doi: 10.3934/amc.2021034
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New type i binary [72, 36, 12] self-dual codes from composite matrices and R1 lifts

1. 

Department of Mathematical and Physical Sciences, University of Chester, Chester, England

2. 

Faculty of Engineering, Department of Natural and Mathematical Sciences, Tarsus University, Mersin, Turkey

3. 

Faculty of Engineering, Department of Computer Engineering, Tarsus University, Mersin, Turkey

* Corresponding author: Adrian Korban

Received  June 2020 Revised  June 2021 Early access August 2021

In this work, we define three composite matrices derived from group rings. We employ these composite matrices to create generator matrices of the form $ [I_n \ | \ \Omega(v)], $ where $ I_n $ is the identity matrix and $ \Omega(v) $ is a composite matrix and search for binary self-dual codes with parameters $ [36,18, 6 \ \text{or} \ 8]. $ We next lift these codes over the ring $ R_1 = \mathbb{F}_2+u\mathbb{F}_2 $ to obtain codes whose binary images are self-dual codes with parameters $ [72,36,12]. $ Many of these codes turn out to have weight enumerators with parameters that were not known in the literature before. In particular, we find $ 30 $ new Type I binary self-dual codes with parameters $ [72,36,12]. $

Citation: Adrian Korban, Serap Şahinkaya, Deniz Ustun. New type i binary [72, 36, 12] self-dual codes from composite matrices and R1 lifts. Advances in Mathematics of Communications, doi: 10.3934/amc.2021034
References:
[1]

M. Borello, On automorphism groups of binary linear codes, Topics in Finite Fields, 632 (2015), 29-41.  doi: 10.1090/conm/632/12617.  Google Scholar

[2]

W. BosmaJ. 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

[3]

I. Bouyukliev, V. Fack and J. Winna, Hadamard matrices of order 36, European Conference on Combinatorics, Graph Theory and Applications, (2005), 93–98. Google Scholar

[4]

R. Dontcheva, New binary self-dual $[70, 35, 12]$ and binary $[72, 36, 12]$ self-dual doubly-even codes, Serdica Math. J., 27 (2002), 287-302.   Google Scholar

[5]

S. T. DoughertyP. GaboritM. 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

[6]

S. T. DoughertyJ. Gildea and A. Korban, Extending an established isomorphism between group rings and a subring of the $n \times n$ matrices, Internat. J. Algebra Comput., 31 (2021), 471-490.  doi: 10.1142/S0218196721500223.  Google Scholar

[7]

S. T. DoughertyJ. GildeaA. 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. Commun., 14 (2020), 677-702.  doi: 10.3934/amc.2020037.  Google Scholar

[8]

S. T. DoughertyJ. GildeaA. Korban and A. Kaya, New extremal self-dual binary codes of length 68 via composite construction, $\mathbb{F}_2+u\mathbb{F}_2$ lifts, extensions and neighbours, Int. J. Inf. Coding Theory, 5 (2020), 211-226.   Google Scholar

[9]

S. T. DoughertyJ. GildeaA. Kaya and A. Korban, Composite matrices from group rings, composite $G$-codes and constructions of self-dual codes, Des. Codes Cryptogr., 89 (2021), 1615-1638.  doi: 10.1007/s10623-021-00882-8.  Google Scholar

[10]

S. T. DoughertyJ. GildeaR. Taylor and A. Tylshchak, Group rings, G-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7.  Google Scholar

[11]

S. T. DoughertyT. A. Gulliver and M. Harada, Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.  doi: 10.1109/18.641574.  Google Scholar

[12]

S. T. DoughertyS. 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

[13]

S. T. DoughertyJ-L. Kim and P. Sole, Double circulant codes from two class association schemes, Adv. Math. Commun., 1 (2007), 45-64.  doi: 10.3934/amc.2007.1.45.  Google Scholar

[14]

S. T. DoughertyJ. L. Kim and P. Sole, Open problems in coding theory, Contemp. Math., 634 (2015), 79-99.  doi: 10.1090/conm/634/12692.  Google Scholar

[15]

T. A. Gulliver and M. Harada, On double circulant doubly-even self-dual $[72, 36, 12]$ codes and their neighbors, Austalas. J. Comb., 40 (2008), 137-144.   Google Scholar

[16]

M. Gurel and N. Yankov, Self-dual codes with an automorphism of order 17, Math. Commun., 21 (2016), 97-107.   Google Scholar

[17]

T. Hurley, Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335.   Google Scholar

[18]

A. KayaB. Yildiz and I. Siap, New extremal binary self-dual codes of length 68 from quadratic residue codes over $\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$, Finite Fields Appl., 29 (2014), 160-177.  doi: 10.1016/j.ffa.2014.04.009.  Google Scholar

[19]

A. Korban, All known type I and type II $[72, 36, 12]$ binary self-dual codes, available online at https://sites.google.com/view/adriankorban/binary-self-dual-codes. Google Scholar

[20]

A. Korban, S. Sahinkaya, D. Ustun, A novel genetic search scheme based on nature – inspired evolutionary algorithms for self-dual codes, arXiv: 2012.12248. Google Scholar

[21]

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

[22]

N. Tufekci and B. Yildiz, On codes over $R_{k, m}$ and constructions for new binary self-dual codes, Math. Slovaca, 66 (2016), 1511-1526.  doi: 10.1515/ms-2016-0240.  Google Scholar

[23]

N. YankovM. H. LeeM. 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

[24]

A. Zhdanov, New self-dual codes of length 72, arXiv: 1705.05779. Google Scholar

[25]

A. Zhdanov, Convolutional encoding of 60, 64, 68, 72-bit self-dual codes, arXiv: 1702.05153. Google Scholar

show all references

References:
[1]

M. Borello, On automorphism groups of binary linear codes, Topics in Finite Fields, 632 (2015), 29-41.  doi: 10.1090/conm/632/12617.  Google Scholar

[2]

W. BosmaJ. 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

[3]

I. Bouyukliev, V. Fack and J. Winna, Hadamard matrices of order 36, European Conference on Combinatorics, Graph Theory and Applications, (2005), 93–98. Google Scholar

[4]

R. Dontcheva, New binary self-dual $[70, 35, 12]$ and binary $[72, 36, 12]$ self-dual doubly-even codes, Serdica Math. J., 27 (2002), 287-302.   Google Scholar

[5]

S. T. DoughertyP. GaboritM. 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

[6]

S. T. DoughertyJ. Gildea and A. Korban, Extending an established isomorphism between group rings and a subring of the $n \times n$ matrices, Internat. J. Algebra Comput., 31 (2021), 471-490.  doi: 10.1142/S0218196721500223.  Google Scholar

[7]

S. T. DoughertyJ. GildeaA. 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. Commun., 14 (2020), 677-702.  doi: 10.3934/amc.2020037.  Google Scholar

[8]

S. T. DoughertyJ. GildeaA. Korban and A. Kaya, New extremal self-dual binary codes of length 68 via composite construction, $\mathbb{F}_2+u\mathbb{F}_2$ lifts, extensions and neighbours, Int. J. Inf. Coding Theory, 5 (2020), 211-226.   Google Scholar

[9]

S. T. DoughertyJ. GildeaA. Kaya and A. Korban, Composite matrices from group rings, composite $G$-codes and constructions of self-dual codes, Des. Codes Cryptogr., 89 (2021), 1615-1638.  doi: 10.1007/s10623-021-00882-8.  Google Scholar

[10]

S. T. DoughertyJ. GildeaR. Taylor and A. Tylshchak, Group rings, G-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7.  Google Scholar

[11]

S. T. DoughertyT. A. Gulliver and M. Harada, Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.  doi: 10.1109/18.641574.  Google Scholar

[12]

S. T. DoughertyS. 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

[13]

S. T. DoughertyJ-L. Kim and P. Sole, Double circulant codes from two class association schemes, Adv. Math. Commun., 1 (2007), 45-64.  doi: 10.3934/amc.2007.1.45.  Google Scholar

[14]

S. T. DoughertyJ. L. Kim and P. Sole, Open problems in coding theory, Contemp. Math., 634 (2015), 79-99.  doi: 10.1090/conm/634/12692.  Google Scholar

[15]

T. A. Gulliver and M. Harada, On double circulant doubly-even self-dual $[72, 36, 12]$ codes and their neighbors, Austalas. J. Comb., 40 (2008), 137-144.   Google Scholar

[16]

M. Gurel and N. Yankov, Self-dual codes with an automorphism of order 17, Math. Commun., 21 (2016), 97-107.   Google Scholar

[17]

T. Hurley, Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335.   Google Scholar

[18]

A. KayaB. Yildiz and I. Siap, New extremal binary self-dual codes of length 68 from quadratic residue codes over $\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$, Finite Fields Appl., 29 (2014), 160-177.  doi: 10.1016/j.ffa.2014.04.009.  Google Scholar

[19]

A. Korban, All known type I and type II $[72, 36, 12]$ binary self-dual codes, available online at https://sites.google.com/view/adriankorban/binary-self-dual-codes. Google Scholar

[20]

A. Korban, S. Sahinkaya, D. Ustun, A novel genetic search scheme based on nature – inspired evolutionary algorithms for self-dual codes, arXiv: 2012.12248. Google Scholar

[21]

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

[22]

N. Tufekci and B. Yildiz, On codes over $R_{k, m}$ and constructions for new binary self-dual codes, Math. Slovaca, 66 (2016), 1511-1526.  doi: 10.1515/ms-2016-0240.  Google Scholar

[23]

N. YankovM. H. LeeM. 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

[24]

A. Zhdanov, New self-dual codes of length 72, arXiv: 1705.05779. Google Scholar

[25]

A. Zhdanov, Convolutional encoding of 60, 64, 68, 72-bit self-dual codes, arXiv: 1702.05153. Google Scholar

Table 1.  Type I $ [36,18,6-8] $ Codes from Theorem 4.1
Type $ r_B $ $ r_C $ $ |Aut(C_i)| $
$ C_1 $ $ [36,18,6] $ $ (0,0,0,0,0,1,0,1,1) $ $ (1,0,1,1,1,0,1,0,1) $ $ 2^2 \cdot 3^2 $
$ C_2 $ $ [36,18,6] $ $ (0,0,0,0,1,1,0,1,1) $ $ (1,0,0,1,0,1,1,1,0) $ $ 2^2 \cdot 3^2 $
$ C_3 $ $ [36,18,8] $ $ (0,0,1,0,0,1,0,0,1) $ $ (1,0,0,1,1,0,1,1,1) $ $ 2^2 \cdot 3^2 $
$ C_4 $ $ [36,18,8] $ $ (0,1,0,0,0,1,0,1,1) $ $ (1,0,0,1,0,0,1,1,1) $ $ 2^2 \cdot 3^2 $
Type $ r_B $ $ r_C $ $ |Aut(C_i)| $
$ C_1 $ $ [36,18,6] $ $ (0,0,0,0,0,1,0,1,1) $ $ (1,0,1,1,1,0,1,0,1) $ $ 2^2 \cdot 3^2 $
$ C_2 $ $ [36,18,6] $ $ (0,0,0,0,1,1,0,1,1) $ $ (1,0,0,1,0,1,1,1,0) $ $ 2^2 \cdot 3^2 $
$ C_3 $ $ [36,18,8] $ $ (0,0,1,0,0,1,0,0,1) $ $ (1,0,0,1,1,0,1,1,1) $ $ 2^2 \cdot 3^2 $
$ C_4 $ $ [36,18,8] $ $ (0,1,0,0,0,1,0,1,1) $ $ (1,0,0,1,0,0,1,1,1) $ $ 2^2 \cdot 3^2 $
Table 2.  New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_1 $
Type $ r_B $ $ r_C $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_1 $ $ W_{72,1} $ $ (u,0,u,u,u,1,u,u + 1,1) $ $ (1,0,1,1,u + 1,0,1,u,1) $ $ 0 $ $ 192 $ $ 36 $
$ \mathcal{C}_2 $ $ W_{72,1} $ $ (u,0,0,u,0,1,u,1,1) $ $ (1,0,u + 1,1,u + 1,0,1,u,u + 1) $ $ 0 $ $ 198 $ $ 36 $
$ \mathcal{C}_3 $ $ W_{72,1} $ $ (u,u,0,u,u,1,u,1,u + 1) $ $ (1,u,u + 1,1,u + 1,0,1,0,u + 1) $ $ 0 $ $ 336 $ $ 36 $
$ \mathcal{C}_4 $ $ W_{72,1} $ $ (0,u,0,0,0,1,0,1,u + 1) $ $ (1,u,u + 1,1,u + 1,0,1,0,u + 1) $ $ 18 $ $ 234 $ $ 36 $
$ \mathcal{C}_5 $ $ W_{72,1} $ $ (u,u,0,u,0,1,u,u + 1,1) $ $ (1,u,u + 1,1,1,u,1,0,u + 1) $ $ 18 $ $ 345 $ $ 36 $
$ \mathcal{C}_6 $ $ W_{72,1} $ $ (0,u,0,0,0,1,0,u + 1,1) $ $ (1,u,1,1,u + 1,u,1,u,1) $ $ 18 $ $ 378 $ $ 36 $
$ \mathcal{C}_7 $ $ W_{72,1} $ $ (u,u,u,u,0,1,u,u + 1,u + 1) $ $ (1,u,1,1,u + 1,0,1,0,1) $ $ 18 $ $ 396 $ $ 36 $
$ \mathcal{C}_8 $ $ W_{72,1} $ $ (0,0,u,0,u,1,0,1,u + 1) $ $ (1,u,1,1,1,0,1,u,1) $ $ 18 $ $ 441 $ $ 36 $
$ \mathcal{C}_9 $ $ W_{72,1} $ $ (u,u,0,u,0,1,u,1,u + 1) $ $ (1,u,1,1,1,0,1,u,1) $ $ 18 $ $ 453 $ $ 36 $
Type $ r_B $ $ r_C $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_1 $ $ W_{72,1} $ $ (u,0,u,u,u,1,u,u + 1,1) $ $ (1,0,1,1,u + 1,0,1,u,1) $ $ 0 $ $ 192 $ $ 36 $
$ \mathcal{C}_2 $ $ W_{72,1} $ $ (u,0,0,u,0,1,u,1,1) $ $ (1,0,u + 1,1,u + 1,0,1,u,u + 1) $ $ 0 $ $ 198 $ $ 36 $
$ \mathcal{C}_3 $ $ W_{72,1} $ $ (u,u,0,u,u,1,u,1,u + 1) $ $ (1,u,u + 1,1,u + 1,0,1,0,u + 1) $ $ 0 $ $ 336 $ $ 36 $
$ \mathcal{C}_4 $ $ W_{72,1} $ $ (0,u,0,0,0,1,0,1,u + 1) $ $ (1,u,u + 1,1,u + 1,0,1,0,u + 1) $ $ 18 $ $ 234 $ $ 36 $
$ \mathcal{C}_5 $ $ W_{72,1} $ $ (u,u,0,u,0,1,u,u + 1,1) $ $ (1,u,u + 1,1,1,u,1,0,u + 1) $ $ 18 $ $ 345 $ $ 36 $
$ \mathcal{C}_6 $ $ W_{72,1} $ $ (0,u,0,0,0,1,0,u + 1,1) $ $ (1,u,1,1,u + 1,u,1,u,1) $ $ 18 $ $ 378 $ $ 36 $
$ \mathcal{C}_7 $ $ W_{72,1} $ $ (u,u,u,u,0,1,u,u + 1,u + 1) $ $ (1,u,1,1,u + 1,0,1,0,1) $ $ 18 $ $ 396 $ $ 36 $
$ \mathcal{C}_8 $ $ W_{72,1} $ $ (0,0,u,0,u,1,0,1,u + 1) $ $ (1,u,1,1,1,0,1,u,1) $ $ 18 $ $ 441 $ $ 36 $
$ \mathcal{C}_9 $ $ W_{72,1} $ $ (u,u,0,u,0,1,u,1,u + 1) $ $ (1,u,1,1,1,0,1,u,1) $ $ 18 $ $ 453 $ $ 36 $
Table 3.  New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_2 $
Type $ r_B $ $ r_C $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_{10} $ $ W_{72,1} $ $ (0,u,0,0,1,1,0,1,u + 1) $ $ (1,u,u,1,u,u + 1,1,1,0) $ $ 0 $ $ 219 $ $ 36 $
$ \mathcal{C}_{11} $ $ W_{72,1} $ $ (u,0,u,u,1,1,u,u + 1,u + 1) $ $ (1,u,0,1,0,1,1,u + 1,u) $ $ 0 $ $ 345 $ $ 36 $
$ \mathcal{C}_{12} $ $ W_{72,1} $ $ (0,0,u,0,1,1,0,1,u + 1) $ $ (1,u,0,1,0,1,1,1,0) $ $ 0 $ $ 408 $ $ 36 $
$ \mathcal{C}_{13} $ $ W_{72,1} $ $ (u,0,0,u,1,u + 1,u,u + 1,u + 1) $ $ (1,0,u,1,0,u + 1,1,u + 1,0) $ $ 18 $ $ 261 $ $ 36 $
$ \mathcal{C}_{14} $ $ W_{72,1} $ $ (u,u,u,u,1,1,u,1,u + 1) $ $ (1,0,0,1,u,1,1,u + 1,0) $ $ 18 $ $ 270 $ $ 36 $
$ \mathcal{C}_{15} $ $ W_{72,1} $ $ (u,0,u,u,1,u + 1,u,1,u + 1) $ $ (1,0,u,1,u,1,1,u + 1,0) $ $ 18 $ $ 357 $ $ 36 $
Type $ r_B $ $ r_C $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_{10} $ $ W_{72,1} $ $ (0,u,0,0,1,1,0,1,u + 1) $ $ (1,u,u,1,u,u + 1,1,1,0) $ $ 0 $ $ 219 $ $ 36 $
$ \mathcal{C}_{11} $ $ W_{72,1} $ $ (u,0,u,u,1,1,u,u + 1,u + 1) $ $ (1,u,0,1,0,1,1,u + 1,u) $ $ 0 $ $ 345 $ $ 36 $
$ \mathcal{C}_{12} $ $ W_{72,1} $ $ (0,0,u,0,1,1,0,1,u + 1) $ $ (1,u,0,1,0,1,1,1,0) $ $ 0 $ $ 408 $ $ 36 $
$ \mathcal{C}_{13} $ $ W_{72,1} $ $ (u,0,0,u,1,u + 1,u,u + 1,u + 1) $ $ (1,0,u,1,0,u + 1,1,u + 1,0) $ $ 18 $ $ 261 $ $ 36 $
$ \mathcal{C}_{14} $ $ W_{72,1} $ $ (u,u,u,u,1,1,u,1,u + 1) $ $ (1,0,0,1,u,1,1,u + 1,0) $ $ 18 $ $ 270 $ $ 36 $
$ \mathcal{C}_{15} $ $ W_{72,1} $ $ (u,0,u,u,1,u + 1,u,1,u + 1) $ $ (1,0,u,1,u,1,1,u + 1,0) $ $ 18 $ $ 357 $ $ 36 $
Table 4.  New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_3 $
Type $ r_B $ $ r_C $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_{16} $ $ W_{72,1} $ $ (u,u,1,u,0,1,u,0,1) $ $ (1,u,0,1,u + 1,u,1,1,u + 1) $ $ 0 $ $ 120 $ $ 36 $
$ \mathcal{C}_{17} $ $ W_{72,1} $ $ (u,u,1,u,0,1,u,u,1) $ $ (1,0,u,1,1,0,1,1,u + 1) $ $ 0 $ $ 282 $ $ 36 $
$ \mathcal{C}_{18} $ $ W_{72,1} $ $ (u,u,1,u,0,u + 1,u,0,1) $ $ (1,u,0,1,u + 1,u,1,1,u + 1) $ $ 0 $ $ 300 $ $ 36 $
$ \mathcal{C}_{19} $ $ W_{72,1} $ $ (u,u,1,u,0,u + 1,u,0,1) $ $ (1,u,u,1,1,0,1,1,1) $ $ 18 $ $ 336 $ $ 36 $
$ \mathcal{C}_{20} $ $ W_{72,1} $ $ (u,0,1,u,0,1,u,u,1) $ $ (1,0,0,1,1,0,1,u + 1,u + 1) $ $ 36 $ $ 435 $ $ 36 $
Type $ r_B $ $ r_C $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_{16} $ $ W_{72,1} $ $ (u,u,1,u,0,1,u,0,1) $ $ (1,u,0,1,u + 1,u,1,1,u + 1) $ $ 0 $ $ 120 $ $ 36 $
$ \mathcal{C}_{17} $ $ W_{72,1} $ $ (u,u,1,u,0,1,u,u,1) $ $ (1,0,u,1,1,0,1,1,u + 1) $ $ 0 $ $ 282 $ $ 36 $
$ \mathcal{C}_{18} $ $ W_{72,1} $ $ (u,u,1,u,0,u + 1,u,0,1) $ $ (1,u,0,1,u + 1,u,1,1,u + 1) $ $ 0 $ $ 300 $ $ 36 $
$ \mathcal{C}_{19} $ $ W_{72,1} $ $ (u,u,1,u,0,u + 1,u,0,1) $ $ (1,u,u,1,1,0,1,1,1) $ $ 18 $ $ 336 $ $ 36 $
$ \mathcal{C}_{20} $ $ W_{72,1} $ $ (u,0,1,u,0,1,u,u,1) $ $ (1,0,0,1,1,0,1,u + 1,u + 1) $ $ 36 $ $ 435 $ $ 36 $
Table 5.  New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_4 $
Type $ r_B $ $ r_C $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_{21} $ $ W_{72,1} $ $ (0,1,u,0,u,1,0,u + 1,u + 1) $ $ (1,u,u,1,u,u,1,u + 1,u + 1) $ $ 0 $ $ 366 $ $ 36 $
$ \mathcal{C}_{22} $ $ W_{72,1} $ $ (u,1,u,u,u,1,u,1,u + 1) $ $ (1,u,0,1,0,0,1,u + 1,1) $ $ 0 $ $ 372 $ $ 36 $
$ \mathcal{C}_{23} $ $ W_{72,1} $ $ (0,1,u,0,u,1,0,u + 1,u + 1) $ $ (1,0,0,1,0,0,1,u + 1,u + 1) $ $ 0 $ $ 384 $ $ 36 $
$ \mathcal{C}_{24} $ $ W_{72,1} $ $ (u,1,u,u,u,1,u,1,u + 1) $ $ (1,u,u,1,u,0,1,u + 1,1) $ $ 0 $ $ 390 $ $ 36 $
$ \mathcal{C}_{25} $ $ W_{72,1} $ $ (u,1,0,u,0,1,u,u + 1,1) $ $ (1,u,0,1,0,0,1,1,u + 1) $ $ 0 $ $ 399 $ $ 36 $
$ \mathcal{C}_{26} $ $ W_{72,1} $ $ (u,1,u,u,u,1,u,u + 1,1) $ $ (1,0,u,1,0,0,1,u + 1,u + 1) $ $ 18 $ $ 264 $ $ 36 $
$ \mathcal{C}_{27} $ $ W_{72,1} $ $ (u,1,0,u,u,u + 1,u,u + 1,1) $ $ (1,0,u,1,0,u,1,1,u + 1) $ $ 18 $ $ 285 $ $ 36 $
$ \mathcal{C}_{28} $ $ W_{72,1} $ $ (0,1,u,0,0,u + 1,0,1,1) $ $ (1,u,u,1,u,0,1,1,1) $ $ 18 $ $ 300 $ $ 36 $
Type $ r_B $ $ r_C $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_{21} $ $ W_{72,1} $ $ (0,1,u,0,u,1,0,u + 1,u + 1) $ $ (1,u,u,1,u,u,1,u + 1,u + 1) $ $ 0 $ $ 366 $ $ 36 $
$ \mathcal{C}_{22} $ $ W_{72,1} $ $ (u,1,u,u,u,1,u,1,u + 1) $ $ (1,u,0,1,0,0,1,u + 1,1) $ $ 0 $ $ 372 $ $ 36 $
$ \mathcal{C}_{23} $ $ W_{72,1} $ $ (0,1,u,0,u,1,0,u + 1,u + 1) $ $ (1,0,0,1,0,0,1,u + 1,u + 1) $ $ 0 $ $ 384 $ $ 36 $
$ \mathcal{C}_{24} $ $ W_{72,1} $ $ (u,1,u,u,u,1,u,1,u + 1) $ $ (1,u,u,1,u,0,1,u + 1,1) $ $ 0 $ $ 390 $ $ 36 $
$ \mathcal{C}_{25} $ $ W_{72,1} $ $ (u,1,0,u,0,1,u,u + 1,1) $ $ (1,u,0,1,0,0,1,1,u + 1) $ $ 0 $ $ 399 $ $ 36 $
$ \mathcal{C}_{26} $ $ W_{72,1} $ $ (u,1,u,u,u,1,u,u + 1,1) $ $ (1,0,u,1,0,0,1,u + 1,u + 1) $ $ 18 $ $ 264 $ $ 36 $
$ \mathcal{C}_{27} $ $ W_{72,1} $ $ (u,1,0,u,u,u + 1,u,u + 1,1) $ $ (1,0,u,1,0,u,1,1,u + 1) $ $ 18 $ $ 285 $ $ 36 $
$ \mathcal{C}_{28} $ $ W_{72,1} $ $ (0,1,u,0,0,u + 1,0,1,1) $ $ (1,u,u,1,u,0,1,1,1) $ $ 18 $ $ 300 $ $ 36 $
Table 6.  Type I $ [36,18,6-8] $ Codes from Theorem 4.2
Type $ r_B $ $ r_C $ $ r_D $ $ |Aut(C_i)| $
$ C_5 $ $ [36,18,6] $ $ (0,0,0,0,0,1) $ $ (1,1,1,0,0,1) $ $ (1,1,1,0,1,0) $ $ 2^5 \cdot 3^4 \cdot 5 $
$ C_6 $ $ [36,18,6] $ $ (0,0,0,0,1,1) $ $ (0,0,0,0,1,1) $ $ (1,1,1,1,0,1) $ $ 2^5 \cdot 3^4 \cdot 5 $
$ C_7 $ $ [36,18,6] $ $ (0,0,0,0,1,1) $ $ (0,1,1,0,1,1) $ $ (0,1,1,1,0,0) $ $ 2^5 \cdot 3^2 $
$ C_8 $ $ [36,18,6] $ $ (0,0,1,0,0,1) $ $ (0,0,1,1,1,0) $ $ (1,1,1,0,0,1) $ $ 2^5 \cdot 3^2 $
Type $ r_B $ $ r_C $ $ r_D $ $ |Aut(C_i)| $
$ C_5 $ $ [36,18,6] $ $ (0,0,0,0,0,1) $ $ (1,1,1,0,0,1) $ $ (1,1,1,0,1,0) $ $ 2^5 \cdot 3^4 \cdot 5 $
$ C_6 $ $ [36,18,6] $ $ (0,0,0,0,1,1) $ $ (0,0,0,0,1,1) $ $ (1,1,1,1,0,1) $ $ 2^5 \cdot 3^4 \cdot 5 $
$ C_7 $ $ [36,18,6] $ $ (0,0,0,0,1,1) $ $ (0,1,1,0,1,1) $ $ (0,1,1,1,0,0) $ $ 2^5 \cdot 3^2 $
$ C_8 $ $ [36,18,6] $ $ (0,0,1,0,0,1) $ $ (0,0,1,1,1,0) $ $ (1,1,1,0,0,1) $ $ 2^5 \cdot 3^2 $
Table 7.  New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_7 $
Type $ r_B $ $ r_C $ $ r_D $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_{29} $ $ W_{72,1} $ $ (0,0,0,u,1,1) $ $ (u,1,u + 1,u,1,1) $ $ (u,u + 1,1,u + 1,0,0) $ $ 0 $ $ 471 $ $ 144 $
Type $ r_B $ $ r_C $ $ r_D $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_{29} $ $ W_{72,1} $ $ (0,0,0,u,1,1) $ $ (u,1,u + 1,u,1,1) $ $ (u,u + 1,1,u + 1,0,0) $ $ 0 $ $ 471 $ $ 144 $
Table 8.  Type I $ [36,18,6-8] $ Codes from Theorem 4.3
Type $ r_B $ $ r_C $ $ r_D $ $ |Aut(C_i)| $
$ C_9 $ $ [36,18,6] $ $ (0,0,0,0,0,1) $ $ (0,1,1,0,1,1) $ $ (1,0,1,1,0,1) $ $ 2^5 \cdot 3^2 $
$ C_{10} $ $ [36,18,6] $ $ (0,0,0,0,0,1) $ $ (1,1,1,0,0,1) $ $ (1,1,1,0,1,0) $ $ 2^5 \cdot 3^4 \cdot 5 $
$ C_{11} $ $ [36,18,6] $ $ (0,0,0,0,1,1) $ $ (0,0,0,0,1,1) $ $ (1,1,1,1,0,1) $ $ 2^5 \cdot 3^4 \cdot 5 $
$ C_{12} $ $ [36,18,6] $ $ (0,0,0,0,1,1) $ $ (0,1,1,0,0,1) $ $ (1,1,0,1,0,1) $ $ 2^5 \cdot 3^2 $
Type $ r_B $ $ r_C $ $ r_D $ $ |Aut(C_i)| $
$ C_9 $ $ [36,18,6] $ $ (0,0,0,0,0,1) $ $ (0,1,1,0,1,1) $ $ (1,0,1,1,0,1) $ $ 2^5 \cdot 3^2 $
$ C_{10} $ $ [36,18,6] $ $ (0,0,0,0,0,1) $ $ (1,1,1,0,0,1) $ $ (1,1,1,0,1,0) $ $ 2^5 \cdot 3^4 \cdot 5 $
$ C_{11} $ $ [36,18,6] $ $ (0,0,0,0,1,1) $ $ (0,0,0,0,1,1) $ $ (1,1,1,1,0,1) $ $ 2^5 \cdot 3^4 \cdot 5 $
$ C_{12} $ $ [36,18,6] $ $ (0,0,0,0,1,1) $ $ (0,1,1,0,0,1) $ $ (1,1,0,1,0,1) $ $ 2^5 \cdot 3^2 $
Table 9.  New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_9 $
Type $ r_B $ $ r_C $ $ r_D $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_{30} $ $ W_{72,1} $ $ (0,u,u,u,u,1) $ $ (u,1,1,u,u + 1,1) $ $ (u + 1,u,u + 1,1,u,u + 1) $ $ 0 $ $ 621 $ $ 432 $
Type $ r_B $ $ r_C $ $ r_D $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ \mathcal{C}_{30} $ $ W_{72,1} $ $ (0,u,u,u,u,1) $ $ (u,1,1,u,u + 1,1) $ $ (u + 1,u,u + 1,1,u,u + 1) $ $ 0 $ $ 621 $ $ 432 $
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