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. ArayaM. 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. 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.

[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. DoughertyJ. GildeaA. KayaA. KorbanA. 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. DoughertyJ. GildeaA. 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. 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.

[11]

S. T. DoughertyJ. GildeaA. 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. DoughertyJ. GildeaR. 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. DoughertyT. 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. DoughertyJ. L. KimB. OzkayaL. 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. DoughertyJ.-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. GalvezJ.-L. KimN. LeeY. 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. GildeaA. KayaR. 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. GildeaA. KorbanA. 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.

show all references

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. ArayaM. 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. 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.

[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. DoughertyJ. GildeaA. KayaA. KorbanA. 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. DoughertyJ. GildeaA. 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. 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.

[11]

S. T. DoughertyJ. GildeaA. 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. DoughertyJ. GildeaR. 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. DoughertyT. 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. DoughertyJ. L. KimB. OzkayaL. 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. DoughertyJ.-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. GalvezJ.-L. KimN. LeeY. 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. GildeaA. KayaR. 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. GildeaA. KorbanA. 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.

Table 1.  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 $
Table 2.  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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