# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022036
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## Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme

 1 University of Scranton, Department of Physical, Mathematical and Engineering Sciences, Scranton, PA, 18518, USA 2 University of Chester, Department of Physical, Mathematical and Engineering Sciences University of Chester, England 3 Tarsus University, Faculty of Engineering, Department of Natural and Mathematical Sciences, Turkey 4 Tarsus University, Faculty of Engineering, Department of Computer Engineering, Turkey

*Corresponding author: Serap Șahinkaya

Received  January 2022 Revised  April 2022 Early access May 2022

Fund Project: The third author is supported by TUBITAK grant no: 1059B192000947

We present a generator matrix of the form $[ \sigma(v_1) \ | \ \sigma(v_2)]$, where $v_1 \in RG$ and $v_2\in RH$, for finite groups $G$ and $H$ of order $n$ for constructing self-dual codes and linear complementary dual codes over the finite Frobenius ring $R$. In general, many of the constructions to produce self-dual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find self-dual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve $10$ of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain $82$ new binary linear complementary dual codes, $50$ of which are either optimal or near optimal of lengths $41 \leq n \leq 61$ which are new to the literature.

Citation: Steven Dougherty, Adrian Korban, Serap Șahinkaya, Deniz Ustun. Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme. Advances in Mathematics of Communications, doi: 10.3934/amc.2022036
##### References:
 [1] M. Araya and M. Harada, On the minimum weights of binary linear complementary dual codes, Cryptogr. Commun., 12 (2020), 285-300.  doi: 10.1007/s12095-019-00402-5. [2] M. Araya, M. Harada and K. Saito, Characterization and classification of optimal LCD codes, Des. Codes Cryptogr., 89 (2021), 617-640.  doi: 10.1007/s10623-020-00834-8. [3] 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. [4] S. Bouyuklieva, Optimal binary LCD codes, Des. Codes Cryptogr., 89 (2021), 2445-2461.  doi: 10.1007/s10623-021-00929-w. [5] R. Dontcheva, New binary self-dual $[70, 35, 12]$ and binary $[72, 36, 12]$ self-dual doubly-even codes, Serdica Math. J., 27 (2022), 287-302. [6] S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2. [7] S. T. Dougherty, J. Gildea and A. Kaya, $2^n$ Bordered constructions of self-dual codes from group rings, Finite Fields Appl., 67 (2020), 101692, 17 pp. doi: 10.1016/j.ffa.2020.101692. [8] S. T. Dougherty, J. Gildea, A. Kaya, A. Korban, A. Tylyshchak and B. Yildiz, Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes, Finite Fields Appl., 57 (2019), 108-127.  doi: 10.1016/j.ffa.2019.02.004. [9] S. T. Dougherty, J. Gildea, A. Kaya and B. Yildiz, New self-dual and formally self-dual codes from group ring constructions, Adv. Math. Commun., 14 (2020), 11-22.  doi: 10.3934/amc.2020002. [10] 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. [11] S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, 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. [12] 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 and Cryptog., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7. [13] 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. [14] S. T. Dougherty, J. L. Kim, B. Ozkaya, L. Sok and P. Sole, The combinatorics of LCD codes: Linear programming bound and orthogonal matrices, Int. J. Inf. Coding Theory, 4 (2017), 116-128.  doi: 10.1504/IJICOT.2017.083827. [15] S. T. Dougherty, J.-L. Kim and P. Sole, Double circulant codes from two class association schemes, Adv. Math. of Commun., 1 (2007), 45-64.  doi: 10.3934/amc.2007.1.45. [16] S. T. Dougherty, A. Korban, S. Sahinkaya and D. Ustun, Group matrix ring codes and constructions of self-dual codes, Appl. Algebra Engrg. Comm. Comput., (2021), https://doi.org/10.1007/s00200-021-00504-9. [17] L. Galvez, J.-L. Kim, N. Lee, Y. G. Roe and B.-S. Won, Some bounds on binary LCD codes, Cryptogr. Commun., 10 (2018), 719-728.  doi: 10.1007/s12095-017-0258-1. [18] 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. [19] J. Gildea, A. Korban, A. Kaya and B. Yildiz, Constructing self-dual codes from group rings and reverse circulant matrices, Adv. in Math. Commun., 15 (2021), 471-485.  doi: 10.3934/amc.2020077. [20] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, (2022), http://www.codetables.de. [21] 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. [22] K. Ishizuka, K. Saito, Construction for both self-dual codes and LCD codes, arXiv: 2108.12544. [23] 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. [24] A. Korban, S. Sahinkaya and D. Ustun, A novel genetic search scheme based on nature-inspired evolutionary algorithms for self-dual codes, Adv. Math. Commun.. [25] J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342.  doi: 10.1016/0012-365X(92)90563-U. [26] E. M. Rains and N. J. A. Sloane, The shadow theory of modular and unimodular lattices, J. Number Theory, 73 (1998), 359-389.  doi: 10.1006/jnth.1998.2306. [27] S. Sahinkaya, available online at https://sites.google.com/view/serap-sahinkaya/generator-matrices.

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##### References:
 [1] M. Araya and M. Harada, On the minimum weights of binary linear complementary dual codes, Cryptogr. Commun., 12 (2020), 285-300.  doi: 10.1007/s12095-019-00402-5. [2] M. Araya, M. Harada and K. Saito, Characterization and classification of optimal LCD codes, Des. Codes Cryptogr., 89 (2021), 617-640.  doi: 10.1007/s10623-020-00834-8. [3] 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. [4] S. Bouyuklieva, Optimal binary LCD codes, Des. Codes Cryptogr., 89 (2021), 2445-2461.  doi: 10.1007/s10623-021-00929-w. [5] R. Dontcheva, New binary self-dual $[70, 35, 12]$ and binary $[72, 36, 12]$ self-dual doubly-even codes, Serdica Math. J., 27 (2022), 287-302. [6] S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2. [7] S. T. Dougherty, J. Gildea and A. Kaya, $2^n$ Bordered constructions of self-dual codes from group rings, Finite Fields Appl., 67 (2020), 101692, 17 pp. doi: 10.1016/j.ffa.2020.101692. [8] S. T. Dougherty, J. Gildea, A. Kaya, A. Korban, A. Tylyshchak and B. Yildiz, Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes, Finite Fields Appl., 57 (2019), 108-127.  doi: 10.1016/j.ffa.2019.02.004. [9] S. T. Dougherty, J. Gildea, A. Kaya and B. Yildiz, New self-dual and formally self-dual codes from group ring constructions, Adv. Math. Commun., 14 (2020), 11-22.  doi: 10.3934/amc.2020002. [10] 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. [11] S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, 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. [12] 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 and Cryptog., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7. [13] 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. [14] S. T. Dougherty, J. L. Kim, B. Ozkaya, L. Sok and P. Sole, The combinatorics of LCD codes: Linear programming bound and orthogonal matrices, Int. J. Inf. Coding Theory, 4 (2017), 116-128.  doi: 10.1504/IJICOT.2017.083827. [15] S. T. Dougherty, J.-L. Kim and P. Sole, Double circulant codes from two class association schemes, Adv. Math. of Commun., 1 (2007), 45-64.  doi: 10.3934/amc.2007.1.45. [16] S. T. Dougherty, A. Korban, S. Sahinkaya and D. Ustun, Group matrix ring codes and constructions of self-dual codes, Appl. Algebra Engrg. Comm. Comput., (2021), https://doi.org/10.1007/s00200-021-00504-9. [17] L. Galvez, J.-L. Kim, N. Lee, Y. G. Roe and B.-S. Won, Some bounds on binary LCD codes, Cryptogr. Commun., 10 (2018), 719-728.  doi: 10.1007/s12095-017-0258-1. [18] 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. [19] J. Gildea, A. Korban, A. Kaya and B. Yildiz, Constructing self-dual codes from group rings and reverse circulant matrices, Adv. in Math. Commun., 15 (2021), 471-485.  doi: 10.3934/amc.2020077. [20] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, (2022), http://www.codetables.de. [21] 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. [22] K. Ishizuka, K. Saito, Construction for both self-dual codes and LCD codes, arXiv: 2108.12544. [23] 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. [24] A. Korban, S. Sahinkaya and D. Ustun, A novel genetic search scheme based on nature-inspired evolutionary algorithms for self-dual codes, Adv. Math. Commun.. [25] J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342.  doi: 10.1016/0012-365X(92)90563-U. [26] E. M. Rains and N. J. A. Sloane, The shadow theory of modular and unimodular lattices, J. Number Theory, 73 (1998), 359-389.  doi: 10.1006/jnth.1998.2306. [27] S. Sahinkaya, available online at https://sites.google.com/view/serap-sahinkaya/generator-matrices.
New Binary Type Ⅰ Codes of Length 72
 Generator Matrix Type $\gamma$ $\beta$ $|Aut(C_i)|$ $C_{1}$ $\mathcal{G}_{1}$ $W_{72, 1}$ $2$ $226$ $1$ $C_{2}$ $\mathcal{G}_{1}$ $W_{72, 1}$ $4$ $230$ $1$ $C_{3}$ $\mathcal{G}_{2}$ $W_{72, 1}$ $14$ $248$ $1$ $C_{4}$ $\mathcal{G}_{2}$ $W_{72, 1}$ $1$ $184$ $1$ $C_{5}$ $\mathcal{G}_{3}$ $W_{72, 1}$ $10$ $318$ $4$ $C_{6}$ $\mathcal{G}_{3}$ $W_{72, 1}$ $4$ $217$ $4$ $C_{7}$ $\mathcal{G}_{4}$ $W_{72, 1}$ $12$ $315$ $6$ $C_{8}$ $\mathcal{G}_{4}$ $W_{72, 1}$ $6$ $295$ $12$ $C_{9}$ $\mathcal{G}_{5}$ $W_{72, 1}$ $2$ $229$ $2$ $C_{10}$ $\mathcal{G}_{5}$ $W_{72, 1}$ $14$ $267$ $2$ $C_{11}$ $\mathcal{G}_{6}$ $W_{72, 1}$ $2$ $225$ $2$ $C_{12}$ $\mathcal{G}_{6}$ $W_{72, 1}$ $4$ $281$ $4$ $C_{13}$ $\mathcal{G}_{6}$ $W_{72, 1}$ $4$ $236$ $4$ $C_{14}$ $\mathcal{G}_{7}$ $W_{72, 1}$ $18$ $297$ $18$
 Generator Matrix Type $\gamma$ $\beta$ $|Aut(C_i)|$ $C_{1}$ $\mathcal{G}_{1}$ $W_{72, 1}$ $2$ $226$ $1$ $C_{2}$ $\mathcal{G}_{1}$ $W_{72, 1}$ $4$ $230$ $1$ $C_{3}$ $\mathcal{G}_{2}$ $W_{72, 1}$ $14$ $248$ $1$ $C_{4}$ $\mathcal{G}_{2}$ $W_{72, 1}$ $1$ $184$ $1$ $C_{5}$ $\mathcal{G}_{3}$ $W_{72, 1}$ $10$ $318$ $4$ $C_{6}$ $\mathcal{G}_{3}$ $W_{72, 1}$ $4$ $217$ $4$ $C_{7}$ $\mathcal{G}_{4}$ $W_{72, 1}$ $12$ $315$ $6$ $C_{8}$ $\mathcal{G}_{4}$ $W_{72, 1}$ $6$ $295$ $12$ $C_{9}$ $\mathcal{G}_{5}$ $W_{72, 1}$ $2$ $229$ $2$ $C_{10}$ $\mathcal{G}_{5}$ $W_{72, 1}$ $14$ $267$ $2$ $C_{11}$ $\mathcal{G}_{6}$ $W_{72, 1}$ $2$ $225$ $2$ $C_{12}$ $\mathcal{G}_{6}$ $W_{72, 1}$ $4$ $281$ $4$ $C_{13}$ $\mathcal{G}_{6}$ $W_{72, 1}$ $4$ $236$ $4$ $C_{14}$ $\mathcal{G}_{7}$ $W_{72, 1}$ $18$ $297$ $18$
LCD Codes, where $41 \leq n \leq 61$ and $15 \leq k \leq 30$
 Generator Matrix $[n, k, d]$ Optimal/Near Optimal Generator Matrix $[n, k, d]$ Optimal/Near Optimal $\mathcal{G}_8$ $[30, 15, 7]$ Near Optimal $\mathcal{G}_8$ $[32, 16, 8]$ Optimal Extended $\mathcal{G}_8$ $[33, 16, 8]$ Optimal Punctered $\mathcal{G}_8$ $[31, 16, 7]$ Near Optimal Shortened $\mathcal{G}_8$ $[31, 15, 8]$ Optimal $\mathcal{G}_8$ $[34, 17, 8]$ Optimal Punctered $\mathcal{G}_8$ $[33, 17, 7]$ Near Optimal Shortened $\mathcal{G}_8$ $[33, 16, 8]$ Optimal $\mathcal{G}_8$ $[36, 18, 8]$ Optimal Shortened $\mathcal{G}_8$ $[35, 17, 8]$ Optimal Extended $\mathcal{G}_8$ $[41, 16, 10]$ $-$ $\mathcal{G}_8$ $[42, 18, 10]$ - Extended $\mathcal{G}_8$ $[43, 18, 10]$ $-$ Punctered $\mathcal{G}_8$ $[41, 18, 9]$ - $\mathcal{G}_8$ $[42, 20, 10]$ Optimal Extended $\mathcal{G}_8$ $[43, 20, 10]$ Near Optimal Punctered $\mathcal{G}_8$ $[41, 20, 9]$ Near Optimal $\mathcal{G}_8$ $[42, 19, 9]$ - Punctered $\mathcal{G}_8$ $[41, 19, 8]$ - $\mathcal{G}_8$ $[42, 21, 9]$ Near Optimal Punctered $\mathcal{G}_8$ $[41, 21, 8]$ Near Optimal $\mathcal{G}_8$ $[44, 20, 10]$ - Extended $\mathcal{G}_8$ $[45, 20, 10]$ - Punctered $\mathcal{G}_8$ $[43, 20, 9]$ - $\mathcal{G}_8$ $[44, 22, 9]$ Near Optimal Extended $\mathcal{G}_8$ $[45, 22, 10]$ Near Optimal Punctered $\mathcal{G}_8$ $[43, 22, 8]$ Near Optimal Shortened $\mathcal{G}_8$ $[43, 21, 9]$ Near Optimal $\mathcal{G}_9$ $[46, 22, 10]$ - Extended $\mathcal{G}_8$ $[47, 22, 10]$ - Punctered $\mathcal{G}_8$ $[45, 22, 9]$ - $\mathcal{G}_8$ $[46, 23, 9]$ - Punctered $\mathcal{G}_8$ $[45, 23, 8]$ - $\mathcal{G}_8$ $[48, 24, 9]$ - Extended $\mathcal{G}_8$ $[49, 24, 10]$ - Punctered $\mathcal{G}_8$ $[47, 24, 8]$ - Shortened $\mathcal{G}_8$ $[47, 23, 9]$ - $\mathcal{G}_8$ $[50, 21, 12]$ Optimal Punctered $\mathcal{G}_8$ $[49, 21, 11]$ Optimal Shortened $\mathcal{G}_8$ $[49, 20, 12]$ Optimal $\mathcal{G}_8$ $[50, 24, 10]$ - Extended $\mathcal{G}_8$ $[51, 24, 10]$ - Punctered $\mathcal{G}_8$ $[49, 24, 9]$ - $\mathcal{G}_8$ $[50, 25, 10]$ Optimal Punctered $\mathcal{G}_8$ $[49, 25, 9]$ Near Optimal Shortened $\mathcal{G}_8$ $[49, 24, 10]$ - $\mathcal{G}_8$ $[52, 24, 12]$ Optimal Extended $\mathcal{G}_8$ $[53, 24, 12]$ Optimal Punctered $\mathcal{G}_8$ $[51, 24, 11]$ Optimal $\mathcal{G}_8$ $[52, 26, 10]$ Optimal Extended $\mathcal{G}_8$ $[53, 26, 10]$ Near Optimal Punctered $\mathcal{G}_9$ $[51, 26, 9]$ Near Optimal Shortened $\mathcal{G}_8$ $[51, 25, 10]$ Near Optimal $\mathcal{G}_8$ $[54, 26, 12]$ Optimal Extended $\mathcal{G}_8$ $[55, 26, 12]$ Optimal Punctered $\mathcal{G}_8$ $[53, 26, 11]$ Optimal $\mathcal{G}_8$ $[54, 24, 12]$ Optimal Extended $\mathcal{G}_8$ $[55, 24, 12]$ Optimal Punctered $\mathcal{G}_8$ $[53, 24, 11]$ Near Optimal $\mathcal{G}_8$ $[54, 25, 11]$ Near Optimal Punctered $\mathcal{G}_8$ $[53, 25, 10]$ - Shortened $\mathcal{G}_8$ $[53, 24, 11]$ Near Optimal $\mathcal{G}_8$ $[54, 27, 10]$ Near Optimal Punctered $\mathcal{G}_8$ $[53, 27, 9]$ Near Optimal Shortened $\mathcal{G}_8$ $[53, 26, 10]$ Near Optimal $\mathcal{G}_8$ $[56, 24, 12]$ Optimal Extended $\mathcal{G}_8$ $[57, 24, 12]$ Near Optimal Punctered $\mathcal{G}_8$ $[55, 24, 11]$ Near Optimal $\mathcal{G}_8$ $[56, 28, 11]$ Near Optimal Extended $\mathcal{G}_8$ $[57, 28, 12]$ Optimal Punctered $\mathcal{G}_8$ $[55, 28, 10]$ Near Optimal Shortened $\mathcal{G}_8$ $[55, 27, 11]$ Near Optimal $\mathcal{G}_8$ $[58, 28, 12]$ Optimal Extended $\mathcal{G}_8$ $[59, 28, 12]$ Optimal Punctered $\mathcal{G}_8$ $[57, 28, 11]$ Optimal $\mathcal{G}_8$ $[58, 29, 11]$ Near Optimal Punctered $\mathcal{G}_8$ $[57, 29, 10]$ - Shortened $\mathcal{G}_8$ $[57, 28, 11]$ Near Optimal $\mathcal{G}_8$ $[60, 28, 12]$ Optimal Extended $\mathcal{G}_8$ $[61, 28, 12]$ Near Optimal Punctered $\mathcal{G}_8$ $[59, 28, 11]$ Near Optimal $\mathcal{G}_8$ $[60, 30, 11]$ Near Optimal Extended $\mathcal{G}_8$ $[61, 30, 12]$ Optimal Punctured $\mathcal{G}_8$ $[59, 30, 10]$ - Shortened $\mathcal{G}_8$ $[59, 29, 11]$ Near Optimal $\mathcal{G}_8$ $[60, 26, 11]$ - Extended $\mathcal{G}_8$ $[61, 26, 12]$ - Punctered $\mathcal{G}_8$ $[59, 26, 10]$ - Shortened $\mathcal{G}_8$ $[59, 25, 11]$ - $\mathcal{G}_9$ $[60, 24, 10]$ - Extended $\mathcal{G}_9$ $[61, 24, 10]$ - Punctered $\mathcal{G}_9$ $[59, 24, 10]$ -
 Generator Matrix $[n, k, d]$ Optimal/Near Optimal Generator Matrix $[n, k, d]$ Optimal/Near Optimal $\mathcal{G}_8$ $[30, 15, 7]$ Near Optimal $\mathcal{G}_8$ $[32, 16, 8]$ Optimal Extended $\mathcal{G}_8$ $[33, 16, 8]$ Optimal Punctered $\mathcal{G}_8$ $[31, 16, 7]$ Near Optimal Shortened $\mathcal{G}_8$ $[31, 15, 8]$ Optimal $\mathcal{G}_8$ $[34, 17, 8]$ Optimal Punctered $\mathcal{G}_8$ $[33, 17, 7]$ Near Optimal Shortened $\mathcal{G}_8$ $[33, 16, 8]$ Optimal $\mathcal{G}_8$ $[36, 18, 8]$ Optimal Shortened $\mathcal{G}_8$ $[35, 17, 8]$ Optimal Extended $\mathcal{G}_8$ $[41, 16, 10]$ $-$ $\mathcal{G}_8$ $[42, 18, 10]$ - Extended $\mathcal{G}_8$ $[43, 18, 10]$ $-$ Punctered $\mathcal{G}_8$ $[41, 18, 9]$ - $\mathcal{G}_8$ $[42, 20, 10]$ Optimal Extended $\mathcal{G}_8$ $[43, 20, 10]$ Near Optimal Punctered $\mathcal{G}_8$ $[41, 20, 9]$ Near Optimal $\mathcal{G}_8$ $[42, 19, 9]$ - Punctered $\mathcal{G}_8$ $[41, 19, 8]$ - $\mathcal{G}_8$ $[42, 21, 9]$ Near Optimal Punctered $\mathcal{G}_8$ $[41, 21, 8]$ Near Optimal $\mathcal{G}_8$ $[44, 20, 10]$ - Extended $\mathcal{G}_8$ $[45, 20, 10]$ - Punctered $\mathcal{G}_8$ $[43, 20, 9]$ - $\mathcal{G}_8$ $[44, 22, 9]$ Near Optimal Extended $\mathcal{G}_8$ $[45, 22, 10]$ Near Optimal Punctered $\mathcal{G}_8$ $[43, 22, 8]$ Near Optimal Shortened $\mathcal{G}_8$ $[43, 21, 9]$ Near Optimal $\mathcal{G}_9$ $[46, 22, 10]$ - Extended $\mathcal{G}_8$ $[47, 22, 10]$ - Punctered $\mathcal{G}_8$ $[45, 22, 9]$ - $\mathcal{G}_8$ $[46, 23, 9]$ - Punctered $\mathcal{G}_8$ $[45, 23, 8]$ - $\mathcal{G}_8$ $[48, 24, 9]$ - Extended $\mathcal{G}_8$ $[49, 24, 10]$ - Punctered $\mathcal{G}_8$ $[47, 24, 8]$ - Shortened $\mathcal{G}_8$ $[47, 23, 9]$ - $\mathcal{G}_8$ $[50, 21, 12]$ Optimal Punctered $\mathcal{G}_8$ $[49, 21, 11]$ Optimal Shortened $\mathcal{G}_8$ $[49, 20, 12]$ Optimal $\mathcal{G}_8$ $[50, 24, 10]$ - Extended $\mathcal{G}_8$ $[51, 24, 10]$ - Punctered $\mathcal{G}_8$ $[49, 24, 9]$ - $\mathcal{G}_8$ $[50, 25, 10]$ Optimal Punctered $\mathcal{G}_8$ $[49, 25, 9]$ Near Optimal Shortened $\mathcal{G}_8$ $[49, 24, 10]$ - $\mathcal{G}_8$ $[52, 24, 12]$ Optimal Extended $\mathcal{G}_8$ $[53, 24, 12]$ Optimal Punctered $\mathcal{G}_8$ $[51, 24, 11]$ Optimal $\mathcal{G}_8$ $[52, 26, 10]$ Optimal Extended $\mathcal{G}_8$ $[53, 26, 10]$ Near Optimal Punctered $\mathcal{G}_9$ $[51, 26, 9]$ Near Optimal Shortened $\mathcal{G}_8$ $[51, 25, 10]$ Near Optimal $\mathcal{G}_8$ $[54, 26, 12]$ Optimal Extended $\mathcal{G}_8$ $[55, 26, 12]$ Optimal Punctered $\mathcal{G}_8$ $[53, 26, 11]$ Optimal $\mathcal{G}_8$ $[54, 24, 12]$ Optimal Extended $\mathcal{G}_8$ $[55, 24, 12]$ Optimal Punctered $\mathcal{G}_8$ $[53, 24, 11]$ Near Optimal $\mathcal{G}_8$ $[54, 25, 11]$ Near Optimal Punctered $\mathcal{G}_8$ $[53, 25, 10]$ - Shortened $\mathcal{G}_8$ $[53, 24, 11]$ Near Optimal $\mathcal{G}_8$ $[54, 27, 10]$ Near Optimal Punctered $\mathcal{G}_8$ $[53, 27, 9]$ Near Optimal Shortened $\mathcal{G}_8$ $[53, 26, 10]$ Near Optimal $\mathcal{G}_8$ $[56, 24, 12]$ Optimal Extended $\mathcal{G}_8$ $[57, 24, 12]$ Near Optimal Punctered $\mathcal{G}_8$ $[55, 24, 11]$ Near Optimal $\mathcal{G}_8$ $[56, 28, 11]$ Near Optimal Extended $\mathcal{G}_8$ $[57, 28, 12]$ Optimal Punctered $\mathcal{G}_8$ $[55, 28, 10]$ Near Optimal Shortened $\mathcal{G}_8$ $[55, 27, 11]$ Near Optimal $\mathcal{G}_8$ $[58, 28, 12]$ Optimal Extended $\mathcal{G}_8$ $[59, 28, 12]$ Optimal Punctered $\mathcal{G}_8$ $[57, 28, 11]$ Optimal $\mathcal{G}_8$ $[58, 29, 11]$ Near Optimal Punctered $\mathcal{G}_8$ $[57, 29, 10]$ - Shortened $\mathcal{G}_8$ $[57, 28, 11]$ Near Optimal $\mathcal{G}_8$ $[60, 28, 12]$ Optimal Extended $\mathcal{G}_8$ $[61, 28, 12]$ Near Optimal Punctered $\mathcal{G}_8$ $[59, 28, 11]$ Near Optimal $\mathcal{G}_8$ $[60, 30, 11]$ Near Optimal Extended $\mathcal{G}_8$ $[61, 30, 12]$ Optimal Punctured $\mathcal{G}_8$ $[59, 30, 10]$ - Shortened $\mathcal{G}_8$ $[59, 29, 11]$ Near Optimal $\mathcal{G}_8$ $[60, 26, 11]$ - Extended $\mathcal{G}_8$ $[61, 26, 12]$ - Punctered $\mathcal{G}_8$ $[59, 26, 10]$ - Shortened $\mathcal{G}_8$ $[59, 25, 11]$ - $\mathcal{G}_9$ $[60, 24, 10]$ - Extended $\mathcal{G}_9$ $[61, 24, 10]$ - Punctered $\mathcal{G}_9$ $[59, 24, 10]$ -
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