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August  2021, 15(3): 471-485. doi: 10.3934/amc.2020077

Constructing self-dual codes from group rings and reverse circulant matrices

1. 

University of Chester, Department of Mathematical and Physical Sciences, Thornton Science Park, Pool Ln, Chester CH2 4NU, England

2. 

Sampoerna University, Department of Engineering Fundamentals, 12780, Jakarta, Indonesia

3. 

Northern Arizona University, Department of Mathematics & Statistics, Flagstaff, AZ 86001, USA

* Corresponding author: Adrian Korban

Received  October 2019 Revised  January 2020 Published  April 2020

In this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary self-dual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92.

Citation: Joe Gildea, Adrian Korban, Abidin Kaya, Bahattin Yildiz. Constructing self-dual codes from group rings and reverse circulant matrices. Advances in Mathematics of Communications, 2021, 15 (3) : 471-485. doi: 10.3934/amc.2020077
References:
[1]

K. BetsumiyaS. GeorgiouT. A. GulliverM. Harada and C. Koukouvinos, On self-dual codes over some prime fields, Discrete Math., 262 (2003), 37-58.  doi: 10.1016/S0012-365X(02)00520-4.  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]

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.  Google Scholar

[4]

J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.  doi: 10.1109/18.59931.  Google Scholar

[5]

G. Dorfer and H. Maharaj, Generalized AG codes and generalized duality, Finite Fields Appl., 9 (2003), 194-210.  doi: 10.1016/S1071-5797(02)00027-8.  Google Scholar

[6]

S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings, Springer Briefs in Mathematics. Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.  Google Scholar

[7]

S. T. DoughertyP. GaboritM. Harada and P. Solé, 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

[8]

S. T. Dougherty, J. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings over Frobenius rings, Cryptogr. Commun., 12 (2019), 127–146. https://doi.org/10.1007/s12095-019-00380-8. doi: 10.1007/s12095-019-00380-8.  Google Scholar

[9]

S. T. DoughertyJ. GildeaA. KorbanA. KayaA. Tylshchak 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.  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 Crypt., 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. DoughertyJ. L. KimH. Kulosman and H. W. Liu, Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 14-26.  doi: 10.1016/j.ffa.2009.11.004.  Google Scholar

[13]

P. GaboritV. PlessP. Solé and O. Atkin, Type Ⅱ codes over $\mathbb{F}_4$, Finite Fields Appl., 8 (2002), 171-183.  doi: 10.1006/ffta.2001.0333.  Google Scholar

[14]

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.  Google Scholar

[15]

J. Gildea, A. Kaya, A. Tylyshchak and B. Yildiz, A group induced four-circulant construction for self-dual codes and new extremal binary self-dual codes, Available online at: https://arXiv.org/abs/1912.11758. Google Scholar

[16]

J. Gildea, A. Kaya and B. Yildiz, New binary self-dual codes via a generalization of the four circulant construction, Available online at: https://arXiv.org/abs/1912.11754. Google Scholar

[17]

J. Gildea, A. Korban, A. Kaya and B. Yildiz, Binary generator matrices of new self-dual binary codes of lengths 64, 68, 80 and 92, available online at http://abidinkaya.wix.com/math/adrian. Google Scholar

[18]

T. A. Gulliver and M. Harada, Classification of extremal double circulant self-dual codes of lengths 74-88, Discr. Math., 306 (2006), 2064-2072.  doi: 10.1016/j.disc.2006.05.004.  Google Scholar

[19]

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.  Google Scholar

[20]

M. Harada and T. Nishimura, An extremal singly even self-dual codes of length 88, Advances in Mathematics of Communications, 1 (2007), 261-267.  doi: 10.3934/amc.2007.1.261.  Google Scholar

[21]

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

[22]

A. Kaya, New extremal binary self-dual codes of lengths 64 and 66 from $R_{2}$-lifts, Finite Fields Appl., 46 (2017), 271-279.  doi: 10.1016/j.ffa.2017.04.003.  Google Scholar

[23]

S. Ling and P. Solé, Type Ⅱ codes over $\mathbb{F}_4+u \mathbb{F}_4$, Europ. J. Combinatorics, 22 (2001), 983-997.  doi: 10.1006/eujc.2001.0509.  Google Scholar

[24]

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

[25]

N. YankovD. Anev and M. Gürel, Self-dual codes with an automorphism of order 13, Advances in Mathematics of Communications, 11 (2017), 635-645.  doi: 10.3934/amc.2017047.  Google Scholar

[26]

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

[27]

N. YankovM. Ivanova and M. H. Lee, Self-dual codes with an automorphism of order 7 and $s$-extremal codes of length 68, Finite Fields Appl., 51 (2018), 17-30.  doi: 10.1016/j.ffa.2017.12.001.  Google Scholar

[28]

N. Yankov and D. Anev, On the self-dual codes with an automorphism of order 5, AAECC, (2019). https://doi.org/10.1007/s00200-019-00403-0. Google Scholar

show all references

References:
[1]

K. BetsumiyaS. GeorgiouT. A. GulliverM. Harada and C. Koukouvinos, On self-dual codes over some prime fields, Discrete Math., 262 (2003), 37-58.  doi: 10.1016/S0012-365X(02)00520-4.  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]

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.  Google Scholar

[4]

J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.  doi: 10.1109/18.59931.  Google Scholar

[5]

G. Dorfer and H. Maharaj, Generalized AG codes and generalized duality, Finite Fields Appl., 9 (2003), 194-210.  doi: 10.1016/S1071-5797(02)00027-8.  Google Scholar

[6]

S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings, Springer Briefs in Mathematics. Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.  Google Scholar

[7]

S. T. DoughertyP. GaboritM. Harada and P. Solé, 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

[8]

S. T. Dougherty, J. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings over Frobenius rings, Cryptogr. Commun., 12 (2019), 127–146. https://doi.org/10.1007/s12095-019-00380-8. doi: 10.1007/s12095-019-00380-8.  Google Scholar

[9]

S. T. DoughertyJ. GildeaA. KorbanA. KayaA. Tylshchak 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.  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 Crypt., 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. DoughertyJ. L. KimH. Kulosman and H. W. Liu, Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 14-26.  doi: 10.1016/j.ffa.2009.11.004.  Google Scholar

[13]

P. GaboritV. PlessP. Solé and O. Atkin, Type Ⅱ codes over $\mathbb{F}_4$, Finite Fields Appl., 8 (2002), 171-183.  doi: 10.1006/ffta.2001.0333.  Google Scholar

[14]

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.  Google Scholar

[15]

J. Gildea, A. Kaya, A. Tylyshchak and B. Yildiz, A group induced four-circulant construction for self-dual codes and new extremal binary self-dual codes, Available online at: https://arXiv.org/abs/1912.11758. Google Scholar

[16]

J. Gildea, A. Kaya and B. Yildiz, New binary self-dual codes via a generalization of the four circulant construction, Available online at: https://arXiv.org/abs/1912.11754. Google Scholar

[17]

J. Gildea, A. Korban, A. Kaya and B. Yildiz, Binary generator matrices of new self-dual binary codes of lengths 64, 68, 80 and 92, available online at http://abidinkaya.wix.com/math/adrian. Google Scholar

[18]

T. A. Gulliver and M. Harada, Classification of extremal double circulant self-dual codes of lengths 74-88, Discr. Math., 306 (2006), 2064-2072.  doi: 10.1016/j.disc.2006.05.004.  Google Scholar

[19]

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.  Google Scholar

[20]

M. Harada and T. Nishimura, An extremal singly even self-dual codes of length 88, Advances in Mathematics of Communications, 1 (2007), 261-267.  doi: 10.3934/amc.2007.1.261.  Google Scholar

[21]

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

[22]

A. Kaya, New extremal binary self-dual codes of lengths 64 and 66 from $R_{2}$-lifts, Finite Fields Appl., 46 (2017), 271-279.  doi: 10.1016/j.ffa.2017.04.003.  Google Scholar

[23]

S. Ling and P. Solé, Type Ⅱ codes over $\mathbb{F}_4+u \mathbb{F}_4$, Europ. J. Combinatorics, 22 (2001), 983-997.  doi: 10.1006/eujc.2001.0509.  Google Scholar

[24]

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

[25]

N. YankovD. Anev and M. Gürel, Self-dual codes with an automorphism of order 13, Advances in Mathematics of Communications, 11 (2017), 635-645.  doi: 10.3934/amc.2017047.  Google Scholar

[26]

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

[27]

N. YankovM. Ivanova and M. H. Lee, Self-dual codes with an automorphism of order 7 and $s$-extremal codes of length 68, Finite Fields Appl., 51 (2018), 17-30.  doi: 10.1016/j.ffa.2017.12.001.  Google Scholar

[28]

N. Yankov and D. Anev, On the self-dual codes with an automorphism of order 5, AAECC, (2019). https://doi.org/10.1007/s00200-019-00403-0. Google Scholar

Table 1.  Self-dual codes over $\mathbb{F}_{4}+u\mathbb{F}_{4}$ of length $64$ from $C_{2, 2}$
$\mathcal{C}_{i}$ $r_{\sigma(v_1)}$ $r_{\sigma(v_2)}$ $r_C$ $|Aut(\mathcal{C}_i)|$ $\beta$
$1$ $(0, 9, 2, 1)$ $(0, 0, A, 4)$ $(3, C, 3, 3)$ $2^{4}$ $0$
$2$ $(0, 9, 4, F)$ $(0, 0, 0, 6)$ $(2, 5, 2, 2)$ $2^{5}$ $0$
$\mathcal{C}_{i}$ $r_{\sigma(v_1)}$ $r_{\sigma(v_2)}$ $r_C$ $|Aut(\mathcal{C}_i)|$ $\beta$
$1$ $(0, 9, 2, 1)$ $(0, 0, A, 4)$ $(3, C, 3, 3)$ $2^{4}$ $0$
$2$ $(0, 9, 4, F)$ $(0, 0, 0, 6)$ $(2, 5, 2, 2)$ $2^{5}$ $0$
Table 2.  Extremal Self-dual codes of length $68$ from Theorem 2.1
$\mathcal{M}_{68, i}$ $\mathcal{C}_i$ $c$ $X$ $\gamma$ $\beta$ $|Aut(\mathcal{M}_{68, i})|$
$1$ $1$ $1$ $(3u0u01u103103030u01u0u0301u10013)$ $\bf{0}$ $\bf{40}$ $2$
$2$ $2$ $u+1$ $(33313311u3uu13110u1030u1u31u31u3)$ $\bf{3}$ $\bf{77}$ $2$
$\mathcal{M}_{68, i}$ $\mathcal{C}_i$ $c$ $X$ $\gamma$ $\beta$ $|Aut(\mathcal{M}_{68, i})|$
$1$ $1$ $1$ $(3u0u01u103103030u01u0u0301u10013)$ $\bf{0}$ $\bf{40}$ $2$
$2$ $2$ $u+1$ $(33313311u3uu13110u1030u1u31u31u3)$ $\bf{3}$ $\bf{77}$ $2$
Table 3.  Self-dual codes over $ \mathbb{F}_{2}+u\mathbb{F}_{2} $ of length $ 64 $ from $ C_{4,2} $
$ \mathcal{E}_{i} $ $ r_{\sigma(v_1)} $ $ r_{\sigma(v_2)} $ $ r_C $ $ |Aut(\mathcal{E}_i)| $ $ \beta $
$ 1 $ $ (u,0,1,1,u,u,u,u) $ $ (u,u,1,1,0,u,u,3) $ $ (0,1,0,1,0,1,0,1) $ $ 2^5 $ $ 0 $
$ 2 $ $ (u,u,1,3,u,u,u,u) $ $ (u,u,1,1,u,u,0,3) $ $ (u,3,u,3,u,3,u,3) $ $ 2^6 $ $ 0 $
$ 3 $ $ (u,0,u,u,0,0,1,3) $ $ (u,u,0,1,u,0,3,1) $ $ (3,3,3,3,3,3,3,3) $ $ 2^7 $ $ 80 $
$ \mathcal{E}_{i} $ $ r_{\sigma(v_1)} $ $ r_{\sigma(v_2)} $ $ r_C $ $ |Aut(\mathcal{E}_i)| $ $ \beta $
$ 1 $ $ (u,0,1,1,u,u,u,u) $ $ (u,u,1,1,0,u,u,3) $ $ (0,1,0,1,0,1,0,1) $ $ 2^5 $ $ 0 $
$ 2 $ $ (u,u,1,3,u,u,u,u) $ $ (u,u,1,1,u,u,0,3) $ $ (u,3,u,3,u,3,u,3) $ $ 2^6 $ $ 0 $
$ 3 $ $ (u,0,u,u,0,0,1,3) $ $ (u,u,0,1,u,0,3,1) $ $ (3,3,3,3,3,3,3,3) $ $ 2^7 $ $ 80 $
Table 4.  New codes of length 64 as neighbors
$ \mathcal{L}_{64,i} $ $ \mathcal{E}_{i} $ $ (x_{33},...,x_{64}) $ $ W_{64,i} $ $ \beta $ $ |Aut(\mathcal{L}_{64,i})| $
$ 1 $ $ 3 $ $ (01110001001001101000011001011111) $ $ 1 $ $ \bf{58} $ $ 2^2 $
$ 2 $ $ 3 $ $ (11011001001101110110010110011010) $ $ 2 $ $ \bf{54} $ $ 2^3 $
$ 3 $ $ 3 $ $ (11111100101011111001111001010010) $ $ 2 $ $ \bf{62} $ $ 2 $
$ \mathcal{L}_{64,i} $ $ \mathcal{E}_{i} $ $ (x_{33},...,x_{64}) $ $ W_{64,i} $ $ \beta $ $ |Aut(\mathcal{L}_{64,i})| $
$ 1 $ $ 3 $ $ (01110001001001101000011001011111) $ $ 1 $ $ \bf{58} $ $ 2^2 $
$ 2 $ $ 3 $ $ (11011001001101110110010110011010) $ $ 2 $ $ \bf{54} $ $ 2^3 $
$ 3 $ $ 3 $ $ (11111100101011111001111001010010) $ $ 2 $ $ \bf{62} $ $ 2 $
Table 5.  Extremal Self-dual codes of length $ 68 $ from Theorem 2.1
$ \mathcal{N}_{68,i} $ $ \mathcal{E}_i $ $ c $ $ X $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{N}_{68,i})| $
$ 1 $ $ 1 $ $ 3 $ $ (01330u3131uuu3330uuuu000333u1u1u) $ $ \bf{0} $ $ \bf{39} $ $ 2 $
$ 2 $ $ 2 $ $ 1 $ $ (0013u1111uu1u0uuuu101u1333330130) $ $ \bf{3} $ $ \bf{79} $ $ 2 $
$ 3 $ $ 2 $ $ 1 $ $ (u30u1u03u10uu113uuu01131u111u030) $ $ \bf{3} $ $ \bf{85} $ $ 2 $
$ \mathcal{N}_{68,i} $ $ \mathcal{E}_i $ $ c $ $ X $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{N}_{68,i})| $
$ 1 $ $ 1 $ $ 3 $ $ (01330u3131uuu3330uuuu000333u1u1u) $ $ \bf{0} $ $ \bf{39} $ $ 2 $
$ 2 $ $ 2 $ $ 1 $ $ (0013u1111uu1u0uuuu101u1333330130) $ $ \bf{3} $ $ \bf{79} $ $ 2 $
$ 3 $ $ 2 $ $ 1 $ $ (u30u1u03u10uu113uuu01131u111u030) $ $ \bf{3} $ $ \bf{85} $ $ 2 $
Table 6.  $ [80,40,14] $ Self-dual codes over $ \mathbb{F}_{4}+u \mathbb{F}_{4} $ from $ C_{5} $
$ \mathcal{D}_{i} $ $ r_{\sigma(v_1)} $ $ r_{\sigma(v_2)} $ $ r_C $ $ |Aut(\mathcal{D}_i)| $ $ (\beta,\alpha) $
$ 1 $ $ (A,A,A,1,3) $ $ (0,2,1,3,E) $ $ (7,7,7,7,7) $ $ 2^3 \cdot 5 $ $ (0,-120) $
$ 2 $ $ (0,A,2,6,F) $ $ (2,1,E,2,1) $ $ (6,6,6,6,6) $ $ 2^2 \cdot 5 $ $ (0,-125) $
$ 3 $ $ (A,A,0,4,F) $ $ (2,A,6,2,F) $ $ (1,1,1,1,1) $ $ 2^2 \cdot 5 $ $ (0,-150) $
$ 4 $ $ (0,A,A,4,5) $ $ (0,3,6,A,B) $ $ (E,E,E,E,E) $ $ 2^2 \cdot 5 $ $ (0,-155) $
$ 5 $ $ (2,0,A,4,5) $ $ (2,A,4,0,5) $ $ (B,B,B,B,B) $ $ 2^2 \cdot 5 $ $ (0,-180) $
$ 6 $ $ (0,A,B,B,E) $ $ (0,2,1,3,1) $ $ (6,6,6,6,6) $ $ 2^2 \cdot 5 $ $ (0,-190) $
$ 7 $ $ (0,A,2,6,F) $ $ (2,2,6,2,7) $ $ (B,B,B,B,B) $ $ 2^2 \cdot 5 $ $ (0,-200) $
$ 8 $ $ (0,0,A,6,F) $ $ (2,1,E,0,3) $ $ (6,6,6,6,6) $ $ 2^2 \cdot 5 $ $ (0,-215) $
$ 9 $ $ (A,0,1,4,7) $ $ (0,3,E,2,7) $ $ (B,B,B,B,B) $ $ 2^2 \cdot 5 $ $ (0,-230) $
$ 10 $ $ (A,2,A,1,4) $ $ (0,0,7,1,F) $ $ (7,7,7,7,7) $ $ 2^2 \cdot 5 $ $ (0,-250) $
$ 11 $ $ (A,A,3,B,4) $ $ (0,A,4,0,7) $ $ (4,4,4,4,4) $ $ 2^2 \cdot 5 $ $ (0,-275) $
$ 12 $ $ (0,2,B,1,E) $ $ (0,0,1,1,3) $ $ (4,4,4,4,4) $ $ 2^2 \cdot 5 $ $ (10,-370) $
$ \mathcal{D}_{i} $ $ r_{\sigma(v_1)} $ $ r_{\sigma(v_2)} $ $ r_C $ $ |Aut(\mathcal{D}_i)| $ $ (\beta,\alpha) $
$ 1 $ $ (A,A,A,1,3) $ $ (0,2,1,3,E) $ $ (7,7,7,7,7) $ $ 2^3 \cdot 5 $ $ (0,-120) $
$ 2 $ $ (0,A,2,6,F) $ $ (2,1,E,2,1) $ $ (6,6,6,6,6) $ $ 2^2 \cdot 5 $ $ (0,-125) $
$ 3 $ $ (A,A,0,4,F) $ $ (2,A,6,2,F) $ $ (1,1,1,1,1) $ $ 2^2 \cdot 5 $ $ (0,-150) $
$ 4 $ $ (0,A,A,4,5) $ $ (0,3,6,A,B) $ $ (E,E,E,E,E) $ $ 2^2 \cdot 5 $ $ (0,-155) $
$ 5 $ $ (2,0,A,4,5) $ $ (2,A,4,0,5) $ $ (B,B,B,B,B) $ $ 2^2 \cdot 5 $ $ (0,-180) $
$ 6 $ $ (0,A,B,B,E) $ $ (0,2,1,3,1) $ $ (6,6,6,6,6) $ $ 2^2 \cdot 5 $ $ (0,-190) $
$ 7 $ $ (0,A,2,6,F) $ $ (2,2,6,2,7) $ $ (B,B,B,B,B) $ $ 2^2 \cdot 5 $ $ (0,-200) $
$ 8 $ $ (0,0,A,6,F) $ $ (2,1,E,0,3) $ $ (6,6,6,6,6) $ $ 2^2 \cdot 5 $ $ (0,-215) $
$ 9 $ $ (A,0,1,4,7) $ $ (0,3,E,2,7) $ $ (B,B,B,B,B) $ $ 2^2 \cdot 5 $ $ (0,-230) $
$ 10 $ $ (A,2,A,1,4) $ $ (0,0,7,1,F) $ $ (7,7,7,7,7) $ $ 2^2 \cdot 5 $ $ (0,-250) $
$ 11 $ $ (A,A,3,B,4) $ $ (0,A,4,0,7) $ $ (4,4,4,4,4) $ $ 2^2 \cdot 5 $ $ (0,-275) $
$ 12 $ $ (0,2,B,1,E) $ $ (0,0,1,1,3) $ $ (4,4,4,4,4) $ $ 2^2 \cdot 5 $ $ (10,-370) $
Table 7.  Self-dual codes over $ \mathbb{F}_{2} $ of length $ 68 $ from $ C_{17} $
$ \mathcal{C}_{i} $ $ r_{\sigma(v_1)} $ $ r_{\sigma(v_2)} $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ 1 $ $ (0,0,0,0,0,0,0,1,1,0,1,1,0,1,1,1,1) $ $ (0,0,0,1,0,0,0,1,1,1,0,0,1,0,1,1,1) $ $ 0 $ $ 255 $ $ 2 \cdot 17 $
$ 2 $ $ (0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,1,1) $ $ (0,0,0,1,0,1,1,1,1,0,1,1,0,1,1,1,1) $ $ 0 $ $ 272 $ $ 2^2 \cdot 17 $
$ \mathcal{C}_{i} $ $ r_{\sigma(v_1)} $ $ r_{\sigma(v_2)} $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $
$ 1 $ $ (0,0,0,0,0,0,0,1,1,0,1,1,0,1,1,1,1) $ $ (0,0,0,1,0,0,0,1,1,1,0,0,1,0,1,1,1) $ $ 0 $ $ 255 $ $ 2 \cdot 17 $
$ 2 $ $ (0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,1,1) $ $ (0,0,0,1,0,1,1,1,1,0,1,1,0,1,1,1,1) $ $ 0 $ $ 272 $ $ 2^2 \cdot 17 $
Table 8.  New codes of length 68 as neighbors
$ \mathcal{N}_{68,i} $ $ \mathcal{C}_{i} $ $ (x_{35},x_{36},...,x_{68}) $ $ \gamma $ $ \beta $ $ \mathcal{N}_{68,i} $ $ \mathcal{C}_{i} $ $ (x_{35},x_{36},...,x_{68}) $ $ \gamma $ $ \beta $
$ \mathcal{N}_{68,1} $ $ \mathcal{C}_{1} $ $ (1001110010101010011000001000111011) $ $ \boldsymbol{0} $ $ \boldsymbol{183} $ $ \mathcal{N}_{68,2} $ $ \mathcal{C}_{1} $ $ (1101000000010001110101011010100001) $ $ \boldsymbol{0} $ $ \boldsymbol{185} $
$ \mathcal{N}_{68,3} $ $ \mathcal{C}_{1} $ $ (0001000010011000110101100010101000) $ $ \boldsymbol{0} $ $ \boldsymbol{189} $ $ \mathcal{N}_{68,4} $ $ \mathcal{C}_{1} $ $ (0100000001110110100011110011101111) $ $ \boldsymbol{0} $ $ \boldsymbol{191} $
$ \mathcal{N}_{68,5} $ $ \mathcal{C}_{1} $ $ (0110110001001000110010110111100001) $ $ \boldsymbol{0} $ $ \boldsymbol{193} $ $ \mathcal{N}_{68,6} $ $ \mathcal{C}_{2} $ $ (0000001110101000111001011000001101) $ $ \boldsymbol{0} $ $ \boldsymbol{195} $
$ \mathcal{N}_{68,7} $ $ \mathcal{C}_{2} $ $ (1001000111000100110010000111111111) $ $ \boldsymbol{0} $ $ \boldsymbol{197} $ $ \mathcal{N}_{68,8} $ $ \mathcal{C}_{2} $ $ (0110100100000001010000101001011100) $ $ \boldsymbol{0} $ $ \boldsymbol{199} $
$ \mathcal{N}_{68,9} $ $ \mathcal{C}_{2} $ $ (1010111001110010001010100100011010) $ $ \boldsymbol{0} $ $ \boldsymbol{200} $ $ \mathcal{N}_{68,10} $ $ \mathcal{C}_{2} $ $ (0000000100000111100111110000110110) $ $ \boldsymbol{0} $ $ \boldsymbol{203} $
$ \mathcal{N}_{68,11} $ $ \mathcal{C}_{1} $ $ (1001010000011000011101100011101101) $ $ \boldsymbol{1} $ $ \boldsymbol{189} $ $ \mathcal{N}_{68,12} $ $ \mathcal{C}_{1} $ $ (0110100111000110000001001001100011) $ $ \boldsymbol{1} $ $ \boldsymbol{201} $
$ \mathcal{N}_{68,13} $ $ \mathcal{C}_{1} $ $ (1010011111110001111001110111001110) $ $ \boldsymbol{1} $ $ \boldsymbol{203} $ $ \mathcal{N}_{68,14} $ $ \mathcal{C}_{1} $ $ (1111011111101101100101100000010101) $ $ \boldsymbol{1} $ $ \boldsymbol{205} $
$ \mathcal{N}_{68,15} $ $ \mathcal{C}_{1} $ $ (1011110111111110101101111111101111) $ $ \boldsymbol{1} $ $ \boldsymbol{213} $ $ \mathcal{N}_{68,16} $ $ \mathcal{C}_{2} $ $ (1010001111110100000010100011101001) $ $ \boldsymbol{1} $ $ \boldsymbol{216} $
$ \mathcal{N}_{68,17} $ $ \mathcal{C}_{1} $ $ (1011110011111011001101111100111101) $ $ \boldsymbol{1} $ $ \boldsymbol{217} $ $ \mathcal{N}_{68,18} $ $ \mathcal{C}_{2} $ $ (0000010011001100100101011101110101) $ $ \boldsymbol{1} $ $ \boldsymbol{233} $
$ \mathcal{N}_{68,i} $ $ \mathcal{C}_{i} $ $ (x_{35},x_{36},...,x_{68}) $ $ \gamma $ $ \beta $ $ \mathcal{N}_{68,i} $ $ \mathcal{C}_{i} $ $ (x_{35},x_{36},...,x_{68}) $ $ \gamma $ $ \beta $
$ \mathcal{N}_{68,1} $ $ \mathcal{C}_{1} $ $ (1001110010101010011000001000111011) $ $ \boldsymbol{0} $ $ \boldsymbol{183} $ $ \mathcal{N}_{68,2} $ $ \mathcal{C}_{1} $ $ (1101000000010001110101011010100001) $ $ \boldsymbol{0} $ $ \boldsymbol{185} $
$ \mathcal{N}_{68,3} $ $ \mathcal{C}_{1} $ $ (0001000010011000110101100010101000) $ $ \boldsymbol{0} $ $ \boldsymbol{189} $ $ \mathcal{N}_{68,4} $ $ \mathcal{C}_{1} $ $ (0100000001110110100011110011101111) $ $ \boldsymbol{0} $ $ \boldsymbol{191} $
$ \mathcal{N}_{68,5} $ $ \mathcal{C}_{1} $ $ (0110110001001000110010110111100001) $ $ \boldsymbol{0} $ $ \boldsymbol{193} $ $ \mathcal{N}_{68,6} $ $ \mathcal{C}_{2} $ $ (0000001110101000111001011000001101) $ $ \boldsymbol{0} $ $ \boldsymbol{195} $
$ \mathcal{N}_{68,7} $ $ \mathcal{C}_{2} $ $ (1001000111000100110010000111111111) $ $ \boldsymbol{0} $ $ \boldsymbol{197} $ $ \mathcal{N}_{68,8} $ $ \mathcal{C}_{2} $ $ (0110100100000001010000101001011100) $ $ \boldsymbol{0} $ $ \boldsymbol{199} $
$ \mathcal{N}_{68,9} $ $ \mathcal{C}_{2} $ $ (1010111001110010001010100100011010) $ $ \boldsymbol{0} $ $ \boldsymbol{200} $ $ \mathcal{N}_{68,10} $ $ \mathcal{C}_{2} $ $ (0000000100000111100111110000110110) $ $ \boldsymbol{0} $ $ \boldsymbol{203} $
$ \mathcal{N}_{68,11} $ $ \mathcal{C}_{1} $ $ (1001010000011000011101100011101101) $ $ \boldsymbol{1} $ $ \boldsymbol{189} $ $ \mathcal{N}_{68,12} $ $ \mathcal{C}_{1} $ $ (0110100111000110000001001001100011) $ $ \boldsymbol{1} $ $ \boldsymbol{201} $
$ \mathcal{N}_{68,13} $ $ \mathcal{C}_{1} $ $ (1010011111110001111001110111001110) $ $ \boldsymbol{1} $ $ \boldsymbol{203} $ $ \mathcal{N}_{68,14} $ $ \mathcal{C}_{1} $ $ (1111011111101101100101100000010101) $ $ \boldsymbol{1} $ $ \boldsymbol{205} $
$ \mathcal{N}_{68,15} $ $ \mathcal{C}_{1} $ $ (1011110111111110101101111111101111) $ $ \boldsymbol{1} $ $ \boldsymbol{213} $ $ \mathcal{N}_{68,16} $ $ \mathcal{C}_{2} $ $ (1010001111110100000010100011101001) $ $ \boldsymbol{1} $ $ \boldsymbol{216} $
$ \mathcal{N}_{68,17} $ $ \mathcal{C}_{1} $ $ (1011110011111011001101111100111101) $ $ \boldsymbol{1} $ $ \boldsymbol{217} $ $ \mathcal{N}_{68,18} $ $ \mathcal{C}_{2} $ $ (0000010011001100100101011101110101) $ $ \boldsymbol{1} $ $ \boldsymbol{233} $
Table 9.  Self-dual codes over $ \mathbb{F}_{2} $ of length $ 92 $ from $ C_{23} $
$ \mathcal{C}_{i} $ $ r_{\sigma(v_1)} $ $ r_{\sigma(v_2)} $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $ Type
$ 1 $ $ (0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,0,1,1,1) $ $ (0,0,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,0,1,1,1) $ $ 0 $ $ \textbf{759} $ $ 2 \cdot 23 $ $ W_{92,1} $
$ 3 $ $ (0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,1) $ $ (0,0,0,0,1,1,0,0,0,1,1,0,1,0,0,1,1,0,1,0,1,1,1) $ $ 0 $ $ \textbf{1012} $ $ 2 \cdot 23 $ $ W_{92,1} $
$ 13 $ $ (0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,1,1) $ $ (0,0,0,0,1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,1) $ $ -46 $ $ \textbf{1564} $ $ 2^2 \cdot 23 $ $ W_{92,1} $
$ 16 $ $ (0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,1,1) $ $ (0,0,0,0,0,0,1,0,0,0,0,1,1,0,1,0,1,1,1,0,0,0,1) $ $ -46 $ $ \textbf{1978} $ $ 2 \cdot 23 $ $ W_{92,1} $
$ \mathcal{C}_{i} $ $ r_{\sigma(v_1)} $ $ r_{\sigma(v_2)} $ $ \gamma $ $ \beta $ $ |Aut(\mathcal{C}_i)| $ Type
$ 1 $ $ (0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,0,1,1,1) $ $ (0,0,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,0,1,1,1) $ $ 0 $ $ \textbf{759} $ $ 2 \cdot 23 $ $ W_{92,1} $
$ 3 $ $ (0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,1) $ $ (0,0,0,0,1,1,0,0,0,1,1,0,1,0,0,1,1,0,1,0,1,1,1) $ $ 0 $ $ \textbf{1012} $ $ 2 \cdot 23 $ $ W_{92,1} $
$ 13 $ $ (0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,1,1) $ $ (0,0,0,0,1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,1) $ $ -46 $ $ \textbf{1564} $ $ 2^2 \cdot 23 $ $ W_{92,1} $
$ 16 $ $ (0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,1,1) $ $ (0,0,0,0,0,0,1,0,0,0,0,1,1,0,1,0,1,1,1,0,0,0,1) $ $ -46 $ $ \textbf{1978} $ $ 2 \cdot 23 $ $ W_{92,1} $
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