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

Maria Bortos; Joe Gildea; Abidin Kaya; Adrian Korban; Alexander Tylyshchak

doi:10.3934/amc.2020111

Article Contents

2022, Volume 16, Issue 2: 269-284. Doi: 10.3934/amc.2020111

This issue Previous Article Two constructions of low-hit-zone frequency-hopping sequence sets Next Article On the equivalence of several classes of quaternary sequences with optimal autocorrelation and length $ 2p$

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: February 29, 2020

Published: May 2022

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Abstract

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.

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Mathematics Subject Classification: 94B05, 16S34.

Citation:

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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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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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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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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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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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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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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} $

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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} $

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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} $

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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} $

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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} $

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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} $

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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} $

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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} $

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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} $

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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} $

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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} $

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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} $

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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} $

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