# American Institute of Mathematical Sciences

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

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. [2] 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. [3] I. Bouyukliev, V. Fack and J. Winna, Hadamard matrices of order 36, European Conference on Combinatorics, Graph Theory and Applications, (2005), 93–98. [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. [5] 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. [6] S. T. Dougherty, J. 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. [7] 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. Commun., 14 (2020), 677-702.  doi: 10.3934/amc.2020037. [8] S. T. Dougherty, J. Gildea, A. 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. [9] S. T. Dougherty, J. Gildea, A. 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. [10] 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 Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7. [11] S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.  doi: 10.1109/18.641574. [12] 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. [13] S. T. Dougherty, J-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. [14] S. T. Dougherty, J. L. Kim and P. Sole, Open problems in coding theory, Contemp. Math., 634 (2015), 79-99.  doi: 10.1090/conm/634/12692. [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. [16] M. Gurel and N. Yankov, Self-dual codes with an automorphism of order 17, Math. Commun., 21 (2016), 97-107. [17] T. Hurley, Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335. [18] A. Kaya, B. 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. [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. [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. [21] E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000. [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. [23] 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. [24] A. Zhdanov, New self-dual codes of length 72, arXiv: 1705.05779. [25] A. Zhdanov, Convolutional encoding of 60, 64, 68, 72-bit self-dual codes, arXiv: 1702.05153.

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. [2] 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. [3] I. Bouyukliev, V. Fack and J. Winna, Hadamard matrices of order 36, European Conference on Combinatorics, Graph Theory and Applications, (2005), 93–98. [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. [5] 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. [6] S. T. Dougherty, J. 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. [7] 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. Commun., 14 (2020), 677-702.  doi: 10.3934/amc.2020037. [8] S. T. Dougherty, J. Gildea, A. 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. [9] S. T. Dougherty, J. Gildea, A. 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. [10] 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 Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7. [11] S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.  doi: 10.1109/18.641574. [12] 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. [13] S. T. Dougherty, J-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. [14] S. T. Dougherty, J. L. Kim and P. Sole, Open problems in coding theory, Contemp. Math., 634 (2015), 79-99.  doi: 10.1090/conm/634/12692. [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. [16] M. Gurel and N. Yankov, Self-dual codes with an automorphism of order 17, Math. Commun., 21 (2016), 97-107. [17] T. Hurley, Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335. [18] A. Kaya, B. 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. [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. [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. [21] E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000. [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. [23] 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. [24] A. Zhdanov, New self-dual codes of length 72, arXiv: 1705.05779. [25] A. Zhdanov, Convolutional encoding of 60, 64, 68, 72-bit self-dual codes, arXiv: 1702.05153.
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 $r_B$ $r_C$ $|Aut(C_i)|$ $C_1$ $[36,18,6]$ $(0,0,0,0,0,1,0,1,1)$ $(1,0,1,1,1,0,1,0,1)$ $2^2 \cdot 3^2$ $C_2$ $[36,18,6]$ $(0,0,0,0,1,1,0,1,1)$ $(1,0,0,1,0,1,1,1,0)$ $2^2 \cdot 3^2$ $C_3$ $[36,18,8]$ $(0,0,1,0,0,1,0,0,1)$ $(1,0,0,1,1,0,1,1,1)$ $2^2 \cdot 3^2$ $C_4$ $[36,18,8]$ $(0,1,0,0,0,1,0,1,1)$ $(1,0,0,1,0,0,1,1,1)$ $2^2 \cdot 3^2$
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 $r_B$ $r_C$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_1$ $W_{72,1}$ $(u,0,u,u,u,1,u,u + 1,1)$ $(1,0,1,1,u + 1,0,1,u,1)$ $0$ $192$ $36$ $\mathcal{C}_2$ $W_{72,1}$ $(u,0,0,u,0,1,u,1,1)$ $(1,0,u + 1,1,u + 1,0,1,u,u + 1)$ $0$ $198$ $36$ $\mathcal{C}_3$ $W_{72,1}$ $(u,u,0,u,u,1,u,1,u + 1)$ $(1,u,u + 1,1,u + 1,0,1,0,u + 1)$ $0$ $336$ $36$ $\mathcal{C}_4$ $W_{72,1}$ $(0,u,0,0,0,1,0,1,u + 1)$ $(1,u,u + 1,1,u + 1,0,1,0,u + 1)$ $18$ $234$ $36$ $\mathcal{C}_5$ $W_{72,1}$ $(u,u,0,u,0,1,u,u + 1,1)$ $(1,u,u + 1,1,1,u,1,0,u + 1)$ $18$ $345$ $36$ $\mathcal{C}_6$ $W_{72,1}$ $(0,u,0,0,0,1,0,u + 1,1)$ $(1,u,1,1,u + 1,u,1,u,1)$ $18$ $378$ $36$ $\mathcal{C}_7$ $W_{72,1}$ $(u,u,u,u,0,1,u,u + 1,u + 1)$ $(1,u,1,1,u + 1,0,1,0,1)$ $18$ $396$ $36$ $\mathcal{C}_8$ $W_{72,1}$ $(0,0,u,0,u,1,0,1,u + 1)$ $(1,u,1,1,1,0,1,u,1)$ $18$ $441$ $36$ $\mathcal{C}_9$ $W_{72,1}$ $(u,u,0,u,0,1,u,1,u + 1)$ $(1,u,1,1,1,0,1,u,1)$ $18$ $453$ $36$
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 $r_B$ $r_C$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_{10}$ $W_{72,1}$ $(0,u,0,0,1,1,0,1,u + 1)$ $(1,u,u,1,u,u + 1,1,1,0)$ $0$ $219$ $36$ $\mathcal{C}_{11}$ $W_{72,1}$ $(u,0,u,u,1,1,u,u + 1,u + 1)$ $(1,u,0,1,0,1,1,u + 1,u)$ $0$ $345$ $36$ $\mathcal{C}_{12}$ $W_{72,1}$ $(0,0,u,0,1,1,0,1,u + 1)$ $(1,u,0,1,0,1,1,1,0)$ $0$ $408$ $36$ $\mathcal{C}_{13}$ $W_{72,1}$ $(u,0,0,u,1,u + 1,u,u + 1,u + 1)$ $(1,0,u,1,0,u + 1,1,u + 1,0)$ $18$ $261$ $36$ $\mathcal{C}_{14}$ $W_{72,1}$ $(u,u,u,u,1,1,u,1,u + 1)$ $(1,0,0,1,u,1,1,u + 1,0)$ $18$ $270$ $36$ $\mathcal{C}_{15}$ $W_{72,1}$ $(u,0,u,u,1,u + 1,u,1,u + 1)$ $(1,0,u,1,u,1,1,u + 1,0)$ $18$ $357$ $36$
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 $r_B$ $r_C$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_{16}$ $W_{72,1}$ $(u,u,1,u,0,1,u,0,1)$ $(1,u,0,1,u + 1,u,1,1,u + 1)$ $0$ $120$ $36$ $\mathcal{C}_{17}$ $W_{72,1}$ $(u,u,1,u,0,1,u,u,1)$ $(1,0,u,1,1,0,1,1,u + 1)$ $0$ $282$ $36$ $\mathcal{C}_{18}$ $W_{72,1}$ $(u,u,1,u,0,u + 1,u,0,1)$ $(1,u,0,1,u + 1,u,1,1,u + 1)$ $0$ $300$ $36$ $\mathcal{C}_{19}$ $W_{72,1}$ $(u,u,1,u,0,u + 1,u,0,1)$ $(1,u,u,1,1,0,1,1,1)$ $18$ $336$ $36$ $\mathcal{C}_{20}$ $W_{72,1}$ $(u,0,1,u,0,1,u,u,1)$ $(1,0,0,1,1,0,1,u + 1,u + 1)$ $36$ $435$ $36$
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 $r_B$ $r_C$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_{21}$ $W_{72,1}$ $(0,1,u,0,u,1,0,u + 1,u + 1)$ $(1,u,u,1,u,u,1,u + 1,u + 1)$ $0$ $366$ $36$ $\mathcal{C}_{22}$ $W_{72,1}$ $(u,1,u,u,u,1,u,1,u + 1)$ $(1,u,0,1,0,0,1,u + 1,1)$ $0$ $372$ $36$ $\mathcal{C}_{23}$ $W_{72,1}$ $(0,1,u,0,u,1,0,u + 1,u + 1)$ $(1,0,0,1,0,0,1,u + 1,u + 1)$ $0$ $384$ $36$ $\mathcal{C}_{24}$ $W_{72,1}$ $(u,1,u,u,u,1,u,1,u + 1)$ $(1,u,u,1,u,0,1,u + 1,1)$ $0$ $390$ $36$ $\mathcal{C}_{25}$ $W_{72,1}$ $(u,1,0,u,0,1,u,u + 1,1)$ $(1,u,0,1,0,0,1,1,u + 1)$ $0$ $399$ $36$ $\mathcal{C}_{26}$ $W_{72,1}$ $(u,1,u,u,u,1,u,u + 1,1)$ $(1,0,u,1,0,0,1,u + 1,u + 1)$ $18$ $264$ $36$ $\mathcal{C}_{27}$ $W_{72,1}$ $(u,1,0,u,u,u + 1,u,u + 1,1)$ $(1,0,u,1,0,u,1,1,u + 1)$ $18$ $285$ $36$ $\mathcal{C}_{28}$ $W_{72,1}$ $(0,1,u,0,0,u + 1,0,1,1)$ $(1,u,u,1,u,0,1,1,1)$ $18$ $300$ $36$
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 $r_B$ $r_C$ $r_D$ $|Aut(C_i)|$ $C_5$ $[36,18,6]$ $(0,0,0,0,0,1)$ $(1,1,1,0,0,1)$ $(1,1,1,0,1,0)$ $2^5 \cdot 3^4 \cdot 5$ $C_6$ $[36,18,6]$ $(0,0,0,0,1,1)$ $(0,0,0,0,1,1)$ $(1,1,1,1,0,1)$ $2^5 \cdot 3^4 \cdot 5$ $C_7$ $[36,18,6]$ $(0,0,0,0,1,1)$ $(0,1,1,0,1,1)$ $(0,1,1,1,0,0)$ $2^5 \cdot 3^2$ $C_8$ $[36,18,6]$ $(0,0,1,0,0,1)$ $(0,0,1,1,1,0)$ $(1,1,1,0,0,1)$ $2^5 \cdot 3^2$
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 $r_B$ $r_C$ $r_D$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_{29}$ $W_{72,1}$ $(0,0,0,u,1,1)$ $(u,1,u + 1,u,1,1)$ $(u,u + 1,1,u + 1,0,0)$ $0$ $471$ $144$
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 $r_B$ $r_C$ $r_D$ $|Aut(C_i)|$ $C_9$ $[36,18,6]$ $(0,0,0,0,0,1)$ $(0,1,1,0,1,1)$ $(1,0,1,1,0,1)$ $2^5 \cdot 3^2$ $C_{10}$ $[36,18,6]$ $(0,0,0,0,0,1)$ $(1,1,1,0,0,1)$ $(1,1,1,0,1,0)$ $2^5 \cdot 3^4 \cdot 5$ $C_{11}$ $[36,18,6]$ $(0,0,0,0,1,1)$ $(0,0,0,0,1,1)$ $(1,1,1,1,0,1)$ $2^5 \cdot 3^4 \cdot 5$ $C_{12}$ $[36,18,6]$ $(0,0,0,0,1,1)$ $(0,1,1,0,0,1)$ $(1,1,0,1,0,1)$ $2^5 \cdot 3^2$
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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