American Institute of Mathematical Sciences

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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References:
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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. Bouyuklieva, R. 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. Çomak, J.-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. 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Gildea, A. 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. Dougherty, J. Gildea, A. 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. Dougherty, J. Gildea, A. Korban, A. Kaya, A. 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. Dougherty, T. 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. 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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 $0$ $\texttt{0}$ $1$ $\texttt{1}$ $u$ $\texttt{2}$ $1+u$ $\texttt{3}$
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 $\texttt{(0111000101101)}$ $\texttt{(1101110000100)}$ $\texttt{(0101111110011)}$ $\texttt{(001101)}$
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 $23$ $3$
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 $36$ $1$ $2$
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 $\texttt{1}$ $\texttt{1}$ $\texttt{(01323103)}$ $\texttt{(20022123)}$ $\texttt{(00300222)}$ $\texttt{(013332)}$
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 $37$ $1$ $1$
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 $-854$ $5$ $1$
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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