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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
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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.
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 $ | Show Table
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Table 2. Codes of length 64 from
$ R_1 $ $ C_1 $ $ 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 $ | Show Table
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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 $ | Show Table
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Table 4. Codes of length 64 from
$ R_1 $ $ 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 $ | Show Table
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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} $ | Show Table
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Table 6.
$ i^{th} $ $ \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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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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} $ | Show Table
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