100101001010)} $ $ \texttt{(101010)} $
3 $ \texttt{(00011110011001011110)} $ $ \texttt{(01010101010011110100)} $ $ \texttt{(10101110111000111011)} $ $ \texttt{(101010)} $
4 $ \texttt{(00000110100111111111)} $ $ \texttt{(00110110000111101000)} $ $ \texttt{(11111011010111011000)} $ $ \texttt{(101001)} $
5 $ \texttt{(11100011011110101011)} $ $ \texttt{(11110001101100110011)} $ $ \texttt{(00100010100000001010)} $ $ \texttt{(101010)} $
6 $ \texttt{(11111110010110010010)} $ $ \texttt{(10001001101001001110)} $ $ \texttt{(01111010111110011001)} $ $ \texttt{(101001)} $
7 $ \texttt{(00111010001011010100)} $ $ \texttt{(11001010111101110001)} $ $ \texttt{(10001100011010110001)} $ $ \texttt{(101010)} $
8 $ \texttt{(00110011011011110001)} $ $ \texttt{(00101110100101000100)} $ $ \texttt{(10110001110000000001)} $ $ \texttt{(101110)} $
9 $ \texttt{(10000011001000100011)} $ $ \texttt{(00110001010001110100)} $ $ \texttt{(00010001110001000101)} $ $ \texttt{(101101)} $
10 $ \texttt{(11101110100101100010)} $ $ \texttt{(01110011001100110001)} $ $ \texttt{(00010100000110011010)} $ $ \texttt{(101101)} $
11 $ \texttt{(00011011111101000011)} $ $ \texttt{(11000000001100111001)} $ $ \texttt{(10100000101010010010)} $ $ \texttt{(101110)} $
12 $ \texttt{(00011110101110000110)} $ $ \texttt{(11000011010011000101)} $ $ \texttt{(01001010001111101110)} $ $ \texttt{(101110)} $
13 $ \texttt{(00100000101100010000)} $ $ \texttt{(11010101010010100011)} $ $ \texttt{(01011101110000111001)} $ $ \texttt{(101101)} $
14 $ \texttt{(10001111010001011100)} $ $ \texttt{(00000001010010011000)} $ $ \texttt{(01101011111010000110)} $ $ \texttt{(101101)} $
15 $ \texttt{(10011111001010110001)} $ $ \texttt{(11000010101110010110)} $ $ \texttt{(01000011001011110111)} $ $ \texttt{(101110)} $
16 $ \texttt{(11100100001011100001)} $ $ \texttt{(00101100110000110100)} $ $ \texttt{(00011111001001111100)} $ $ \texttt{(101101)} $
17 $ \texttt{(10001110110000101100)} $ $ \texttt{(00111010000111110010)} $ $ \texttt{(01110111101001100001)} $ $ \texttt{(101110)} $
18 $ \texttt{(00001101111100100101)} $ $ \texttt{(00011001110100011111)} $ $ \texttt{(01001100001011101111)} $ $ \texttt{(101110)} $
$\mathcal{C}_{82,i}$ $W_{82,j}$ $\alpha$ $\beta$ $|\text{Aut}({\mathcal{C}_{82,i}})|$
1 2 $-738$ $18$ $1$
2 2 $-736$ $18$ $1$
3 2 $-734$ $18$ $1$
4 2 $-714$ $18$ $1$
5 2 $-706$ $18$ $1$
6 2 $-688$ $18$ $1$
7 2 $-662$ $18$ $1$
8 3 $-828$ $0$ $1$
9 3 $-816$ $0$ $1$
10 3 $-812$ $0$ $1$
11 3 $-798$ $0$ $1$
12 3 $-786$ $0$ $1$
13 3 $-778$ $0$ $1$
14 3 $-776$ $0$ $1$
15 3 $-818$ $1$ $1$
16 3 $-838$ $2$ $1$
17 3 $-818$ $2$ $1$
18 3 $-854$ $5$ $1$
Table 8.  New binary self-dual $ [94,47,16] $ codes from Theorem 3.1 over $ \mathbb{F}_2 $, where $ \boldsymbol{{\xi}} = (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) $
$ \mathcal{C}_{94,i} $ $ {\bf{{a}}} $ $ {\bf{{b}}} $ $ {\bf{{c}}} $ $ \boldsymbol{{\xi}} $
1 $ \texttt{(01111111111001110101110)} $ $ \texttt{(01101101000111011010001)} $ $ \texttt{(00001000000000000000000)} $ $ \texttt{(001110)} $
2 $ \texttt{(10010111111101010000010)} $ $ \texttt{(11100100111001001111001)} $ $ \texttt{(00001000000000000000000)} $ $ \texttt{(001110)} $
3 $ \texttt{(01100111001001011111010)} $ $ \texttt{(10110101001111101000010)} $ $ \texttt{(11010111010100010110011)} $ $ \texttt{(001110)} $
4 $ \texttt{(10010101100111000001101)} $ $ \texttt{(11010000110110110000001)} $ $ \texttt{(01010001111011001010111)} $ $ \texttt{(110010)} $
5 $ \texttt{(00000111101001000010100)} $ $ \texttt{(11110100110110100111000)} $ $ \texttt{(01001111001111101100100)} $ $ \texttt{(001101)} $
6 $ \texttt{(11011110010100111000000)} $ $ \texttt{(01110100011001101101111)} $ $ \texttt{(01110000001111000111111)} $ $ \texttt{(001101)} $
7 $ \texttt{(01011011110110010001110)} $ $ \texttt{(10010110110110001100101)} $ $ \texttt{(00000100000000000000000)} $ $ \texttt{(110010)} $
8 $ \texttt{(01100001100001100101010)} $ $ \texttt{(11111101000110000010101)} $ $ \texttt{(00100000000000000000000)} $ $ \texttt{(001101)} $
9 $ \texttt{(00000111001111011011110)} $ $ \texttt{(11100000000100010011010)} $ $ \texttt{(01101111110111000010001)} $ $ \texttt{(110010)} $
10 $ \texttt{(01101101011111000010001)} $ $ \texttt{(10100110011101001101101)} $ $ \texttt{(01011000110000010010101)} $ $ \texttt{(110010)} $
11 $ \texttt{(11010010011100001111011)} $ $ \texttt{(10001110000000010001110)} $ $ \texttt{(11101110011100011101000)} $ $ \texttt{(110010)} $
12 $ \texttt{(10101100011011001010111)} $ $ \texttt{(00010010000011111000010)} $ $ \texttt{(00111100000011101111110)} $ $ \texttt{(001101)} $
$ \mathcal{C}_{94,i} $ $ W_{94,j} $ $ \alpha $ $ \beta $ $ |\text{Aut}({\mathcal{C}_{94,i}})| $
1 1 $ 4646 $ $ -92 $ $ 2\cdot 23 $
2 1 $ 3450 $ $ -46 $ $ 2\cdot 23 $
3 1 $ 3680 $ $ -46 $ $ 23 $
4 1 $ 3772 $ $ -46 $ $ 23 $
5 1 $ 4186 $ $ -46 $ $ 23 $
6 1 $ 2944 $ $ -23 $ $ 23 $
7 1 $ 3680 $ $ -23 $ $ 23 $
8 1 $ 2346 $ $ 0 $ $ 2\cdot 23 $
9 1 $ 2530 $ $ 0 $ $ 23 $
10 1 $ 2576 $ $ 0 $ $ 23 $
11 1 $ 3496 $ $ 0 $ $ 23 $
12 1 $ 3588 $ $ 0 $ $ 23 $
$ \mathcal{C}_{94,i} $ $ {\bf{{a}}} $ $ {\bf{{b}}} $ $ {\bf{{c}}} $ $ \boldsymbol{{\xi}} $
1 $ \texttt{(01111111111001110101110)} $ $ \texttt{(01101101000111011010001)} $ $ \texttt{(00001000000000000000000)} $ $ \texttt{(001110)} $
2 $ \texttt{(10010111111101010000010)} $ $ \texttt{(11100100111001001111001)} $ $ \texttt{(00001000000000000000000)} $ $ \texttt{(001110)} $
3 $ \texttt{(01100111001001011111010)} $ $ \texttt{(10110101001111101000010)} $ $ \texttt{(11010111010100010110011)} $ $ \texttt{(001110)} $
4 $ \texttt{(10010101100111000001101)} $ $ \texttt{(11010000110110110000001)} $ $ \texttt{(01010001111011001010111)} $ $ \texttt{(110010)} $
5 $ \texttt{(00000111101001000010100)} $ $ \texttt{(11110100110110100111000)} $ $ \texttt{(01001111001111101100100)} $ $ \texttt{(001101)} $
6 $ \texttt{(11011110010100111000000)} $ $ \texttt{(01110100011001101101111)} $ $ \texttt{(01110000001111000111111)} $ $ \texttt{(001101)} $
7 $ \texttt{(01011011110110010001110)} $ $ \texttt{(10010110110110001100101)} $ $ \texttt{(00000100000000000000000)} $ $ \texttt{(110010)} $
8 $ \texttt{(01100001100001100101010)} $ $ \texttt{(11111101000110000010101)} $ $ \texttt{(00100000000000000000000)} $ $ \texttt{(001101)} $
9 $ \texttt{(00000111001111011011110)} $ $ \texttt{(11100000000100010011010)} $ $ \texttt{(01101111110111000010001)} $ $ \texttt{(110010)} $
10 $ \texttt{(01101101011111000010001)} $ $ \texttt{(10100110011101001101101)} $ $ \texttt{(01011000110000010010101)} $ $ \texttt{(110010)} $
11 $ \texttt{(11010010011100001111011)} $ $ \texttt{(10001110000000010001110)} $ $ \texttt{(11101110011100011101000)} $ $ \texttt{(110010)} $
12 $ \texttt{(10101100011011001010111)} $ $ \texttt{(00010010000011111000010)} $ $ \texttt{(00111100000011101111110)} $ $ \texttt{(001101)} $
$ \mathcal{C}_{94,i} $ $ W_{94,j} $ $ \alpha $ $ \beta $ $ |\text{Aut}({\mathcal{C}_{94,i}})| $
1 1 $ 4646 $ $ -92 $ $ 2\cdot 23 $
2 1 $ 3450 $ $ -46 $ $ 2\cdot 23 $
3 1 $ 3680 $ $ -46 $ $ 23 $
4 1 $ 3772 $ $ -46 $ $ 23 $
5 1 $ 4186 $ $ -46 $ $ 23 $
6 1 $ 2944 $ $ -23 $ $ 23 $
7 1 $ 3680 $ $ -23 $ $ 23 $
8 1 $ 2346 $ $ 0 $ $ 2\cdot 23 $
9 1 $ 2530 $ $ 0 $ $ 23 $
10 1 $ 2576 $ $ 0 $ $ 23 $
11 1 $ 3496 $ $ 0 $ $ 23 $
12 1 $ 3588 $ $ 0 $ $ 23 $
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