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doi: 10.3934/amc.2022033
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## A novel genetic search scheme based on nature-inspired evolutionary algorithms for binary self-dual codes

 1 University of Chester, Department of Physical, Mathematical and Engineering Sciences University of Chester, England 2 Tarsus University, Faculty of Engineering, Department of Natural and Mathematical Sciences, Turkey 3 Tarsus University, Faculty of Engineering, Department of Computer Engineering, Turkey

* Corresponding author: Deniz Ustun

Received  December 2021 Revised  April 2022 Early access April 2022

In this paper, a genetic algorithm, one of the evolutionary algorithm optimization methods, is used for the first time for the problem of computing extremal binary self-dual codes. We present a comparison of the computational times between the genetic algorithm and a linear search for different size search spaces and show that the genetic algorithm is capable of computing binary self-dual codes significantly faster than the linear search. Moreover, by employing a known matrix construction together with the genetic algorithm, we are able to obtain new binary self-dual codes of lengths 68 and 72 in a significantly short time. In particular, we obtain 11 new binary self-dual codes of length 68 and 17 new binary self-dual codes of length 72.

Citation: Adrian Korban, Serap Şahinkaya, Deniz Ustun. A novel genetic search scheme based on nature-inspired evolutionary algorithms for binary self-dual codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2022033
##### References:
 [1] K. B. Ajitha Shenoy, S. Biswas and P. P. Kurur, Efficacy of the metropolis algorithm for the minimum-weight codeword problem using codeword and generator search spaces, IEEE Transactions on Evolutionary Computation, 24, (2020), 664–678 [2] P. E. Black, Big-O notation, dictionary of algorithms and data structures [online], U.S. National Institute of Standards and Technology, 2008 (accessed 26 November 2009). Available from: http://www.itl.nist.gov/div897/sqg/dads/HTML/bigOnotation.html [3] J. A. Bland, Local search optimisation applied to the minimum distance problem, Adv. Eng. Informat., 21 (2007), 391-397. [4] J. A. Bland and A. T. Baylis, A tabu search approach to the minimum distance of error-correcting codes, Int. J. Electron., 79 (1995), 829-837. [5] M. Bortos, J. Gildea, A. Kaya, A. Korban and A. Tylyshchak, New self-dual codes of length 68 from a $2 \times 2$ block matrix construction and group rings, Adv. Math. Commun., 16 (2022), 269-284.  doi: 10.3934/amc.2020111. [6] 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. [7] H. Bouzkraoui, A. Azouaoui and Y. Hadi, New ant Colony Optimization for Searching the Minimum Distance for Linear Codes, Advanced Communication Technologies and Networking (CommNet), International Conference on. IEEE, 2018. [8] I. Bouyukliev, V. Fack and J. Winna, Hadamard matrices of order 36, European Conference on Combinatorics, Graph Theory and Applications, (2005), 93–98. [9] 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. [10] S. Carbas, A. Toktas and D. Ustun, Nature-Inspired Metaheuristic Algorithms for Engineering Optimization Applications, 1, Springer Singapore, 2021. doi: 10.1007/978-981-33-6773-9. [11] N. Chen and Z. Yan, Complexity analysis of ReedSolomon decoding over GF(2m) without using syndromes, EURASIP Journal on Wireless Communications and Networking, Article ID 843634, 2008. doi: 10.1155/2008/843634. [12] R. Dontcheva, New binary self-dual $[70, 35, 12]$ and binary $[72, 36, 12]$ self-dual doubly-even codes, Serdica Math. J., 27 (2001), 287-302. [13] R. A. Dontcheva, A. J. van Zanten and S. M. Dodunekov, Binary self-dual codes with automorphism of composite order, IEEE Trans. Inform. Theory, 50 (2004), 311-318.  doi: 10.1109/TIT.2003.822598. [14] S. T. Dougherty, Combinatorics and Finite Geometry, Springer Undergraduate Mathematics Series (SUMS), 2020. doi: 10.1007/978-3-030-56395-0. [15] 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. [16] S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, New extremal self-dual codes of length 68 via composite construction, $\mathbb{F}_2+u\mathbb{F}_2$ lifts, extensions and neighbours, Int. J. Inf. Coding Theory, 5 (2018), 211-226. [17] 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. [18] 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. [19] S. T. Dougherty, J.-L. Kim and P. Solé, Double circulant codes from two class association schemes, Adv. Math. Commun., 1 (2007), 45-64.  doi: 10.3934/amc.2007.1.45. [20] J. Gildea, A. Kaya, A. Korban and A. Tylyshchak, Self-dual codes using bisymmetric matrices and group rings, Discrete Math., 343 (2020), 112085, 10 pp. doi: 10.1016/j.disc.2020.112085. [21] 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. [22] J. Gildea, A. Korban, A. Kaya and B. Yildiz, Constructing self-dual codes from group rings and reverse circulant matrices, Adv. Math. Commun., 15 (2021), 471-485.  doi: 10.3934/amc.2020077. [23] D. E. Goldberg, Genetic Algorithm in Search, Optimization, and Machine Learning, Addison Wesley Publishing Company, 1989. [24] D. E. Goldberg, Genetic and evolutionary algorithms come of age, Communications of the ACM, 37 (1994), 113-119. [25] T. A. Gulliver and M. Harada, Classification of extremal double circulant self-dual codes of lengths $64$ to $72$, Des. Codes Cryptogr., 13 (1998), 257-269.  doi: 10.1023/A:1008249924142. [26] T. A. Gulliver and M. Harada, On double circulant doubly even self-dual $[72, 36, 12]$ codes and their neighbors, Australas. J. Combin., 40 (2008), 137-144. [27] M. Gürel and N. Yankov, Self-dual codes with an automorphism of order 17, Math. Commun., 21 (2016), 97-101. [28] 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. [29] H. S. He, Forest land scape models: Definitions, characterization, and classification, Forest Ecologyand Management., 254 (2008), 484-498. [30] J. H. Holland, Adaptation in Natural und Artificial Systems, The University of Michigan Press, Ann Arbor, 1975. [31] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math., 31 (2006), 319-335. [32] A. Kaya and B. Yildiz, New extremal binary self-dual codes from a Baumert-Hall array, Discrete Appl. Math., 271 (2019), 74-83.  doi: 10.1016/j.dam.2019.08.003. [33] 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. [34] D. E. Knuth, Big omicron and big Omega and big theta, SIGACT News, 8 (1976), 18-24. [35] A. Korban, S. Sahinkaya and D. Ustun, A novel genetic search scheme based on nature-inspired evolutionary algorithms for binary self-dual codes, A Source Code for Magma of the Genetic Algorithm, available online at https://drive.google.com/file/d/11PhB7u92ti8OSKjXJv2aMpDzqGKloH5p/view?usp=sharing [36] F. Y. Kuo, Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces, Journal of Complexity, 19 (2003), 301-320.  doi: 10.1016/S0885-064X(03)00006-2. [37] E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000. [38] 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. [39] B. J. Waterhouse, F. Y. Kuo and I. H. Sloan, Randomly shifted lattice rules on the unit cube for unbounded integrands in high dimensions, J. Complexity, 22 (2006), 71-101.  doi: 10.1016/j.jco.2005.06.004. [40] N. Yankov, M. H. Lee, M. Gürel 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. [41] A. Zhdanov, New self-dual codes of length 72, arXiv: 1705.05779. [42] A. Zhdanov, Convolutional encoding of 60, 64, 68, 72-bit self-dual codes, arXiv: 1702.05153.

show all references

##### References:
 [1] K. B. Ajitha Shenoy, S. Biswas and P. P. Kurur, Efficacy of the metropolis algorithm for the minimum-weight codeword problem using codeword and generator search spaces, IEEE Transactions on Evolutionary Computation, 24, (2020), 664–678 [2] P. E. Black, Big-O notation, dictionary of algorithms and data structures [online], U.S. National Institute of Standards and Technology, 2008 (accessed 26 November 2009). Available from: http://www.itl.nist.gov/div897/sqg/dads/HTML/bigOnotation.html [3] J. A. Bland, Local search optimisation applied to the minimum distance problem, Adv. Eng. Informat., 21 (2007), 391-397. [4] J. A. Bland and A. T. Baylis, A tabu search approach to the minimum distance of error-correcting codes, Int. J. Electron., 79 (1995), 829-837. [5] M. Bortos, J. Gildea, A. Kaya, A. Korban and A. Tylyshchak, New self-dual codes of length 68 from a $2 \times 2$ block matrix construction and group rings, Adv. Math. Commun., 16 (2022), 269-284.  doi: 10.3934/amc.2020111. [6] 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. [7] H. Bouzkraoui, A. Azouaoui and Y. Hadi, New ant Colony Optimization for Searching the Minimum Distance for Linear Codes, Advanced Communication Technologies and Networking (CommNet), International Conference on. IEEE, 2018. [8] I. Bouyukliev, V. Fack and J. Winna, Hadamard matrices of order 36, European Conference on Combinatorics, Graph Theory and Applications, (2005), 93–98. [9] 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. [10] S. Carbas, A. Toktas and D. Ustun, Nature-Inspired Metaheuristic Algorithms for Engineering Optimization Applications, 1, Springer Singapore, 2021. doi: 10.1007/978-981-33-6773-9. [11] N. Chen and Z. Yan, Complexity analysis of ReedSolomon decoding over GF(2m) without using syndromes, EURASIP Journal on Wireless Communications and Networking, Article ID 843634, 2008. doi: 10.1155/2008/843634. [12] R. Dontcheva, New binary self-dual $[70, 35, 12]$ and binary $[72, 36, 12]$ self-dual doubly-even codes, Serdica Math. J., 27 (2001), 287-302. [13] R. A. Dontcheva, A. J. van Zanten and S. M. Dodunekov, Binary self-dual codes with automorphism of composite order, IEEE Trans. Inform. Theory, 50 (2004), 311-318.  doi: 10.1109/TIT.2003.822598. [14] S. T. Dougherty, Combinatorics and Finite Geometry, Springer Undergraduate Mathematics Series (SUMS), 2020. doi: 10.1007/978-3-030-56395-0. [15] 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. [16] S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, New extremal self-dual codes of length 68 via composite construction, $\mathbb{F}_2+u\mathbb{F}_2$ lifts, extensions and neighbours, Int. J. Inf. Coding Theory, 5 (2018), 211-226. [17] 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. [18] 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. [19] S. T. Dougherty, J.-L. Kim and P. Solé, Double circulant codes from two class association schemes, Adv. Math. Commun., 1 (2007), 45-64.  doi: 10.3934/amc.2007.1.45. [20] J. Gildea, A. Kaya, A. Korban and A. Tylyshchak, Self-dual codes using bisymmetric matrices and group rings, Discrete Math., 343 (2020), 112085, 10 pp. doi: 10.1016/j.disc.2020.112085. [21] 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. [22] J. Gildea, A. Korban, A. Kaya and B. Yildiz, Constructing self-dual codes from group rings and reverse circulant matrices, Adv. Math. Commun., 15 (2021), 471-485.  doi: 10.3934/amc.2020077. [23] D. E. Goldberg, Genetic Algorithm in Search, Optimization, and Machine Learning, Addison Wesley Publishing Company, 1989. [24] D. E. Goldberg, Genetic and evolutionary algorithms come of age, Communications of the ACM, 37 (1994), 113-119. [25] T. A. Gulliver and M. Harada, Classification of extremal double circulant self-dual codes of lengths $64$ to $72$, Des. Codes Cryptogr., 13 (1998), 257-269.  doi: 10.1023/A:1008249924142. [26] T. A. Gulliver and M. Harada, On double circulant doubly even self-dual $[72, 36, 12]$ codes and their neighbors, Australas. J. Combin., 40 (2008), 137-144. [27] M. Gürel and N. Yankov, Self-dual codes with an automorphism of order 17, Math. Commun., 21 (2016), 97-101. [28] 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. [29] H. S. He, Forest land scape models: Definitions, characterization, and classification, Forest Ecologyand Management., 254 (2008), 484-498. [30] J. H. Holland, Adaptation in Natural und Artificial Systems, The University of Michigan Press, Ann Arbor, 1975. [31] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math., 31 (2006), 319-335. [32] A. Kaya and B. Yildiz, New extremal binary self-dual codes from a Baumert-Hall array, Discrete Appl. Math., 271 (2019), 74-83.  doi: 10.1016/j.dam.2019.08.003. [33] 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. [34] D. E. Knuth, Big omicron and big Omega and big theta, SIGACT News, 8 (1976), 18-24. [35] A. Korban, S. Sahinkaya and D. Ustun, A novel genetic search scheme based on nature-inspired evolutionary algorithms for binary self-dual codes, A Source Code for Magma of the Genetic Algorithm, available online at https://drive.google.com/file/d/11PhB7u92ti8OSKjXJv2aMpDzqGKloH5p/view?usp=sharing [36] F. Y. Kuo, Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces, Journal of Complexity, 19 (2003), 301-320.  doi: 10.1016/S0885-064X(03)00006-2. [37] E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000. [38] 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. [39] B. J. Waterhouse, F. Y. Kuo and I. H. Sloan, Randomly shifted lattice rules on the unit cube for unbounded integrands in high dimensions, J. Complexity, 22 (2006), 71-101.  doi: 10.1016/j.jco.2005.06.004. [40] N. Yankov, M. H. Lee, M. Gürel 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. [41] A. Zhdanov, New self-dual codes of length 72, arXiv: 1705.05779. [42] A. Zhdanov, Convolutional encoding of 60, 64, 68, 72-bit self-dual codes, arXiv: 1702.05153.
The Initial Phase
The Crossover Phase
The Mutation Phase
Pseudo-code of the genetic algorithm
Flowchart of the genetic algorithm
An application of the genetic algorithm to the problem of computing binary self-dual codes of length 72 with minimum distance 12
Time Complexities of Genetic Algorithm and Linear Search for Binary Self-Dual Codes
 Algorithms Genetic Algorithm Linear Search Time Complexities $\mathcal{O}(NPm)=\mathcal{O}(200(500)m)$ $\mathcal{O}(L^m)=\mathcal{O}(2^{m})$
 Algorithms Genetic Algorithm Linear Search Time Complexities $\mathcal{O}(NPm)=\mathcal{O}(200(500)m)$ $\mathcal{O}(L^m)=\mathcal{O}(2^{m})$
New Type I $[68, 34, 12]$ Codes
 Type $r_A$ $r_B$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_{1}$ $W_{68, 2}$ $(0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1)$ $(1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0)$ $0$ $34$ $34$ $C_2$ $W_{68, 2}$ $(1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1)$ $(0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1)$ $0$ $51$ $34$ $C_3$ $W_{68, 2}$ $(1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0)$ $(1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1)$ $0$ $68$ $34$ $C_4$ $W_{68, 2}$ $(0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0)$ $(0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1)$ $0$ $85$ $34$ $C_5$ $W_{68, 2}$ $(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1)$ $(0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1)$ $0$ $119$ $34$ $C_6$ $W_{68, 2}$ $(0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1)$ $(0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0)$ $0$ $153$ $34$ $C_7$ $W_{68, 2}$ $(1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1)$ $(1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0)$ $0$ $136$ $34$ $C_{8}$ $W_{68, 2}$ $(0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1)$ $(1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1)$ $0$ $187$ $34$ $C_{9}$ $W_{68, 2}$ $(0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0)$ $(1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0)$ $0$ $221$ $34$ $C_{10}$ $W_{68, 2}$ $(0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0)$ $(0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1)$ $0$ $238$ $34$ $C_{11}$ $W_{68, 2}$ $(0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1)$ $(1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1)$ $0$ $255$ $34$
 Type $r_A$ $r_B$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_{1}$ $W_{68, 2}$ $(0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1)$ $(1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0)$ $0$ $34$ $34$ $C_2$ $W_{68, 2}$ $(1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1)$ $(0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1)$ $0$ $51$ $34$ $C_3$ $W_{68, 2}$ $(1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0)$ $(1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1)$ $0$ $68$ $34$ $C_4$ $W_{68, 2}$ $(0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0)$ $(0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1)$ $0$ $85$ $34$ $C_5$ $W_{68, 2}$ $(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1)$ $(0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1)$ $0$ $119$ $34$ $C_6$ $W_{68, 2}$ $(0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1)$ $(0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0)$ $0$ $153$ $34$ $C_7$ $W_{68, 2}$ $(1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1)$ $(1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0)$ $0$ $136$ $34$ $C_{8}$ $W_{68, 2}$ $(0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1)$ $(1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1)$ $0$ $187$ $34$ $C_{9}$ $W_{68, 2}$ $(0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0)$ $(1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0)$ $0$ $221$ $34$ $C_{10}$ $W_{68, 2}$ $(0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0)$ $(0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1)$ $0$ $238$ $34$ $C_{11}$ $W_{68, 2}$ $(0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1)$ $(1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1)$ $0$ $255$ $34$
New Type I $[72, 36, 12]$ Codes
 Type $r_A$ $r_B$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_1$ $W_{72, 1}$ $(1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0)$ $(0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0)$ $0$ $201$ $72$ $C_2$ $W_{72, 1}$ $(1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1)$ $(1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0)$ $36$ $471$ $72$
 Type $r_A$ $r_B$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_1$ $W_{72, 1}$ $(1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0)$ $(0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0)$ $0$ $201$ $72$ $C_2$ $W_{72, 1}$ $(1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1)$ $(1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0)$ $36$ $471$ $72$
New Type I $[72, 36, 12]$ Codes
 Type $r_A$ $r_B$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_3$ $W_{72, 1}$ $(1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0)$ $(1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1)$ $72$ $825$ $72$
 Type $r_A$ $r_B$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_3$ $W_{72, 1}$ $(1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0)$ $(1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1)$ $72$ $825$ $72$
New Type I $[72, 36, 12]$ Codes
 Type $r_A$ $r_B$ $r_C$ $r_D$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_4$ $W_{72, 1}$ $(0, 1, 0, 0, 1, 1, 0, 1, 0)$ $(1, 0, 0, 1, 1, 1, 1, 0, 0)$ $(0, 0, 0, 1, 0, 1, 1, 1, 0)$ $(0, 1, 1, 1, 0, 0, 1, 0, 0)$ $36$ $441$ $72$
 Type $r_A$ $r_B$ $r_C$ $r_D$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_4$ $W_{72, 1}$ $(0, 1, 0, 0, 1, 1, 0, 1, 0)$ $(1, 0, 0, 1, 1, 1, 1, 0, 0)$ $(0, 0, 0, 1, 0, 1, 1, 1, 0)$ $(0, 1, 1, 1, 0, 0, 1, 0, 0)$ $36$ $441$ $72$
New Type II $[72, 36, 12]$ Codes
 $r_A$ $r_B$ $r_C$ $r_D$ $\alpha$ $|Aut(C_i)|$ $C_5$ $(0, 0, 0, 1, 0, 1, 1, 1, 0)$ $(1, 0, 0, 1, 1, 0, 1, 1, 0)$ $(1, 1, 1, 1, 0, 0, 1, 0, 0)$ $(1, 0, 1, 0, 1, 1, 0, 1, 0)$ $-2772$ $72$
 $r_A$ $r_B$ $r_C$ $r_D$ $\alpha$ $|Aut(C_i)|$ $C_5$ $(0, 0, 0, 1, 0, 1, 1, 1, 0)$ $(1, 0, 0, 1, 1, 0, 1, 1, 0)$ $(1, 1, 1, 1, 0, 0, 1, 0, 0)$ $(1, 0, 1, 0, 1, 1, 0, 1, 0)$ $-2772$ $72$
New Type I $[72, 36, 12]$ Codes
 Type $r_A$ $r_B$ $r_C$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_{6}$ $W_{72, 1}$ $(1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0)$ $(1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0)$ $(0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1)$ $36$ $456$ $72$
 Type $r_A$ $r_B$ $r_C$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_{6}$ $W_{72, 1}$ $(1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0)$ $(1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0)$ $(0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1)$ $36$ $456$ $72$
New Type II $[72, 36, 12]$ Codes
 $r_A$ $r_B$ $r_C$ $\alpha$ $|Aut(C_i)|$ $C_7$ $(1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1)$ $(1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1)$ $(1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0)$ $-2106$ $144$ $C_8$ $(1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0)$ $(0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0)$ $(0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0)$ $-2322$ $144$ $C_9$ $(0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1)$ $(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1)$ $(0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0)$ $-2472$ $144$ $C_{10}$ $(0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0)$ $(0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0)$ $(0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0)$ $-2520$ $144$ $C_{11}$ $(0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0)$ $(0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0)$ $(0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0)$ $-2550$ $144$ $C_{12}$ $(1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0)$ $(1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0)$ $(0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0)$ $-3030$ $72$ $C_{13}$ $(0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0)$ $(0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0)$ $(1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1)$ $-3954$ $144$
 $r_A$ $r_B$ $r_C$ $\alpha$ $|Aut(C_i)|$ $C_7$ $(1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1)$ $(1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1)$ $(1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0)$ $-2106$ $144$ $C_8$ $(1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0)$ $(0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0)$ $(0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0)$ $-2322$ $144$ $C_9$ $(0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1)$ $(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1)$ $(0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0)$ $-2472$ $144$ $C_{10}$ $(0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0)$ $(0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0)$ $(0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0)$ $-2520$ $144$ $C_{11}$ $(0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0)$ $(0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0)$ $(0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0)$ $-2550$ $144$ $C_{12}$ $(1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0)$ $(1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0)$ $(0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0)$ $-3030$ $72$ $C_{13}$ $(0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0)$ $(0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0)$ $(1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1)$ $-3954$ $144$
New Type I $[72, 36, 12]$ Codes
 Type $r_A$ $r_B$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_{14}$ $W_{72, 1}$ $(1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1)$ $(0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1)$ $0$ $354$ $36$ $C_{15}$ $W_{72, 1}$ $(1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0)$ $(0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1)$ $18$ $273$ $36$ $C_{16}$ $W_{72, 1}$ $(0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0)$ $(0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0)$ $36$ $372$ $36$ $C_{17}$ $W_{72, 1}$ $(0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1)$ $(1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1)$ $54$ $669$ $36$
 Type $r_A$ $r_B$ $\gamma$ $\beta$ $|Aut(C_i)|$ $C_{14}$ $W_{72, 1}$ $(1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1)$ $(0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1)$ $0$ $354$ $36$ $C_{15}$ $W_{72, 1}$ $(1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0)$ $(0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1)$ $18$ $273$ $36$ $C_{16}$ $W_{72, 1}$ $(0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0)$ $(0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0)$ $36$ $372$ $36$ $C_{17}$ $W_{72, 1}$ $(0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1)$ $(1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1)$ $54$ $669$ $36$
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