American Institute of Mathematical Sciences

doi: 10.3934/amc.2020111

New self-dual codes of length 68 from a $2 \times 2$ block matrix construction and group rings

 1 Department of Algebra, Uzhgorod National University, Uzhgorod, Ukraine 2 Department of Mathematical and Physical Sciences, University of Chester, UK 3 Harmony School of Technology, Houston, TX 77038, USA

Received  March 2020 Published  September 2020

Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form $G = (I_n \ | \ A),$ where $I_n$ is the $n \times n$ identity matrix and $A$ is the $n \times n$ matrix fully determined by the first row. In this work, we define a generator matrix in which $A$ is a block matrix, where the blocks come from group rings and also, $A$ is not fully determined by the elements appearing in the first row. By applying our construction over $\mathbb{F}_2+u\mathbb{F}_2$ and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual $[68, 34, 12]$-codes with the rare parameters $\gamma = 7, 8$ and $9$ in $W_{68, 2}.$ In particular, we find 92 new binary self-dual $[68, 34, 12]$-codes.

Citation: Maria Bortos, Joe Gildea, Abidin Kaya, Adrian Korban, Alexander Tylyshchak. New self-dual codes of length 68 from a $2 \times 2$ block matrix construction and group rings. Advances in Mathematics of Communications, doi: 10.3934/amc.2020111
References:
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References:
 [1] 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 [2] 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.  Google Scholar [3] 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 [4] S. T. Dougherty, P. Gaborit, M. Harada and P. Sole, Type II codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45.  doi: 10.1109/18.746770.  Google Scholar [5] S. T. Dougherty, J. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings over Frobenius rings, Cryptogr. Commun., (2019). doi: 10.1007/s12095-019-00380-8.  Google Scholar [6] S. T. Dougherty, J. Gildea, A. 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. Comm., (2019). doi: 10.1016/j.ffa.2020.101692.  Google Scholar [7] S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, A. Tylshchak and B. Yildiz, Bordered constructions of self-dual codes from group rings, Finite Fields Appl., 57 (2019), 108-127.  doi: 10.1016/j.ffa.2019.02.004.  Google Scholar [8] S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak, Group rings, g-codes and constructions of self-dual and formally self-dual codes, Des., Codes and Cryptog., Designs, 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7.  Google Scholar [9] S. T. Dougherty, S. Karadeniz and B. Yildiz, 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 [10] S. T. Dougherty, J. L. Kim, H. Kulosman and H. Liu, Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 14-26.  doi: 10.1016/j.ffa.2009.11.004.  Google Scholar [11] J. Gildea, A. Kaya, A. Korban and B. Yildiz, Constructing self-dual codes from group rings and reverse circulant matrices, Adv. Math. Comm.. doi: 10.3934/amc.2020077.  Google Scholar [12] J. Gildea, A. Kaya, A. Korban and B. Yildiz, New extremal binary self-dual codes of length 68 from generalized neighbours, Finite Fields Appl., (2020). doi: 10.1016/j.ffa.2020.101727.  Google Scholar [13] J. Gildea, A. Kaya, R. 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.  Google Scholar [14] M. Harada and A. Munemasa, Some restrictions on weight enumerators of singly even self-dual codes, IEEE Trans. Inform. Theory, 52 (2006), 1266-1269.  doi: 10.1109/TIT.2005.864416.  Google Scholar [15] T. Hurley, "Group Rings and Rings of Matrices", J. Pure Appl. Math., 31 (2006), 319-335.   Google Scholar [16] S. Karadeniz, B. Yildiz and N. Aydin, Extremal binary self-dual codes of lengths 64 and 66 from four-circulant constructions over $\mathbb{F}_2+u\mathbb{F}_2$, Filomat, 28 (2014), 937-945.  doi: 10.2298/FIL1405937K.  Google Scholar [17] A. Kaya, New extremal binary self-dual codes of lengths 64 and 66 from $R_{2}$-lifts, Finite Fields Appl., 46 (2017), 271-279.  doi: 10.1016/j.ffa.2017.04.003.  Google Scholar [18] A. Kaya and B. Yildiz, Various constructions for self-dual codes over rings and new binary self-dual codes, Discrete Math., 339 (2016), 460-469.  doi: 10.1016/j.disc.2015.09.010.  Google Scholar [19] E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000.  Google Scholar [20] N. Yankov, M. H. Lee, M. Gurel and M. Ivanova, Self-dual codes with an automorphism of order 11, IEEE Trans. Inform. Theory, 61 (2015), 1188-1193.  doi: 10.1109/TIT.2015.2396915.  Google Scholar [21] 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 [22] N. Yankov and D. Anev, On the self-dual codes with an automorphism of order 5, AAECC, (2019). doi: 10.1007/s00200-019-00403-0.  Google Scholar
Codes of length 32 via Theorem 3.1 with the cyclic group $C_8$
 Code $v_1$ $v_2$ $v_3$ $|Aut(C)|$ Type $C_1$ $(0, 0, 0, 0, 0, 1, 1, 1)$ $(0, 0, 0, 0, 0, 1, 0, 1)$ $(0, 1, 0, 1, 0, 0, 1, 0)$ $2^93^25$ $[32, 16, 6]_I$ $C_2$ $(0, 0, 0, 0, 1, 1, 1, 1)$ $(0, 0, 1, 1, 0, 1, 1, 1)$ $(0, 0, 0, 1, 1, 1, 1, 0)$ $2^5$ $[32, 16, 6]_I$
 Code $v_1$ $v_2$ $v_3$ $|Aut(C)|$ Type $C_1$ $(0, 0, 0, 0, 0, 1, 1, 1)$ $(0, 0, 0, 0, 0, 1, 0, 1)$ $(0, 1, 0, 1, 0, 0, 1, 0)$ $2^93^25$ $[32, 16, 6]_I$ $C_2$ $(0, 0, 0, 0, 1, 1, 1, 1)$ $(0, 0, 1, 1, 0, 1, 1, 1)$ $(0, 0, 0, 1, 1, 1, 1, 0)$ $2^5$ $[32, 16, 6]_I$
Codes of length 64 from $R_1$ lifts of $C_1$ and $C_2$
 Code $v_1$ $v_2$ $v_3$ $|Aut(C)|$ $W_{64, 2}$ $I_1$ $C_2$ $(u, 0, 0, u, 1, 1, u+1, 1)$ $(0, 0, 1, u+1, 0, u+1, 1, 1)$ $(0, u, u, u+1, u+1, u+1, 1, 0)$ $2^5$ $\beta=0$ $I_2$ $C_1$ $(0, u, 0, 0, 0, 1, 1, u+1)$ $(0, u, 0, u, u, 1, 0, 1)$ $(0, 1, 0, u+1, 0, 0, 1, u)$ $2^7$ $\beta=80$
 Code $v_1$ $v_2$ $v_3$ $|Aut(C)|$ $W_{64, 2}$ $I_1$ $C_2$ $(u, 0, 0, u, 1, 1, u+1, 1)$ $(0, 0, 1, u+1, 0, u+1, 1, 1)$ $(0, u, u, u+1, u+1, u+1, 1, 0)$ $2^5$ $\beta=0$ $I_2$ $C_1$ $(0, u, 0, 0, 0, 1, 1, u+1)$ $(0, u, 0, u, u, 1, 0, 1)$ $(0, 1, 0, u+1, 0, 0, 1, u)$ $2^7$ $\beta=80$
Codes of length 32 via Theorem 3.1 with the dihedral group $D_8$
 Code $v_1$ $v_2$ $v_3$ $|Aut(C)|$ Type $C_3$ $(0, 0, 0, 1, 0, 0, 1, 1)$ $(0, 0, 1, 1, 0, 1, 0, 1)$ $(1, 1, 0, 1, 0, 1, 1, 0)$ $2^33$ $[32, 16, 6]_I$
 Code $v_1$ $v_2$ $v_3$ $|Aut(C)|$ Type $C_3$ $(0, 0, 0, 1, 0, 0, 1, 1)$ $(0, 0, 1, 1, 0, 1, 0, 1)$ $(1, 1, 0, 1, 0, 1, 1, 0)$ $2^33$ $[32, 16, 6]_I$
Codes of length 64 from $R_1$ lifts of $C_3$
 Code $v_1$ $v_2$ $v_3$ $|Aut(C)|$ $W_{64, 2}$ $I_3$ $C_3$ $(0, u, u, 1, 0, 0, 1, 1)$ $(0, 0, 1, u+1, u, 1, 0, 1)$ $(u+1, u+1, 0, u+1, 0, 1, u+1, 0)$ $2^43$ $\beta=64$
 Code $v_1$ $v_2$ $v_3$ $|Aut(C)|$ $W_{64, 2}$ $I_3$ $C_3$ $(0, u, u, 1, 0, 0, 1, 1)$ $(0, 0, 1, u+1, u, 1, 0, 1)$ $(u+1, u+1, 0, u+1, 0, 1, u+1, 0)$ $2^43$ $\beta=64$
New codes of length 68 from Theorem 2.4
 $C_{68, i}$ Code $(x_{17}, x_{18}, \dots , x_{32})$ $c$ $\gamma$ $\beta \ in\ W_{64, 2}$ $C_{68, 1}$ $I_{3}$ $(u, u, 0, 1, 3, u, u, u, u, 3, u, 1, 0, 0, u, 3, u, 1, 0, 1, 1, 3, 3, 3, u, 0, 3, u, u, 1, 0, 0)$ $1$ $\boldsymbol{0}$ $\boldsymbol{181}$ $C_{68, 2}$ $I_{3}$ $(0, 3, 0, 1, 3, 1, 3, 1, 1, 1, 3, 1, u, 0, u, 0, u, 0, u, 3, 0, 0, 1, 1, 1, 1, u, u, 1, 3, u, u)$ $3$ $\boldsymbol{1}$ $\boldsymbol{185}$ $C_{68, 3}$ $I_{1}$ $(0, 1, 1, u, u, 3, u, 1, 3, 3, 1, 0, 0, 3, 3, u, 1, 3, 3, u, u, 3, 0, u, 3, u, 3, u, 1, 3, 0, 0)$ $3$ $\boldsymbol{2}$ $\boldsymbol{54}$ $C_{68, 4}$ $I_{2}$ $(u, 3, 1, 3, 0, 0, 3, u, 0, 3, 0, u, u, u, 0, u, 3, 1, 0, 3, 0, 3, u, 1, 1, 1, 1, u, 0, 3, 0, 1)$ $1$ $\boldsymbol{2}$ $\boldsymbol{202}$ $C_{68, 5}$ $I_{3}$ $(u, u, u, 3, 0, 0, 1, u, 1, u, 1, 3, u, 0, 0, 3, 0, 1, u, 3, 0, 1, 0, 3, 1, 1, 0, 3, u, 3, 0, 1)$ $1$ $\boldsymbol{3}$ $\boldsymbol{179}$ $C_{68, 6}$ $I_{3}$ $(u, 0, 0, 1, 1, 0, 1, 0, 3, 3, u, 0, 1, 0, 3, 3, 1, 0, 3, 0, 3, 3, 1, u, 1, u, 1, u, 3, 0, 1, u)$ $3$ $\boldsymbol{3}$ $\boldsymbol{189}$ $C_{68, 7}$ $I_{3}$ $(0, 3, 0, 1, u, 3, u, 3, 0, 1, 0, 3, 0, 3, 0, 0, 1, u, u, 1, 0, u, 1, 0, u, 0, u, 1, 3, 0, 1, u)$ $3$ $\boldsymbol{3}$ $\boldsymbol{198}$
 $C_{68, i}$ Code $(x_{17}, x_{18}, \dots , x_{32})$ $c$ $\gamma$ $\beta \ in\ W_{64, 2}$ $C_{68, 1}$ $I_{3}$ $(u, u, 0, 1, 3, u, u, u, u, 3, u, 1, 0, 0, u, 3, u, 1, 0, 1, 1, 3, 3, 3, u, 0, 3, u, u, 1, 0, 0)$ $1$ $\boldsymbol{0}$ $\boldsymbol{181}$ $C_{68, 2}$ $I_{3}$ $(0, 3, 0, 1, 3, 1, 3, 1, 1, 1, 3, 1, u, 0, u, 0, u, 0, u, 3, 0, 0, 1, 1, 1, 1, u, u, 1, 3, u, u)$ $3$ $\boldsymbol{1}$ $\boldsymbol{185}$ $C_{68, 3}$ $I_{1}$ $(0, 1, 1, u, u, 3, u, 1, 3, 3, 1, 0, 0, 3, 3, u, 1, 3, 3, u, u, 3, 0, u, 3, u, 3, u, 1, 3, 0, 0)$ $3$ $\boldsymbol{2}$ $\boldsymbol{54}$ $C_{68, 4}$ $I_{2}$ $(u, 3, 1, 3, 0, 0, 3, u, 0, 3, 0, u, u, u, 0, u, 3, 1, 0, 3, 0, 3, u, 1, 1, 1, 1, u, 0, 3, 0, 1)$ $1$ $\boldsymbol{2}$ $\boldsymbol{202}$ $C_{68, 5}$ $I_{3}$ $(u, u, u, 3, 0, 0, 1, u, 1, u, 1, 3, u, 0, 0, 3, 0, 1, u, 3, 0, 1, 0, 3, 1, 1, 0, 3, u, 3, 0, 1)$ $1$ $\boldsymbol{3}$ $\boldsymbol{179}$ $C_{68, 6}$ $I_{3}$ $(u, 0, 0, 1, 1, 0, 1, 0, 3, 3, u, 0, 1, 0, 3, 3, 1, 0, 3, 0, 3, 3, 1, u, 1, u, 1, u, 3, 0, 1, u)$ $3$ $\boldsymbol{3}$ $\boldsymbol{189}$ $C_{68, 7}$ $I_{3}$ $(0, 3, 0, 1, u, 3, u, 3, 0, 1, 0, 3, 0, 3, 0, 0, 1, u, u, 1, 0, u, 1, 0, u, 0, u, 1, 3, 0, 1, u)$ $3$ $\boldsymbol{3}$ $\boldsymbol{198}$
$i^{th}$ neighbour of $\mathcal{N}_{(0)}$
 $i$ $\mathcal{N}_{(i+1)}$ $x_i$ $\gamma$ $\beta$ $i$ $\mathcal{N}_{(i+1)}$ $x_i$ $\gamma$ $\beta$ $0$ $\mathcal{N}_{(1)}$ $(1110000001001111010001001000010000)$ $3$ $180$ $1$ $\mathcal{N}_{(2)}$ $(0110111111100111000010000110011111)$ $4$ $177$ $2$ $\mathcal{N}_{(3)}$ $(1111110110010011100101001000101111)$ $5$ $169$ $3$ $\mathcal{N}_{(4)}$ $(0100000000110011110000010000011110)$ $6$ $191$ $4$ $\mathcal{N}_{(5)}$ $(0100000000001101110010001110000110)$ $6$ $199$ $5$ $\mathcal{N}_{(6)}$ $(0000100000001100010011001110000111)$ $7$ $199$ $6$ $\mathcal{N}_{(7)}$ $(1011111101001111000101010111111010)$ $\textbf{7}$ $\textbf{209}$ $7$ $\mathcal{N}_{(8)}$ $(1110011111110110000101111101111110)$ $\textbf{7}$ $\textbf{220}$ $8$ $\mathcal{N}_{(9)}$ $(1100101001011011001101000000111100)$ $8$ $212$ $9$ $\mathcal{N}_{(10)}$ $(1110110100111111000010011111011000)$ $\textbf{8}$ $\textbf{226}$ $10$ $\mathcal{N}_{(11)}$ $(1011101011111010111010001000101000)$ $\textbf{8}$ $\textbf{233}$ $11$ $\mathcal{N}_{(12)}$ $(1101100000000101110010111111001110)$ $9$ $213$ $12$ $\mathcal{N}_{(13)}$ $(0111110101110100100110100100000111)$ $9$ $222$ $13$ $\mathcal{N}_{(14)}$ $(1100011000000000100101010101100010)$ $\textbf{9}$ $\textbf{229}$ $14$ $\mathcal{N}_{(15)}$ $(0100111000100000110010100011000100)$ $\textbf{9}$ $\textbf{235}$ $15$ $\mathcal{N}_{(16)}$ $(0000111011111101001101111010001101)$ $\textbf{9}$ $\textbf{236}$ $16$ $\mathcal{N}_{(17)}$ $(0110010111000111101001101101101110)$ $\textbf{9}$ $\textbf{240}$ $17$ $\mathcal{N}_{(18)}$ $(0001101000010111100111111110001001)$ $\textbf{9}$ $\textbf{243}$ $18$ $\mathcal{N}_{(19)}$ $(1111010011001111000010010001010001)$ $\textbf{9}$ $\textbf{247}$ $19$ $\mathcal{N}_{(20)}$ $(0001000000100010001011101011110000)$ $\textbf{8}$ $\textbf{234}$ $20$ $\mathcal{N}_{(21)}$ $(0110110001101011101000111100110001)$ $\textbf{8}$ $\textbf{245}$ $21$ $\mathcal{N}_{(22)}$ $(1000011100010110100110011011000011)$ $\textbf{8}$ $\textbf{250}$
 $i$ $\mathcal{N}_{(i+1)}$ $x_i$ $\gamma$ $\beta$ $i$ $\mathcal{N}_{(i+1)}$ $x_i$ $\gamma$ $\beta$ $0$ $\mathcal{N}_{(1)}$ $(1110000001001111010001001000010000)$ $3$ $180$ $1$ $\mathcal{N}_{(2)}$ $(0110111111100111000010000110011111)$ $4$ $177$ $2$ $\mathcal{N}_{(3)}$ $(1111110110010011100101001000101111)$ $5$ $169$ $3$ $\mathcal{N}_{(4)}$ $(0100000000110011110000010000011110)$ $6$ $191$ $4$ $\mathcal{N}_{(5)}$ $(0100000000001101110010001110000110)$ $6$ $199$ $5$ $\mathcal{N}_{(6)}$ $(0000100000001100010011001110000111)$ $7$ $199$ $6$ $\mathcal{N}_{(7)}$ $(1011111101001111000101010111111010)$ $\textbf{7}$ $\textbf{209}$ $7$ $\mathcal{N}_{(8)}$ $(1110011111110110000101111101111110)$ $\textbf{7}$ $\textbf{220}$ $8$ $\mathcal{N}_{(9)}$ $(1100101001011011001101000000111100)$ $8$ $212$ $9$ $\mathcal{N}_{(10)}$ $(1110110100111111000010011111011000)$ $\textbf{8}$ $\textbf{226}$ $10$ $\mathcal{N}_{(11)}$ $(1011101011111010111010001000101000)$ $\textbf{8}$ $\textbf{233}$ $11$ $\mathcal{N}_{(12)}$ $(1101100000000101110010111111001110)$ $9$ $213$ $12$ $\mathcal{N}_{(13)}$ $(0111110101110100100110100100000111)$ $9$ $222$ $13$ $\mathcal{N}_{(14)}$ $(1100011000000000100101010101100010)$ $\textbf{9}$ $\textbf{229}$ $14$ $\mathcal{N}_{(15)}$ $(0100111000100000110010100011000100)$ $\textbf{9}$ $\textbf{235}$ $15$ $\mathcal{N}_{(16)}$ $(0000111011111101001101111010001101)$ $\textbf{9}$ $\textbf{236}$ $16$ $\mathcal{N}_{(17)}$ $(0110010111000111101001101101101110)$ $\textbf{9}$ $\textbf{240}$ $17$ $\mathcal{N}_{(18)}$ $(0001101000010111100111111110001001)$ $\textbf{9}$ $\textbf{243}$ $18$ $\mathcal{N}_{(19)}$ $(1111010011001111000010010001010001)$ $\textbf{9}$ $\textbf{247}$ $19$ $\mathcal{N}_{(20)}$ $(0001000000100010001011101011110000)$ $\textbf{8}$ $\textbf{234}$ $20$ $\mathcal{N}_{(21)}$ $(0110110001101011101000111100110001)$ $\textbf{8}$ $\textbf{245}$ $21$ $\mathcal{N}_{(22)}$ $(1000011100010110100110011011000011)$ $\textbf{8}$ $\textbf{250}$
Neighbours of $\mathcal{N}_{(7)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $7$ $1$ $(0001101011101000000010101100100000)$ $\textbf{7}$ $\textbf{200}$ $7$ $2$ $(1011110000101000000000000110110010)$ $\textbf{7}$ $\textbf{201}$ $7$ $3$ $(1010110010100001100100011110001110)$ $\textbf{7}$ $\textbf{202}$ $7$ $4$ $(0001001110011000100100000010000111)$ $\textbf{7}$ $\textbf{204}$ $7$ $5$ $(0110000000101001011011010010111000)$ $\textbf{7}$ $\textbf{205}$ $7$ $6$ $(1101010010010101110001000011001000)$ $\textbf{7}$ $\textbf{206}$ $7$ $7$ $(0000101100001000100101011000010101)$ $\textbf{7}$ $\textbf{207}$ $7$ $8$ $(1101100101111100101110100100000011)$ $\textbf{7}$ $\textbf{212}$ $7$ $9$ $(1111011000001111101001111011111111)$ $\textbf{7}$ $\textbf{214}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $7$ $1$ $(0001101011101000000010101100100000)$ $\textbf{7}$ $\textbf{200}$ $7$ $2$ $(1011110000101000000000000110110010)$ $\textbf{7}$ $\textbf{201}$ $7$ $3$ $(1010110010100001100100011110001110)$ $\textbf{7}$ $\textbf{202}$ $7$ $4$ $(0001001110011000100100000010000111)$ $\textbf{7}$ $\textbf{204}$ $7$ $5$ $(0110000000101001011011010010111000)$ $\textbf{7}$ $\textbf{205}$ $7$ $6$ $(1101010010010101110001000011001000)$ $\textbf{7}$ $\textbf{206}$ $7$ $7$ $(0000101100001000100101011000010101)$ $\textbf{7}$ $\textbf{207}$ $7$ $8$ $(1101100101111100101110100100000011)$ $\textbf{7}$ $\textbf{212}$ $7$ $9$ $(1111011000001111101001111011111111)$ $\textbf{7}$ $\textbf{214}$
Neighbours of $\mathcal{N}_{(8)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $8$ $10$ $(0101101001001001100111000010001010)$ $\textbf{6}$ $\textbf{205}$ $8$ $11$ $(0000111000000101110110010000000101)$ $\textbf{6}$ $\textbf{211}$ $8$ $12$ $(1000110101001001010000000111111011)$ $\textbf{7}$ $\textbf{208}$ $8$ $13$ $(1100001010000110010100101000001100)$ $\textbf{7}$ $\textbf{211}$ $8$ $14$ $(0000011000010001001000011101100110)$ $\textbf{7}$ $\textbf{213}$ $8$ $15$ $(0011000110000110101101001101111011)$ $\textbf{7}$ $\textbf{215}$ $8$ $16$ $(0100001111110010110100000101101010)$ $\textbf{7}$ $\textbf{216}$ $8$ $17$ $(1111101111010101000001000100011110)$ $\textbf{7}$ $\textbf{218}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $8$ $10$ $(0101101001001001100111000010001010)$ $\textbf{6}$ $\textbf{205}$ $8$ $11$ $(0000111000000101110110010000000101)$ $\textbf{6}$ $\textbf{211}$ $8$ $12$ $(1000110101001001010000000111111011)$ $\textbf{7}$ $\textbf{208}$ $8$ $13$ $(1100001010000110010100101000001100)$ $\textbf{7}$ $\textbf{211}$ $8$ $14$ $(0000011000010001001000011101100110)$ $\textbf{7}$ $\textbf{213}$ $8$ $15$ $(0011000110000110101101001101111011)$ $\textbf{7}$ $\textbf{215}$ $8$ $16$ $(0100001111110010110100000101101010)$ $\textbf{7}$ $\textbf{216}$ $8$ $17$ $(1111101111010101000001000100011110)$ $\textbf{7}$ $\textbf{218}$
Neighbours of $\mathcal{N}_{(10)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $10$ $18$ $(1000111101011101000010001111000100)$ $\textbf{8}$ $\textbf{222}$ $10$ $19$ $(1000001100101001110001001010110111)$ $\textbf{8}$ $\textbf{223}$ $10$ $20$ $(0000100110010101011101101001100110)$ $\textbf{8}$ $\textbf{227}$ $10$ $21$ $(1011001101010011010111011000101010)$ $\textbf{8}$ $\textbf{229}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $10$ $18$ $(1000111101011101000010001111000100)$ $\textbf{8}$ $\textbf{222}$ $10$ $19$ $(1000001100101001110001001010110111)$ $\textbf{8}$ $\textbf{223}$ $10$ $20$ $(0000100110010101011101101001100110)$ $\textbf{8}$ $\textbf{227}$ $10$ $21$ $(1011001101010011010111011000101010)$ $\textbf{8}$ $\textbf{229}$
Neighbours of $\mathcal{N}_{(11)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $11$ $22$ $(1010110110000101110101111100110110)$ $\textbf{7}$ $\textbf{221}$ $11$ $23$ $(0000001101000001110010110101100000)$ $\textbf{7}$ $\textbf{222}$ $11$ $24$ $(1101010100100000111010001000010011)$ $\textbf{8}$ $\textbf{224}$ $11$ $25$ $(0000010011001000010100011111011111)$ $\textbf{8}$ $\textbf{225}$ $11$ $26$ $(1110111110110010111011101101101110)$ $\textbf{8}$ $\textbf{228}$ $11$ $27$ $(1001100110100111000010100000100101)$ $\textbf{8}$ $\textbf{230}$ $11$ $28$ $(0000110001111000001001000011101000)$ $\textbf{8}$ $\textbf{231}$ $11$ $29$ $(1001011111010011000001100001010000)$ $\textbf{8}$ $\textbf{232}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $11$ $22$ $(1010110110000101110101111100110110)$ $\textbf{7}$ $\textbf{221}$ $11$ $23$ $(0000001101000001110010110101100000)$ $\textbf{7}$ $\textbf{222}$ $11$ $24$ $(1101010100100000111010001000010011)$ $\textbf{8}$ $\textbf{224}$ $11$ $25$ $(0000010011001000010100011111011111)$ $\textbf{8}$ $\textbf{225}$ $11$ $26$ $(1110111110110010111011101101101110)$ $\textbf{8}$ $\textbf{228}$ $11$ $27$ $(1001100110100111000010100000100101)$ $\textbf{8}$ $\textbf{230}$ $11$ $28$ $(0000110001111000001001000011101000)$ $\textbf{8}$ $\textbf{231}$ $11$ $29$ $(1001011111010011000001100001010000)$ $\textbf{8}$ $\textbf{232}$
Neighbours of $\mathcal{N}_{(12)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $12$ $30$ $(1000100110000001010101010100001001)$ $\textbf{9}$ $\textbf{191}$ $12$ $31$ $(0111010100101000000001100101011010)$ $\textbf{9}$ $\textbf{197}$ $12$ $32$ $(1111100000101101001011110111000010)$ $\textbf{9}$ $\textbf{212}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $12$ $30$ $(1000100110000001010101010100001001)$ $\textbf{9}$ $\textbf{191}$ $12$ $31$ $(0111010100101000000001100101011010)$ $\textbf{9}$ $\textbf{197}$ $12$ $32$ $(1111100000101101001011110111000010)$ $\textbf{9}$ $\textbf{212}$
Neighbour of $\mathcal{N}_{(14)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $14$ $33$ $(1011110001101000100111010000010000)$ $\textbf{9}$ $\textbf{227}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $14$ $33$ $(1011110001101000100111010000010000)$ $\textbf{9}$ $\textbf{227}$
Neighbours of $\mathcal{N}_{(15)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $15$ $34$ $(1011000011111110011101011000000101)$ $\textbf{9}$ $\textbf{231}$ $15$ $35$ $(1111110111110000010110000100010011)$ $\textbf{9}$ $\textbf{232}$ $15$ $36$ $(0011000011010010100011010000111001)$ $\textbf{9}$ $\textbf{233}$ $15$ $37$ $(0000000000111100000000101100111101)$ $\textbf{9}$ $\textbf{234}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $15$ $34$ $(1011000011111110011101011000000101)$ $\textbf{9}$ $\textbf{231}$ $15$ $35$ $(1111110111110000010110000100010011)$ $\textbf{9}$ $\textbf{232}$ $15$ $36$ $(0011000011010010100011010000111001)$ $\textbf{9}$ $\textbf{233}$ $15$ $37$ $(0000000000111100000000101100111101)$ $\textbf{9}$ $\textbf{234}$
Neighbours of $\mathcal{N}_{(17)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $17$ $38$ $(0110000100000010110010110000110100)$ $\textbf{9}$ $\textbf{237}$ $17$ $39$ $(0011111001100000111100111101010010)$ $\textbf{9}$ $\textbf{238}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $17$ $38$ $(0110000100000010110010110000110100)$ $\textbf{9}$ $\textbf{237}$ $17$ $39$ $(0011111001100000111100111101010010)$ $\textbf{9}$ $\textbf{238}$
Neighbours of $\mathcal{N}_{(18)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $18$ $40$ $(0110010110000001001110111010011100)$ $\textbf{9}$ $\textbf{239}$ $18$ $41$ $(1111000010111111010100101000111101)$ $\textbf{9}$ $\textbf{241}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $18$ $40$ $(0110010110000001001110111010011100)$ $\textbf{9}$ $\textbf{239}$ $18$ $41$ $(1111000010111111010100101000111101)$ $\textbf{9}$ $\textbf{241}$
Neighbours of $\mathcal{N}_{(19)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $19$ $42$ $(1011110110001100110101001011001010)$ $\textbf{9}$ $\textbf{242}$ $19$ $43$ $(0101010111011010111100000111011110)$ $\textbf{9}$ $\textbf{244}$ $19$ $44$ $(1010110011000110001101001010010000)$ $\textbf{9}$ $\textbf{246}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $19$ $42$ $(1011110110001100110101001011001010)$ $\textbf{9}$ $\textbf{242}$ $19$ $43$ $(0101010111011010111100000111011110)$ $\textbf{9}$ $\textbf{244}$ $19$ $44$ $(1010110011000110001101001010010000)$ $\textbf{9}$ $\textbf{246}$
Neighbours of $\mathcal{N}_{(20)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $20$ $45$ $(1111111110101111010101000110001101)$ $\textbf{8}$ $\textbf{236}$ $20$ $46$ $(1010000010110000100011110101111001)$ $\textbf{8}$ $\textbf{239}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $20$ $45$ $(1111111110101111010101000110001101)$ $\textbf{8}$ $\textbf{236}$ $20$ $46$ $(1010000010110000100011110101111001)$ $\textbf{8}$ $\textbf{239}$
Neighbours of $\mathcal{N}_{(21)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $21$ $47$ $(0100000110001011000001000101101010)$ $\textbf{6}$ $\textbf{208}$ $21$ $48$ $(1101111000000001010010000110110001)$ $\textbf{6}$ $\textbf{209}$ $21$ $49$ $(1011101011101010010101111101000101)$ $\textbf{6}$ $\textbf{212}$ $21$ $50$ $(1111110111000100010010111011100000)$ $\textbf{6}$ $\textbf{214}$ $21$ $51$ $(1011111101010010111011101111111100)$ $\textbf{6}$ $\textbf{215}$ $21$ $52$ $(0000000001100001001001100111011100)$ $\textbf{6}$ $\textbf{218}$ $21$ $53$ $(1111011001110010100001101011011011)$ $\textbf{6}$ $\textbf{220}$ $21$ $54$ $(0100000001010101001001101001000011)$ $\textbf{7}$ $\textbf{219}$ $21$ $55$ $(1100000000000001110100001001100111)$ $\textbf{7}$ $\textbf{223}$ $21$ $56$ $(0000001101000100110101111100001111)$ $\textbf{7}$ $\textbf{225}$ $21$ $57$ $(1111011001111010111110100111110110)$ $\textbf{7}$ $\textbf{226}$ $21$ $58$ $(0010011000011000001000111001000101)$ $\textbf{7}$ $\textbf{227}$ $21$ $59$ $(1001010101101111101110000000000011)$ $\textbf{7}$ $\textbf{230}$ $21$ $60$ $(1111110101100000100011001110100110)$ $\textbf{8}$ $\textbf{235}$ $21$ $61$ $(0110000110110100100100101111100100)$ $\textbf{8}$ $\textbf{238}$ $21$ $62$ $(1010010010111110111001111011100010)$ $\textbf{8}$ $\textbf{240}$ $21$ $63$ $(1101011100111011010011111101111110)$ $\textbf{8}$ $\textbf{241}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $21$ $47$ $(0100000110001011000001000101101010)$ $\textbf{6}$ $\textbf{208}$ $21$ $48$ $(1101111000000001010010000110110001)$ $\textbf{6}$ $\textbf{209}$ $21$ $49$ $(1011101011101010010101111101000101)$ $\textbf{6}$ $\textbf{212}$ $21$ $50$ $(1111110111000100010010111011100000)$ $\textbf{6}$ $\textbf{214}$ $21$ $51$ $(1011111101010010111011101111111100)$ $\textbf{6}$ $\textbf{215}$ $21$ $52$ $(0000000001100001001001100111011100)$ $\textbf{6}$ $\textbf{218}$ $21$ $53$ $(1111011001110010100001101011011011)$ $\textbf{6}$ $\textbf{220}$ $21$ $54$ $(0100000001010101001001101001000011)$ $\textbf{7}$ $\textbf{219}$ $21$ $55$ $(1100000000000001110100001001100111)$ $\textbf{7}$ $\textbf{223}$ $21$ $56$ $(0000001101000100110101111100001111)$ $\textbf{7}$ $\textbf{225}$ $21$ $57$ $(1111011001111010111110100111110110)$ $\textbf{7}$ $\textbf{226}$ $21$ $58$ $(0010011000011000001000111001000101)$ $\textbf{7}$ $\textbf{227}$ $21$ $59$ $(1001010101101111101110000000000011)$ $\textbf{7}$ $\textbf{230}$ $21$ $60$ $(1111110101100000100011001110100110)$ $\textbf{8}$ $\textbf{235}$ $21$ $61$ $(0110000110110100100100101111100100)$ $\textbf{8}$ $\textbf{238}$ $21$ $62$ $(1010010010111110111001111011100010)$ $\textbf{8}$ $\textbf{240}$ $21$ $63$ $(1101011100111011010011111101111110)$ $\textbf{8}$ $\textbf{241}$
Neighbours of $\mathcal{N}_{(22)}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $22$ $63$ $(0011111100111010011001010011100100)$ $\textbf{5}$ $\textbf{207}$ $22$ $64$ $(1011111100101111100110111111111101)$ $\textbf{6}$ $\textbf{213}$ $22$ $65$ $(1001011101001100101011001000110100)$ $\textbf{6}$ $\textbf{217}$ $22$ $66$ $(0100111101000110110111101101111110)$ $\textbf{6}$ $\textbf{219}$ $22$ $68$ $(1000010000111101010101110010010011)$ $\textbf{7}$ $\textbf{229}$ $22$ $69$ $(0100000001011101000011001111110011)$ $\textbf{8}$ $\textbf{237}$ $22$ $70$ $(1111111101001111101100000010100000)$ $\textbf{8}$ $\textbf{242}$ $22$ $71$ $(0010000100001001100001001110111000)$ $\textbf{8}$ $\textbf{243}$ $22$ $72$ $(1110110000001011011001101011011010)$ $\textbf{8}$ $\textbf{247}$
 $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{(i)}$ $\mathcal{M}_{i}$ $(x_{35}, x_{36}, ..., x_{68})$ $\gamma$ $\beta$ $22$ $63$ $(0011111100111010011001010011100100)$ $\textbf{5}$ $\textbf{207}$ $22$ $64$ $(1011111100101111100110111111111101)$ $\textbf{6}$ $\textbf{213}$ $22$ $65$ $(1001011101001100101011001000110100)$ $\textbf{6}$ $\textbf{217}$ $22$ $66$ $(0100111101000110110111101101111110)$ $\textbf{6}$ $\textbf{219}$ $22$ $68$ $(1000010000111101010101110010010011)$ $\textbf{7}$ $\textbf{229}$ $22$ $69$ $(0100000001011101000011001111110011)$ $\textbf{8}$ $\textbf{237}$ $22$ $70$ $(1111111101001111101100000010100000)$ $\textbf{8}$ $\textbf{242}$ $22$ $71$ $(0010000100001001100001001110111000)$ $\textbf{8}$ $\textbf{243}$ $22$ $72$ $(1110110000001011011001101011011010)$ $\textbf{8}$ $\textbf{247}$
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