# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021039
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## New self-dual codes from $2 \times 2$ block circulant matrices, group rings and neighbours of neighbours

 1 Department of Mathematics, Faculty of Science and Engineering, University of Chester, England 2 Harmony Public Schools, Houston, TX, USA 3 Department of Algebra, Uzhgorod National University, Uzhgorod, Ukraine

Received  March 2021 Revised  June 2021 Early access September 2021

In this paper, we construct new self-dual codes from a construction that involves a unique combination; $2 \times 2$ block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields self-dual codes. The theory is supported by the construction of self-dual codes over the rings $\mathbb{F}_2$, $\mathbb{F}_2+u \mathbb{F}_2$ and $\mathbb{F}_4+u \mathbb{F}_4$. Using extensions and neighbours of codes, we construct $32$ new self-dual codes of length $68$. We construct 48 new best known singly-even self-dual codes of length 96.

Citation: Joe Gildea, Abidin Kaya, Adam Michael Roberts, Rhian Taylor, Alexander Tylyshchak. New self-dual codes from $2 \times 2$ block circulant matrices, group rings and neighbours of neighbours. Advances in Mathematics of Communications, doi: 10.3934/amc.2021039
##### References:
 [1] D. Anev, M. Harada and N. Yankov, New extremal singly even self-dual codes of lengths 64 and 66, J. Algebra Comb. Discrete Struct. Appl., 5 (2018), 143-151.  doi: 10.13069/jacodesmath.458601.  Google Scholar [2] E. R. Berlekamp, F. J. MacWilliams and N. J. A. Sloane, Gleason's theorem on self-dual codes, IEEE Trans. Inform. Theory, IT-18 (1972), 409-414.  doi: 10.1109/tit.1972.1054817.  Google Scholar [3] F. Bernhardt, P. Landrock and O. Manz, The extended golay codes considered as ideals, J. Combin. Theory Ser. A, 55 (1990), 235-246.  doi: 10.1016/0097-3165(90)90069-9.  Google Scholar [4] W. Bosma, J. 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.  Google Scholar [5] C. L. Chen, W. W Peterson and E. J. Weldon. Jr, Some results on quasi-cyclic codes, Information and Control, 15 (1969), 407-423.  doi: 10.1016/S0019-9958(69)90497-5.  Google Scholar [6] 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.  Google Scholar [7] P. J. Davis, Circulant Matrices, A Wiley-Interscience Publication; Pure and Applied Mathematics, John Wiley & Sons, New York-Chichester-Brisbane, 1979.  Google Scholar [8] S. T. Dougherty, P. Gaborit, M. Harada and P. Solé, Type ii codes over ${\mathbf f}_2+u{\mathbf f}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45.  doi: 10.1109/18.746770.  Google Scholar [9] 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.  Google Scholar [10] S. T. Dougherty, J. Gildea, R. Taylor and A. Tylyshchak, Group rings, $g$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7.  Google Scholar [11] S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.  doi: 10.1109/18.641574.  Google Scholar [12] S. T. Dougherty, B. Yildiz and S. Karadeniz, Codes over $r_k$, gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219.  doi: 10.1016/j.ffa.2010.11.002.  Google Scholar [13] S. Dougherty, B. Yildiz and S. Karadeniz, Self-dual codes over $r_k$ and binary self-dual codes, Eur. J. Pure Appl. Math., 6 (2013), 89-106.   Google Scholar [14] P. Gaborit, Quadratic double circulant codes over fields, J. Combin. Theory Ser. A, 97 (2002), 85-107.  doi: 10.1006/jcta.2001.3198.  Google Scholar [15] S. D. Georgiou and E. Lappas, Self-dual codes from circulant matrices, Des. Codes Cryptogr., 64 (2012), 129-141.  doi: 10.1007/s10623-011-9510-4.  Google Scholar [16] 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, Information Processing Letters, 157. doi: 10.1016/j.ipl.2020.105927.  Google Scholar [17] J. Gildea, A. Kaya, A. Korban 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.  Google Scholar [18] J. Gildea, A. Korban and A. M. Roberts, New binary self-dual codes of lengths 80, 84 and 96 from composite matrices, 2021, https://arXiv.org/abs/2106.12355. Google Scholar [19] A. M. Gleason, Weight polynomials of self-dual codes and the macwilliams identities, 1971,211–215.  Google Scholar [20] T. A. Gulliver and M. Harada, On extremal double circulant self-dual codes of lengths 90–96, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 403-415.  doi: 10.1007/s00200-019-00381-3.  Google Scholar [21] M. Gürel and N. Yankov, Self-dual codes with an automorphism of order 17, Math. Commun., 21 (2016), 97-107.   Google Scholar [22] T. Hurley, Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335.   Google Scholar [23] S. Karadeniz and B. Yildiz, New extremal binary self-dual codes of length 66 as extensions of self-dual codes over $R_k$, J. Franklin Inst., 350, (2013), 1963–1973 doi: 10.1016/j.jfranklin.2013.05.015.  Google Scholar [24] M. Karlin, New binary coding results by circulants, IEEE Trans. Inform. Theory, IT-15 (1969), 81-92.  doi: 10.1109/tit.1969.1054261.  Google Scholar [25] A. Kaya, B. Yildiz and I. Siap, Quadratic residue codes over $\mathbb{F}_p+v\mathbb{F}_p$ and their Gray images, J. Pure Appl. Algebra, 218 (2014), 1999-2011.  doi: 10.1016/j.jpaa.2014.03.002.  Google Scholar [26] J.-L. Kim, New extremal self-dual codes of lengths 36, 38, and 58, IEEE Trans. Inform. Theory, 47 (2001), 386-393.  doi: 10.1109/18.904540.  Google Scholar [27] S. Ling and P. Solé, Type ii codes over $\mathbf { F_4+u} f_4$, European J. Combin., 22 (2001), 983-997.  doi: 10.1006/eujc.2001.0509.  Google Scholar [28] F. J. MacWilliams, C. L. Mallows and N. J. A Sloane, Generalizations of gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, IT-18 (1972), 794-805.  doi: 10.1109/tit.1972.1054898.  Google Scholar [29] F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson, Good self dual codes exist, Discrete Math., 3 (1972), 153-162.  doi: 10.1016/0012-365X(72)90030-1.  Google Scholar [30] E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000.  Google Scholar [31] A. M. Roberts, Weight Enumerator Parameter Database for Binary Self-Dual Codes, 2021, https://amr-wepd-bsdc.netlify.app Google Scholar [32] N. Yankov and D. Anev, On the self-dual codes with an automorphism of order 5, Appl. Algebra Engrg. Comm. Comput., 32 (2021), 97-111.  doi: 10.1007/s00200-019-00403-0.  Google Scholar [33] N. Yankov, M. Ivanova and M. H. Lee, Self-dual codes with an automorphism of order 7 and $s$-extremal codes of length 68, Finite Fields Appl., 51 (2018), 17-30.  doi: 10.1016/j.ffa.2017.12.001.  Google Scholar [34] R. Yorgova and A. Wassermann, Binary self-dual codes with automorphisms of order 23, Des. Codes Cryptogr., 48 (2008), 155-164.  doi: 10.1007/s10623-007-9152-8.  Google Scholar

show all references

##### References:
 [1] D. Anev, M. Harada and N. Yankov, New extremal singly even self-dual codes of lengths 64 and 66, J. Algebra Comb. Discrete Struct. Appl., 5 (2018), 143-151.  doi: 10.13069/jacodesmath.458601.  Google Scholar [2] E. R. Berlekamp, F. J. MacWilliams and N. J. A. Sloane, Gleason's theorem on self-dual codes, IEEE Trans. Inform. Theory, IT-18 (1972), 409-414.  doi: 10.1109/tit.1972.1054817.  Google Scholar [3] F. Bernhardt, P. Landrock and O. Manz, The extended golay codes considered as ideals, J. Combin. Theory Ser. A, 55 (1990), 235-246.  doi: 10.1016/0097-3165(90)90069-9.  Google Scholar [4] W. Bosma, J. 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.  Google Scholar [5] C. L. Chen, W. W Peterson and E. J. Weldon. Jr, Some results on quasi-cyclic codes, Information and Control, 15 (1969), 407-423.  doi: 10.1016/S0019-9958(69)90497-5.  Google Scholar [6] 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.  Google Scholar [7] P. J. Davis, Circulant Matrices, A Wiley-Interscience Publication; Pure and Applied Mathematics, John Wiley & Sons, New York-Chichester-Brisbane, 1979.  Google Scholar [8] S. T. Dougherty, P. Gaborit, M. Harada and P. Solé, Type ii codes over ${\mathbf f}_2+u{\mathbf f}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45.  doi: 10.1109/18.746770.  Google Scholar [9] 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.  Google Scholar [10] S. T. Dougherty, J. Gildea, R. Taylor and A. Tylyshchak, Group rings, $g$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7.  Google Scholar [11] S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.  doi: 10.1109/18.641574.  Google Scholar [12] S. T. Dougherty, B. Yildiz and S. Karadeniz, Codes over $r_k$, gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219.  doi: 10.1016/j.ffa.2010.11.002.  Google Scholar [13] S. Dougherty, B. Yildiz and S. Karadeniz, Self-dual codes over $r_k$ and binary self-dual codes, Eur. J. Pure Appl. Math., 6 (2013), 89-106.   Google Scholar [14] P. Gaborit, Quadratic double circulant codes over fields, J. Combin. Theory Ser. A, 97 (2002), 85-107.  doi: 10.1006/jcta.2001.3198.  Google Scholar [15] S. D. Georgiou and E. Lappas, Self-dual codes from circulant matrices, Des. Codes Cryptogr., 64 (2012), 129-141.  doi: 10.1007/s10623-011-9510-4.  Google Scholar [16] 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, Information Processing Letters, 157. doi: 10.1016/j.ipl.2020.105927.  Google Scholar [17] J. Gildea, A. Kaya, A. Korban 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.  Google Scholar [18] J. Gildea, A. Korban and A. M. Roberts, New binary self-dual codes of lengths 80, 84 and 96 from composite matrices, 2021, https://arXiv.org/abs/2106.12355. Google Scholar [19] A. M. Gleason, Weight polynomials of self-dual codes and the macwilliams identities, 1971,211–215.  Google Scholar [20] T. A. Gulliver and M. Harada, On extremal double circulant self-dual codes of lengths 90–96, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 403-415.  doi: 10.1007/s00200-019-00381-3.  Google Scholar [21] M. Gürel and N. Yankov, Self-dual codes with an automorphism of order 17, Math. Commun., 21 (2016), 97-107.   Google Scholar [22] T. Hurley, Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335.   Google Scholar [23] S. Karadeniz and B. Yildiz, New extremal binary self-dual codes of length 66 as extensions of self-dual codes over $R_k$, J. Franklin Inst., 350, (2013), 1963–1973 doi: 10.1016/j.jfranklin.2013.05.015.  Google Scholar [24] M. Karlin, New binary coding results by circulants, IEEE Trans. Inform. Theory, IT-15 (1969), 81-92.  doi: 10.1109/tit.1969.1054261.  Google Scholar [25] A. Kaya, B. Yildiz and I. Siap, Quadratic residue codes over $\mathbb{F}_p+v\mathbb{F}_p$ and their Gray images, J. Pure Appl. Algebra, 218 (2014), 1999-2011.  doi: 10.1016/j.jpaa.2014.03.002.  Google Scholar [26] J.-L. Kim, New extremal self-dual codes of lengths 36, 38, and 58, IEEE Trans. Inform. Theory, 47 (2001), 386-393.  doi: 10.1109/18.904540.  Google Scholar [27] S. Ling and P. Solé, Type ii codes over $\mathbf { F_4+u} f_4$, European J. Combin., 22 (2001), 983-997.  doi: 10.1006/eujc.2001.0509.  Google Scholar [28] F. J. MacWilliams, C. L. Mallows and N. J. A Sloane, Generalizations of gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, IT-18 (1972), 794-805.  doi: 10.1109/tit.1972.1054898.  Google Scholar [29] F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson, Good self dual codes exist, Discrete Math., 3 (1972), 153-162.  doi: 10.1016/0012-365X(72)90030-1.  Google Scholar [30] E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000.  Google Scholar [31] A. M. Roberts, Weight Enumerator Parameter Database for Binary Self-Dual Codes, 2021, https://amr-wepd-bsdc.netlify.app Google Scholar [32] N. Yankov and D. Anev, On the self-dual codes with an automorphism of order 5, Appl. Algebra Engrg. Comm. Comput., 32 (2021), 97-111.  doi: 10.1007/s00200-019-00403-0.  Google Scholar [33] N. Yankov, M. Ivanova and M. H. Lee, Self-dual codes with an automorphism of order 7 and $s$-extremal codes of length 68, Finite Fields Appl., 51 (2018), 17-30.  doi: 10.1016/j.ffa.2017.12.001.  Google Scholar [34] R. Yorgova and A. Wassermann, Binary self-dual codes with automorphisms of order 23, Des. Codes Cryptogr., 48 (2008), 155-164.  doi: 10.1007/s10623-007-9152-8.  Google Scholar
Self-dual code over $\mathbb{F}_4+u \mathbb{F}_4$ of length $64$ from $C_{4}$ and $C_{4}$
 $A_{i}$ $v_1 \in C_{4}$ $v_2 \in C_{4}$ $r_A$ $|Aut(A_i)|$ $\beta$ $1$ $(8966)$ $(0000)$ $(A617)$ $2^4$ $0$
 $A_{i}$ $v_1 \in C_{4}$ $v_2 \in C_{4}$ $r_A$ $|Aut(A_i)|$ $\beta$ $1$ $(8966)$ $(0000)$ $(A617)$ $2^4$ $0$
Self-dual code over $\mathbb{F}_2+u \mathbb{F}_2$ of length $64$ from $C_{8}$ and $C_{8}$
 $B_{i}$ $v_1 \in C_{8}$ $v_2 \in C_{8}$ $r_A$ $|Aut(B_i)|$ $\beta$ $1$ $(uuu10311)$ $(uu011uu0)$ $(u0300013)$ $2^3$ $0$
 $B_{i}$ $v_1 \in C_{8}$ $v_2 \in C_{8}$ $r_A$ $|Aut(B_i)|$ $\beta$ $1$ $(uuu10311)$ $(uu011uu0)$ $(u0300013)$ $2^3$ $0$
Self-dual code over $\mathbb{F}_2+u \mathbb{F}_2$ of length $64$ from $C_{2\cdot 4}$ and $C_{2\cdot 4}$
 $C_{i}$ $v_1 \in C_{2\cdot 4}$ $v_2 \in C_{2\cdot 4}$ $r_A$ $|Aut(C_i)|$ $\beta$ $1$ $(uu01u0u1)$ $(u0u11u31)$ $(u3u3u3u0)$ $2^{4}$ $48$
 $C_{i}$ $v_1 \in C_{2\cdot 4}$ $v_2 \in C_{2\cdot 4}$ $r_A$ $|Aut(C_i)|$ $\beta$ $1$ $(uu01u0u1)$ $(u0u11u31)$ $(u3u3u3u0)$ $2^{4}$ $48$
Self-dual code of length $68$ from extensions of $C_1$, $C_2$ and $C_3$
 $D_i$ Code $c$ $X$ $\gamma$ $\beta$ $|Aut(E_i)|$ $1$ $A_1$ $1$ $(0133010303011u1001333u01031uuu1u)$ $4$ $113$ $2$ $2$ $B_1$ $u+1$ $(013011030003013301111030uuu13u10)$ $\textbf{2}$ $\textbf{61}$ $2$ $3$ $C_1$ $u+1$ $(0u10303u110333001103u00130103303)$ $\textbf{1}$ $\textbf{179}$ $2$
 $D_i$ Code $c$ $X$ $\gamma$ $\beta$ $|Aut(E_i)|$ $1$ $A_1$ $1$ $(0133010303011u1001333u01031uuu1u)$ $4$ $113$ $2$ $2$ $B_1$ $u+1$ $(013011030003013301111030uuu13u10)$ $\textbf{2}$ $\textbf{61}$ $2$ $3$ $C_1$ $u+1$ $(0u10303u110333001103u00130103303)$ $\textbf{1}$ $\textbf{179}$ $2$
Self-dual codes over $\mathbb{F}_2$ of length $68$ $(W_{68, 2})$ from $C_{17}$ and $C_{17}$
 $E_{i}$ $v_1 \in C_{17}$ $v_2 \in C_{17}$ $r_A$ $|Aut(D_i)|$ $\gamma$ $\beta$ $1$ (00000000000011011) $(00000000000000000)$ $(00100110010110111)$ $2^2 \cdot 17$ $0$ $238$ $2$ (00000000110001111) $(00000000000000000)$ $(00100100101010101)$ $2^2 \cdot 17$ $0$ $272$
 $E_{i}$ $v_1 \in C_{17}$ $v_2 \in C_{17}$ $r_A$ $|Aut(D_i)|$ $\gamma$ $\beta$ $1$ (00000000000011011) $(00000000000000000)$ $(00100110010110111)$ $2^2 \cdot 17$ $0$ $238$ $2$ (00000000110001111) $(00000000000000000)$ $(00100100101010101)$ $2^2 \cdot 17$ $0$ $272$
New codes of length 68 from neighbours of $E_1$ and $E_2$
 $F_{i}$ $E_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(F_i) |$ $\gamma$ $\beta$ Type $1$ $2$ $(0111011100100011000001001000100110)$ $2$ $\textbf{0}$ $\textbf{208}$ $W_{68, 2}$ $2$ $2$ $(1110000011111000011000011110011000)$ $1$ $\textbf{0}$ $\textbf{214}$ $W_{68, 2}$ $3$ $2$ $(0001000100001110111100001010011010)$ $2$ $\textbf{1}$ $\textbf{191}$ $W_{68, 2}$ $4$ $2$ $(0010111111111110001111001010111001)$ $2$ $\textbf{1}$ $\textbf{202}$ $W_{68, 2}$ $5$ $1$ $(1001101111101110011000101000010110)$ $1$ $\textbf{1}$ $\textbf{210}$ $W_{68, 2}$ $6$ $2$ $(0101001000111001100011110011000101)$ $1$ $\textbf{1}$ $\textbf{211}$ $W_{68, 2}$ $7$ $2$ $(0010101101010100111100000001010001)$ $1$ $\textbf{1}$ $\textbf{229}$ $W_{68, 2}$ $8$ $2$ $(1111111111111111111011101111111111)$ $1$ ${}$ $\textbf{317}$ $W_{68, 1}$
 $F_{i}$ $E_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(F_i) |$ $\gamma$ $\beta$ Type $1$ $2$ $(0111011100100011000001001000100110)$ $2$ $\textbf{0}$ $\textbf{208}$ $W_{68, 2}$ $2$ $2$ $(1110000011111000011000011110011000)$ $1$ $\textbf{0}$ $\textbf{214}$ $W_{68, 2}$ $3$ $2$ $(0001000100001110111100001010011010)$ $2$ $\textbf{1}$ $\textbf{191}$ $W_{68, 2}$ $4$ $2$ $(0010111111111110001111001010111001)$ $2$ $\textbf{1}$ $\textbf{202}$ $W_{68, 2}$ $5$ $1$ $(1001101111101110011000101000010110)$ $1$ $\textbf{1}$ $\textbf{210}$ $W_{68, 2}$ $6$ $2$ $(0101001000111001100011110011000101)$ $1$ $\textbf{1}$ $\textbf{211}$ $W_{68, 2}$ $7$ $2$ $(0010101101010100111100000001010001)$ $1$ $\textbf{1}$ $\textbf{229}$ $W_{68, 2}$ $8$ $2$ $(1111111111111111111011101111111111)$ $1$ ${}$ $\textbf{317}$ $W_{68, 1}$
New codes of length 68 from neighbours of $F_7$ and $F_8$
 $G_{i}$ $F_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(G_i) |$ $\gamma$ $\beta$ Type $1$ $8$ $(0001001101110000000000101011001100)$ $1$ $\textbf{0}$ $\textbf{218}$ $W_{68, 2}$ $2$ $7$ $(0110000010001000111000111000100010)$ $1$ $\textbf{1}$ $\textbf{193}$ $W_{68, 2}$ $3$ $7$ $(1000100101011000011011110011000000)$ $1$ $\textbf{1}$ $\textbf{195}$ $W_{68, 2}$ $4$ $7$ $(0101001010010010000100100101001001)$ $1$ $1$ $233$ $W_{68, 2}$ $5$ $7$ $(0111010010001001001000000100101010)$ $1$ $\textbf{2}$ $\textbf{193}$ $W_{68, 2}$ $6$ $7$ $(1100010011000010110111011101101111)$ $1$ $\textbf{2}$ $\textbf{195}$ $W_{68, 2}$
 $G_{i}$ $F_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(G_i) |$ $\gamma$ $\beta$ Type $1$ $8$ $(0001001101110000000000101011001100)$ $1$ $\textbf{0}$ $\textbf{218}$ $W_{68, 2}$ $2$ $7$ $(0110000010001000111000111000100010)$ $1$ $\textbf{1}$ $\textbf{193}$ $W_{68, 2}$ $3$ $7$ $(1000100101011000011011110011000000)$ $1$ $\textbf{1}$ $\textbf{195}$ $W_{68, 2}$ $4$ $7$ $(0101001010010010000100100101001001)$ $1$ $1$ $233$ $W_{68, 2}$ $5$ $7$ $(0111010010001001001000000100101010)$ $1$ $\textbf{2}$ $\textbf{193}$ $W_{68, 2}$ $6$ $7$ $(1100010011000010110111011101101111)$ $1$ $\textbf{2}$ $\textbf{195}$ $W_{68, 2}$
New codes of length 68 from neighbours of $G_5$
 $H_{i}$ $G_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(H_i) |$ $\gamma$ $\beta$ Type $1$ $5$ $(0010010110011000000010111001111110)$ $1$ $\textbf{1}$ $\textbf{197}$ $W_{68, 2}$ $2$ $5$ $(0100001011001011101010110111011111)$ $1$ $\textbf{1}$ $\textbf{199}$ $W_{68, 2}$ $3$ $5$ $(1101001011101101011111110111100111)$ $1$ $\textbf{2}$ $\textbf{199}$ $W_{68, 2}$ $4$ $5$ $(0011000011001110011000001100000001)$ $1$ $\textbf{2}$ $\textbf{191}$ $W_{68, 2}$ $5$ $5$ $(0001100100110010010101000111100100)$ $1$ $\textbf{2}$ $\textbf{204}$ $W_{68, 2}$ $6$ $5$ $(1011101001000001101001010111011101)$ $1$ $\textbf{2}$ $\textbf{218}$ $W_{68, 2}$
 $H_{i}$ $G_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(H_i) |$ $\gamma$ $\beta$ Type $1$ $5$ $(0010010110011000000010111001111110)$ $1$ $\textbf{1}$ $\textbf{197}$ $W_{68, 2}$ $2$ $5$ $(0100001011001011101010110111011111)$ $1$ $\textbf{1}$ $\textbf{199}$ $W_{68, 2}$ $3$ $5$ $(1101001011101101011111110111100111)$ $1$ $\textbf{2}$ $\textbf{199}$ $W_{68, 2}$ $4$ $5$ $(0011000011001110011000001100000001)$ $1$ $\textbf{2}$ $\textbf{191}$ $W_{68, 2}$ $5$ $5$ $(0001100100110010010101000111100100)$ $1$ $\textbf{2}$ $\textbf{204}$ $W_{68, 2}$ $6$ $5$ $(1011101001000001101001010111011101)$ $1$ $\textbf{2}$ $\textbf{218}$ $W_{68, 2}$
Code of length 68 from the neighbours of $D_1$
 $I_{i}$ $D_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(I_i) |$ $\gamma$ $\beta$ Type $1$ $1$ $(1111000110110011110111001010111101)$ $1$ $5$ $133$ $W_{68, 2}$
 $I_{i}$ $D_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(I_i) |$ $\gamma$ $\beta$ Type $1$ $1$ $(1111000110110011110111001010111101)$ $1$ $5$ $133$ $W_{68, 2}$
Code of length 68 from the neighbours of $I_1$
 $J_{i}$ $I_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(J_i) |$ $\gamma$ $\beta$ Type $1$ $1$ $(0000100001011000111001010100001100$ $1$ $6$ $141$ $W_{68, 2}$
 $J_{i}$ $I_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(J_i) |$ $\gamma$ $\beta$ Type $1$ $1$ $(0000100001011000111001010100001100$ $1$ $6$ $141$ $W_{68, 2}$
New codes of length 68 from the neighbours of $J_1$
 $K_{i}$ $J_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(K_i) |$ $\gamma$ $\beta$ Type $1$ $1$ $(1111111101001100010100001000010100)$ $1$ $\textbf{6}$ $\textbf{131}$ $W_{68, 2}$ $2$ $1$ $(0000001110010111101110011111001111)$ $1$ $\textbf{7}$ $\textbf{158}$ $W_{68, 2}$
 $K_{i}$ $J_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(K_i) |$ $\gamma$ $\beta$ Type $1$ $1$ $(1111111101001100010100001000010100)$ $1$ $\textbf{6}$ $\textbf{131}$ $W_{68, 2}$ $2$ $1$ $(0000001110010111101110011111001111)$ $1$ $\textbf{7}$ $\textbf{158}$ $W_{68, 2}$
New codes of length 68 from the neighbours of $K_2$
 $L_{i}$ $K_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(L_i) |$ $\gamma$ $\beta$ Type $1$ $2$ $(0110111111010100011101010011010101)$ $1$ $\textbf{7}$ $\textbf{155}$ $W_{68, 2}$ $2$ $2$ $(0101010101010001001010011101110010)$ $1$ $\textbf{7}$ $\textbf{156}$ $W_{68, 2}$ $3$ $2$ $(0010011101010101010111011110110110)$ $1$ $\textbf{7}$ $\textbf{157}$ $W_{68, 2}$ $4$ $2$ $(1101111110110111001111110101101100)$ $1$ $\textbf{7}$ $\textbf{159}$ $W_{68, 2}$ $5$ $2$ $(1001011111000110001111101100101110)$ $1$ $\textbf{7}$ $\textbf{160}$ $W_{68, 2}$ $6$ $2$ $(1100000100100000010100101100011010)$ $1$ $\textbf{7}$ $\textbf{162}$ $W_{68, 2}$ $7$ $2$ $(1000010000010110000111110010011111)$ $1$ $\textbf{7}$ $\textbf{164}$ $W_{68, 2}$ $8$ $2$ $(0100001001101111111010000101010001)$ $1$ $\textbf{7}$ $\textbf{165}$ $W_{68, 2}$ $9$ $2$ $(0011101000100011011101001111101111)$ $1$ $\textbf{7}$ $\textbf{167}$ $W_{68, 2}$
 $L_{i}$ $K_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $|Aut(L_i) |$ $\gamma$ $\beta$ Type $1$ $2$ $(0110111111010100011101010011010101)$ $1$ $\textbf{7}$ $\textbf{155}$ $W_{68, 2}$ $2$ $2$ $(0101010101010001001010011101110010)$ $1$ $\textbf{7}$ $\textbf{156}$ $W_{68, 2}$ $3$ $2$ $(0010011101010101010111011110110110)$ $1$ $\textbf{7}$ $\textbf{157}$ $W_{68, 2}$ $4$ $2$ $(1101111110110111001111110101101100)$ $1$ $\textbf{7}$ $\textbf{159}$ $W_{68, 2}$ $5$ $2$ $(1001011111000110001111101100101110)$ $1$ $\textbf{7}$ $\textbf{160}$ $W_{68, 2}$ $6$ $2$ $(1100000100100000010100101100011010)$ $1$ $\textbf{7}$ $\textbf{162}$ $W_{68, 2}$ $7$ $2$ $(1000010000010110000111110010011111)$ $1$ $\textbf{7}$ $\textbf{164}$ $W_{68, 2}$ $8$ $2$ $(0100001001101111111010000101010001)$ $1$ $\textbf{7}$ $\textbf{165}$ $W_{68, 2}$ $9$ $2$ $(0011101000100011011101001111101111)$ $1$ $\textbf{7}$ $\textbf{167}$ $W_{68, 2}$
New singly-even binary self-dual $[96, 48, 16]$ codes from $C_6$ and $C_6$ over $\mathbb{F}_4+u \mathbb{F}_4$
 $C_{96, i}$ $v_1 \in C_{6}$ $v_2 \in C_{6}$ $r_A$ $|Aut(C_{96, i})|$ $\alpha$ $\beta$ $\gamma$ Type $1$ $(17DD00)$ $(DC34EB)$ $(7C111C)$ $2^{4}$ $\textbf{11104}$ $-\textbf{68}$ $\textbf{0}$ $W_{96, 2}$ $2$ $(C00E11)$ $(C8BDA9)$ $(F656F5)$ $2^{4}$ $\textbf{10208}$ $-\textbf{52}$ $\textbf{0}$ $W_{96, 2}$ $3$ $(6482FF)$ $(0D0D0D)$ $(7C111C)$ $2^{4}\cdot 3$ $\textbf{11328}$ $-\textbf{28}$ $\textbf{0}$ $W_{96, 2}$ $4$ $(1236FC)$ $(914FD8)$ $(D4DE6E)$ $2^{4}$ $\textbf{11312}$ $-\textbf{108}$ $\textbf{2}$ $W_{96, 2}$ $5$ $(3E222F)$ $(8EBA97)$ $(D4DE6E)$ $2^{4}$ $\textbf{11728}$ $-\textbf{100}$ $\textbf{2}$ $W_{96, 2}$ $6$ $(C6EB5F)$ $(EA56C1)$ $(7C111C)$ $2^{4}$ $\textbf{11184}$ $-\textbf{84}$ $\textbf{2}$ $W_{96, 2}$ $7$ $(B88D66)$ $(99680F)$ $(7C111C)$ $2^{4}$ $\textbf{10592}$ $-\textbf{80}$ $\textbf{2}$ $W_{96, 2}$ $8$ $(1D271F)$ $(A7870E)$ $(6B6DBD)$ $2^{4}$ $\textbf{11184}$ $-\textbf{76}$ $\textbf{2}$ $W_{96, 2}$ $9$ $(0A7B3D)$ $(126325)$ $(6B6DBD)$ $2^{4}$ $\textbf{11488}$ $-\textbf{72}$ $\textbf{2}$ $W_{96, 2}$ $10$ $(535DD1)$ $(F1CECB)$ $(6B6DBD)$ $2^{4}$ $\textbf{10624}$ $-\textbf{64}$ $\textbf{2}$ $W_{96, 2}$ $11$ $(C2F3D9)$ $(1EDF0A)$ $(6B6DBD)$ $2^{4}$ $\textbf{10944}$ $-\textbf{60}$ $\textbf{2}$ $W_{96, 2}$ $12$ $(D4787D)$ $(9FCD5D)$ $(6B6DBD)$ $2^{4}$ $\textbf{11224}$ $-\textbf{56}$ $\textbf{2}$ $W_{96, 2}$ $13$ $(344A57)$ $(47F231)$ $(7C111C)$ $2^{4}$ $\textbf{10728}$ $-\textbf{48}$ $\textbf{2}$ $W_{96, 2}$ $14$ $(D399AB)$ $(6DB3F0)$ $(D4DE6E)$ $2^{4}$ $\textbf{12320}$ $-\textbf{156}$ $\textbf{4}$ $W_{96, 2}$ $15$ $(F7A016)$ $(AE0EBF)$ $(D4DE6E)$ $2^{4}$ $\textbf{11104}$ $-\textbf{140}$ $\textbf{4}$ $W_{96, 2}$ $16$ $(EF2862)$ $(8867A5)$ $(F656F5)$ $2^{4}$ $\textbf{11528}$ $-\textbf{136}$ $\textbf{4}$ $W_{96, 2}$ $17$ $(A56B03)$ $(317717)$ $(7C111C)$ $2^{4}$ $\textbf{11472}$ $-\textbf{132}$ $\textbf{4}$ $W_{96, 2}$ $18$ $(4250B6)$ $(979C73)$ $(D4DE6E)$ $2^{4}$ $\textbf{11728}$ $-\textbf{120}$ $\textbf{4}$ $W_{96, 2}$ $19$ $(01A176)$ $(CA0455)$ $(F656F5)$ $2^{4}$ $\textbf{11360}$ $-\textbf{116}$ $\textbf{4}$ $W_{96, 2}$ $20$ $(FE26F3)$ $(23B01B)$ $(F656F5)$ $2^{4}$ $\textbf{11160}$ $-\textbf{112}$ $\textbf{4}$ $W_{96, 2}$ $21$ $(6C02AE)$ $(6F098F)$ $(6B6DBD)$ $2^{4}$ $\textbf{11328}$ $-\textbf{112}$ $\textbf{4}$ $W_{96, 2}$ $22$ $(F79924)$ $(AA77C9)$ $(D4DE6E)$ $2^{4}$ $\textbf{11568}$ $-\textbf{112}$ $\textbf{4}$ $W_{96, 2}$ $23$ $(5FFB7B)$ $(4A6DD5)$ $(7C111C)$ $2^{4}$ $\textbf{11088}$ $-\textbf{108}$ $\textbf{4}$ $W_{96, 2}$ $24$ $(3522FB)$ $(C05E9F)$ $(6B6DBD)$ $2^{4}$ $\textbf{11488}$ $-\textbf{108}$ $\textbf{4}$ $W_{96, 2}$ $25$ $(9E88C6)$ $(07DE86)$ $(7C111C)$ $2^{4}$ $\textbf{11072}$ $-\textbf{104}$ $\textbf{4}$ $W_{96, 2}$ $26$ $(088C5F)$ $(77601A)$ $(F656F5)$ $2^{4}$ $\textbf{10672}$ $-\textbf{100}$ $\textbf{4}$ $W_{96, 2}$ $27$ $(313674)$ $(343BD9)$ $(7C111C)$ $2^{4}$ $\textbf{10944}$ $-\textbf{100}$ $\textbf{4}$ $W_{96, 2}$ $28$ $(35EA9C)$ $(930785)$ $(7C111C)$ $2^{4}$ $\textbf{11048}$ $-\textbf{96}$ $\textbf{4}$ $W_{96, 2}$ $29$ $(505084)$ $(57696E)$ $(F656F5)$ $2^{4}$ $\textbf{11064}$ $-\textbf{88}$ $\textbf{4}$ $W_{96, 2}$ $30$ $(6D4401)$ $(92206E)$ $(6B6DBD)$ $2^{4}$ $\textbf{11504}$ $-\textbf{84}$ $\textbf{4}$ $W_{96, 2}$ $31$ $(58263B)$ $(D98510)$ $(6B6DBD)$ $2^{4}$ $\textbf{10888}$ $-\textbf{80}$ $\textbf{4}$ $W_{96, 2}$ $32$ $(9AE7CA)$ $(74D032)$ $(F656F5)$ $2^{4}$ $\textbf{12504}$ $-\textbf{160}$ $\textbf{6}$ $W_{96, 2}$ $33$ $(73A8CF)$ $(D46308)$ $(F656F5)$ $2^{4}$ $\textbf{11552}$ $-\textbf{156}$ $\textbf{6}$ $W_{96, 2}$ $34$ $(F97D3B)$ $(6B7D82)$ $(6B6DBD)$ $2^{4}$ $\textbf{11872}$ $-\textbf{156}$ $\textbf{6}$ $W_{96, 2}$ $35$ $(B4196E)$ $(97B0E5)$ $(D4DE6E)$ $2^{4}$ $\textbf{11376}$ $-\textbf{148}$ $\textbf{6}$ $W_{96, 2}$ $36$ $(47E5CD)$ $(CECECE)$ $(6B6DBD)$ $2^{4}\cdot 3$ $\textbf{11736}$ $-\textbf{148}$ $\textbf{6}$ $W_{96, 2}$ $37$ $(6B78E6)$ $(113CD9)$ $(F656F5)$ $2^{4}$ $\textbf{11576}$ $-\textbf{140}$ $\textbf{6}$ $W_{96, 2}$ $38$ $(B1C856)$ $(F7452D)$ $(D4DE6E)$ $2^{4}$ $\textbf{12448}$ $-\textbf{140}$ $\textbf{6}$ $W_{96, 2}$ $39$ $(FC0863)$ $(18BD3B)$ $(D4DE6E)$ $2^{4}$ $\textbf{11008}$ $-\textbf{132}$ $\textbf{6}$ $W_{96, 2}$ $40$ $(DC4A91)$ $(A58C34)$ $(6B6DBD)$ $2^{4}$ $\textbf{11304}$ $-\textbf{132}$ $\textbf{6}$ $W_{96, 2}$ $41$ $(8798CD)$ $(FD6017)$ $(7C111C)$ $2^{4}$ $\textbf{11312}$ $-\textbf{120}$ $\textbf{6}$ $W_{96, 2}$ $42$ $(9217CF)$ $(DCD676)$ $(7C111C)$ $2^{4}$ $\textbf{12928}$ $-\textbf{192}$ $\textbf{8}$ $W_{96, 2}$ $43$ $(C620D5)$ $(EAE546)$ $(7C111C)$ $2^{4}$ $\textbf{11768}$ $-\textbf{172}$ $\textbf{8}$ $W_{96, 2}$ $44$ $(3617E2)$ $(19B065)$ $(7C111C)$ $2^{4}$ $\textbf{11272}$ $-\textbf{168}$ $\textbf{8}$ $W_{96, 2}$ $45$ $(3BAE33)$ $(5F852E)$ $(7C111C)$ $2^{4}$ $\textbf{11968}$ $-\textbf{168}$ $\textbf{8}$ $W_{96, 2}$ $46$ $(E90589)$ $(D62FE2)$ $(D4DE6E)$ $2^{4}$ $\textbf{12896}$ $-\textbf{260}$ $\textbf{12}$ $W_{96, 2}$ $47$ $(B89454)$ $(F5F331)$ $(D4DE6E)$ $2^{4}$ $\textbf{12288}$ $-\textbf{244}$ $\textbf{12}$ $W_{96, 2}$ $48$ $(E9DA51)$ $(6D030D)$ $(6B6DBD)$ $2^{4}$ $\textbf{12320}$ $-\textbf{244}$ $\textbf{12}$ $W_{96, 2}$
 $C_{96, i}$ $v_1 \in C_{6}$ $v_2 \in C_{6}$ $r_A$ $|Aut(C_{96, i})|$ $\alpha$ $\beta$ $\gamma$ Type $1$ $(17DD00)$ $(DC34EB)$ $(7C111C)$ $2^{4}$ $\textbf{11104}$ $-\textbf{68}$ $\textbf{0}$ $W_{96, 2}$ $2$ $(C00E11)$ $(C8BDA9)$ $(F656F5)$ $2^{4}$ $\textbf{10208}$ $-\textbf{52}$ $\textbf{0}$ $W_{96, 2}$ $3$ $(6482FF)$ $(0D0D0D)$ $(7C111C)$ $2^{4}\cdot 3$ $\textbf{11328}$ $-\textbf{28}$ $\textbf{0}$ $W_{96, 2}$ $4$ $(1236FC)$ $(914FD8)$ $(D4DE6E)$ $2^{4}$ $\textbf{11312}$ $-\textbf{108}$ $\textbf{2}$ $W_{96, 2}$ $5$ $(3E222F)$ $(8EBA97)$ $(D4DE6E)$ $2^{4}$ $\textbf{11728}$ $-\textbf{100}$ $\textbf{2}$ $W_{96, 2}$ $6$ $(C6EB5F)$ $(EA56C1)$ $(7C111C)$ $2^{4}$ $\textbf{11184}$ $-\textbf{84}$ $\textbf{2}$ $W_{96, 2}$ $7$ $(B88D66)$ $(99680F)$ $(7C111C)$ $2^{4}$ $\textbf{10592}$ $-\textbf{80}$ $\textbf{2}$ $W_{96, 2}$ $8$ $(1D271F)$ $(A7870E)$ $(6B6DBD)$ $2^{4}$ $\textbf{11184}$ $-\textbf{76}$ $\textbf{2}$ $W_{96, 2}$ $9$ $(0A7B3D)$ $(126325)$ $(6B6DBD)$ $2^{4}$ $\textbf{11488}$ $-\textbf{72}$ $\textbf{2}$ $W_{96, 2}$ $10$ $(535DD1)$ $(F1CECB)$ $(6B6DBD)$ $2^{4}$ $\textbf{10624}$ $-\textbf{64}$ $\textbf{2}$ $W_{96, 2}$ $11$ $(C2F3D9)$ $(1EDF0A)$ $(6B6DBD)$ $2^{4}$ $\textbf{10944}$ $-\textbf{60}$ $\textbf{2}$ $W_{96, 2}$ $12$ $(D4787D)$ $(9FCD5D)$ $(6B6DBD)$ $2^{4}$ $\textbf{11224}$ $-\textbf{56}$ $\textbf{2}$ $W_{96, 2}$ $13$ $(344A57)$ $(47F231)$ $(7C111C)$ $2^{4}$ $\textbf{10728}$ $-\textbf{48}$ $\textbf{2}$ $W_{96, 2}$ $14$ $(D399AB)$ $(6DB3F0)$ $(D4DE6E)$ $2^{4}$ $\textbf{12320}$ $-\textbf{156}$ $\textbf{4}$ $W_{96, 2}$ $15$ $(F7A016)$ $(AE0EBF)$ $(D4DE6E)$ $2^{4}$ $\textbf{11104}$ $-\textbf{140}$ $\textbf{4}$ $W_{96, 2}$ $16$ $(EF2862)$ $(8867A5)$ $(F656F5)$ $2^{4}$ $\textbf{11528}$ $-\textbf{136}$ $\textbf{4}$ $W_{96, 2}$ $17$ $(A56B03)$ $(317717)$ $(7C111C)$ $2^{4}$ $\textbf{11472}$ $-\textbf{132}$ $\textbf{4}$ $W_{96, 2}$ $18$ $(4250B6)$ $(979C73)$ $(D4DE6E)$ $2^{4}$ $\textbf{11728}$ $-\textbf{120}$ $\textbf{4}$ $W_{96, 2}$ $19$ $(01A176)$ $(CA0455)$ $(F656F5)$ $2^{4}$ $\textbf{11360}$ $-\textbf{116}$ $\textbf{4}$ $W_{96, 2}$ $20$ $(FE26F3)$ $(23B01B)$ $(F656F5)$ $2^{4}$ $\textbf{11160}$ $-\textbf{112}$ $\textbf{4}$ $W_{96, 2}$ $21$ $(6C02AE)$ $(6F098F)$ $(6B6DBD)$ $2^{4}$ $\textbf{11328}$ $-\textbf{112}$ $\textbf{4}$ $W_{96, 2}$ $22$ $(F79924)$ $(AA77C9)$ $(D4DE6E)$ $2^{4}$ $\textbf{11568}$ $-\textbf{112}$ $\textbf{4}$ $W_{96, 2}$ $23$ $(5FFB7B)$ $(4A6DD5)$ $(7C111C)$ $2^{4}$ $\textbf{11088}$ $-\textbf{108}$ $\textbf{4}$ $W_{96, 2}$ $24$ $(3522FB)$ $(C05E9F)$ $(6B6DBD)$ $2^{4}$ $\textbf{11488}$ $-\textbf{108}$ $\textbf{4}$ $W_{96, 2}$ $25$ $(9E88C6)$ $(07DE86)$ $(7C111C)$ $2^{4}$ $\textbf{11072}$ $-\textbf{104}$ $\textbf{4}$ $W_{96, 2}$ $26$ $(088C5F)$ $(77601A)$ $(F656F5)$ $2^{4}$ $\textbf{10672}$ $-\textbf{100}$ $\textbf{4}$ $W_{96, 2}$ $27$ $(313674)$ $(343BD9)$ $(7C111C)$ $2^{4}$ $\textbf{10944}$ $-\textbf{100}$ $\textbf{4}$ $W_{96, 2}$ $28$ $(35EA9C)$ $(930785)$ $(7C111C)$ $2^{4}$ $\textbf{11048}$ $-\textbf{96}$ $\textbf{4}$ $W_{96, 2}$ $29$ $(505084)$ $(57696E)$ $(F656F5)$ $2^{4}$ $\textbf{11064}$ $-\textbf{88}$ $\textbf{4}$ $W_{96, 2}$ $30$ $(6D4401)$ $(92206E)$ $(6B6DBD)$ $2^{4}$ $\textbf{11504}$ $-\textbf{84}$ $\textbf{4}$ $W_{96, 2}$ $31$ $(58263B)$ $(D98510)$ $(6B6DBD)$ $2^{4}$ $\textbf{10888}$ $-\textbf{80}$ $\textbf{4}$ $W_{96, 2}$ $32$ $(9AE7CA)$ $(74D032)$ $(F656F5)$ $2^{4}$ $\textbf{12504}$ $-\textbf{160}$ $\textbf{6}$ $W_{96, 2}$ $33$ $(73A8CF)$ $(D46308)$ $(F656F5)$ $2^{4}$ $\textbf{11552}$ $-\textbf{156}$ $\textbf{6}$ $W_{96, 2}$ $34$ $(F97D3B)$ $(6B7D82)$ $(6B6DBD)$ $2^{4}$ $\textbf{11872}$ $-\textbf{156}$ $\textbf{6}$ $W_{96, 2}$ $35$ $(B4196E)$ $(97B0E5)$ $(D4DE6E)$ $2^{4}$ $\textbf{11376}$ $-\textbf{148}$ $\textbf{6}$ $W_{96, 2}$ $36$ $(47E5CD)$ $(CECECE)$ $(6B6DBD)$ $2^{4}\cdot 3$ $\textbf{11736}$ $-\textbf{148}$ $\textbf{6}$ $W_{96, 2}$ $37$ $(6B78E6)$ $(113CD9)$ $(F656F5)$ $2^{4}$ $\textbf{11576}$ $-\textbf{140}$ $\textbf{6}$ $W_{96, 2}$ $38$ $(B1C856)$ $(F7452D)$ $(D4DE6E)$ $2^{4}$ $\textbf{12448}$ $-\textbf{140}$ $\textbf{6}$ $W_{96, 2}$ $39$ $(FC0863)$ $(18BD3B)$ $(D4DE6E)$ $2^{4}$ $\textbf{11008}$ $-\textbf{132}$ $\textbf{6}$ $W_{96, 2}$ $40$ $(DC4A91)$ $(A58C34)$ $(6B6DBD)$ $2^{4}$ $\textbf{11304}$ $-\textbf{132}$ $\textbf{6}$ $W_{96, 2}$ $41$ $(8798CD)$ $(FD6017)$ $(7C111C)$ $2^{4}$ $\textbf{11312}$ $-\textbf{120}$ $\textbf{6}$ $W_{96, 2}$ $42$ $(9217CF)$ $(DCD676)$ $(7C111C)$ $2^{4}$ $\textbf{12928}$ $-\textbf{192}$ $\textbf{8}$ $W_{96, 2}$ $43$ $(C620D5)$ $(EAE546)$ $(7C111C)$ $2^{4}$ $\textbf{11768}$ $-\textbf{172}$ $\textbf{8}$ $W_{96, 2}$ $44$ $(3617E2)$ $(19B065)$ $(7C111C)$ $2^{4}$ $\textbf{11272}$ $-\textbf{168}$ $\textbf{8}$ $W_{96, 2}$ $45$ $(3BAE33)$ $(5F852E)$ $(7C111C)$ $2^{4}$ $\textbf{11968}$ $-\textbf{168}$ $\textbf{8}$ $W_{96, 2}$ $46$ $(E90589)$ $(D62FE2)$ $(D4DE6E)$ $2^{4}$ $\textbf{12896}$ $-\textbf{260}$ $\textbf{12}$ $W_{96, 2}$ $47$ $(B89454)$ $(F5F331)$ $(D4DE6E)$ $2^{4}$ $\textbf{12288}$ $-\textbf{244}$ $\textbf{12}$ $W_{96, 2}$ $48$ $(E9DA51)$ $(6D030D)$ $(6B6DBD)$ $2^{4}$ $\textbf{12320}$ $-\textbf{244}$ $\textbf{12}$ $W_{96, 2}$
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