American Institute of Mathematical Sciences

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May  2022, 16(2): 269-284. doi: 10.3934/amc.2020111

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

Maria Bortos 1, , Joe Gildea 2, , Abidin Kaya 3, , Adrian Korban 2, and  Alexander Tylyshchak 1,

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  May 2022 Early access  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.

Keywords: Group rings, self-dual codes, linear codes, block matrix construction, codes over rings.
Mathematics Subject Classification: 94B05, 16S34.
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, 2022, 16 (2) : 269-284. doi: 10.3934/amc.2020111
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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

T. Hurley, "Group Rings and Rings of Matrices", J. Pure Appl. Math., 31 (2006), 319-335. 

[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.

[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.

[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.

[19]

E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000.

[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.

[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.

[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.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

T. Hurley, "Group Rings and Rings of Matrices", J. Pure Appl. Math., 31 (2006), 319-335. 

[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.

[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.

[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.

[19]

E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000.

[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.

[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.

[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.

Table 1.  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 $
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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 $
Table 2.  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 $
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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 $
Table 3.  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 $
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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 $
Table 4.  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 $
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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 $
Table 5.  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} $
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$ 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} $
Table 6.  $ 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} $
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$ 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} $
Table 7.  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} $
Table Options

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$ \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} $
Table 8.  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} $
Table Options

Download as excel

$ \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} $
Table 9.  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} $
Table Options

Download as excel

$ \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} $
Table 10.  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} $
Table Options

Download as excel

$ \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} $
Table 11.  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} $
Table Options

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$ \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} $
Table 12.  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} $
Table Options

Download as excel

$ \mathcal{N}_{(i)} $ $ \mathcal{M}_{i} $ $ (x_{35}, x_{36}, ..., x_{68}) $ $ \gamma $ $ \beta $
$ 14 $ $ 33 $ $ (1011110001101000100111010000010000) $ $ \textbf{9} $ $ \textbf{227} $
Table 13.  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} $
Table Options

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$ \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} $
Table 14.  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} $
Table Options

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$ \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} $
Table 15.  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} $
Table Options

Download as excel

$ \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} $
Table 16.  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} $
Table Options

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$ \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} $
Table 17.  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} $
Table Options

Download as excel

$ \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} $
Table 18.  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} $
Table Options

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$ \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} $
Table 19.  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} $
Table Options

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$ \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} $
[1]

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, 2021  doi: 10.3934/amc.2021039
[2]

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