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February  2020, 14(1): 11-22. doi: 10.3934/amc.2020002

New self-dual and formally self-dual codes from group ring constructions

1. 

Department of Mathematics, University of Scranton, Scranton, PA 18510, USA

2. 

Department of Mathematics, University of Chester, Chester, UK

3. 

Department of Mathematics Education, Sampoerna University, 12780, Jakarta, Indonesia

4. 

Department of Mathematics & Statistics, Northern Arizona University, Flagstaff, AZ 86011, USA

* Corresponding author: Bahattin Yildiz

Received  January 2018 Revised  February 2019 Published  August 2019

In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings $ \mathbb{F}_2+u \mathbb{F}_2 $ and $ \mathbb{F}_4+u \mathbb{F}_4 $, we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally self-dual codes with parameters $ [72,36,14] $.

Citation: Steven T. Dougherty, Joe Gildea, Abidin Kaya, Bahattin Yildiz. New self-dual and formally self-dual codes from group ring constructions. Advances in Mathematics of Communications, 2020, 14 (1) : 11-22. doi: 10.3934/amc.2020002
References:
[1]

D. AnevM. Harada and N. Yankov, New extremal singly even self-dual codes of lengths 64 and 66, J. Algebra Comb. Discrete Appl., 5 (2018), 143-151.  doi: 10.13069/jacodesmath.458601.  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]

A. Bovdi and C. A. Szakács, Unitary Subgroup of the group of units of a modular group algebra of a finite abelian $p$-group, Math. Zametki, 45 (1989), 23-29.   Google Scholar

[4]

V. Bovdi and A. L. Rosa, On the order of the unitary subgroup of a modular group algebra, Comm. Algebra, 28 (2000), 1897-1905.  doi: 10.1080/00927870008826934.  Google Scholar

[5]

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

[6]

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

[7]

P. J. Davis, Circulant Matrices, A Wiley-Interscience Publication. Pure and Applied Mathematics. John Wiley & Sons, New York-Chichester-Brisbane, 1979.  Google Scholar

[8]

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

[9]

S. T. DoughertyJ. GildeaR. Taylor and A. Tylyshchak, 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

[10]

S. T. DoughertyB. Yildiz and S. Karadeniz, Codes over $R_k$, Gray maps and their Binary Images, Finite Fields Appl., 17 (2011), 205-219.  doi: 10.1016/j.ffa.2010.11.002.  Google Scholar

[11]

S. T. DoughertyS. Karadeniz and B. Yildiz, Cyclic codes over $R_k$, Des. Codes Crypt., 63 (2012), 113-126.  doi: 10.1007/s10623-011-9539-4.  Google Scholar

[12]

S. DoughertyB. Yildiz and S. Karadeniz, Self-dual codes over $R_k$ and binary self-dual codes, Eur. J. Pure and Applied Math., 6 (2013), 89-106.   Google Scholar

[13]

Binary Generator Matrices of New Extremal Binary Self-Dual Codes of Length 68, Available from: http://www.abidinkaya.wix.com/math/research4. 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]

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

[16]

A. Kaya and B. Yildiz, Various constructions for self-dual codes over rings and new binary self-dual codes, Discrete Math., 339 (2016), 460-469.  doi: 10.1016/j.disc.2015.09.010.  Google Scholar

[17]

A. Kaya and B. Yildiz, Constructing formally self-dual codes from block $\lambda$-circulant matrices, Math. Commun., 24 (2019), 91-105.   Google Scholar

[18]

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.  doi: 10.1353/ajm.1999.0024.  Google Scholar

show all references

References:
[1]

D. AnevM. Harada and N. Yankov, New extremal singly even self-dual codes of lengths 64 and 66, J. Algebra Comb. Discrete Appl., 5 (2018), 143-151.  doi: 10.13069/jacodesmath.458601.  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]

A. Bovdi and C. A. Szakács, Unitary Subgroup of the group of units of a modular group algebra of a finite abelian $p$-group, Math. Zametki, 45 (1989), 23-29.   Google Scholar

[4]

V. Bovdi and A. L. Rosa, On the order of the unitary subgroup of a modular group algebra, Comm. Algebra, 28 (2000), 1897-1905.  doi: 10.1080/00927870008826934.  Google Scholar

[5]

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

[6]

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

[7]

P. J. Davis, Circulant Matrices, A Wiley-Interscience Publication. Pure and Applied Mathematics. John Wiley & Sons, New York-Chichester-Brisbane, 1979.  Google Scholar

[8]

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

[9]

S. T. DoughertyJ. GildeaR. Taylor and A. Tylyshchak, 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

[10]

S. T. DoughertyB. Yildiz and S. Karadeniz, Codes over $R_k$, Gray maps and their Binary Images, Finite Fields Appl., 17 (2011), 205-219.  doi: 10.1016/j.ffa.2010.11.002.  Google Scholar

[11]

S. T. DoughertyS. Karadeniz and B. Yildiz, Cyclic codes over $R_k$, Des. Codes Crypt., 63 (2012), 113-126.  doi: 10.1007/s10623-011-9539-4.  Google Scholar

[12]

S. DoughertyB. Yildiz and S. Karadeniz, Self-dual codes over $R_k$ and binary self-dual codes, Eur. J. Pure and Applied Math., 6 (2013), 89-106.   Google Scholar

[13]

Binary Generator Matrices of New Extremal Binary Self-Dual Codes of Length 68, Available from: http://www.abidinkaya.wix.com/math/research4. 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]

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

[16]

A. Kaya and B. Yildiz, Various constructions for self-dual codes over rings and new binary self-dual codes, Discrete Math., 339 (2016), 460-469.  doi: 10.1016/j.disc.2015.09.010.  Google Scholar

[17]

A. Kaya and B. Yildiz, Constructing formally self-dual codes from block $\lambda$-circulant matrices, Math. Commun., 24 (2019), 91-105.   Google Scholar

[18]

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.  doi: 10.1353/ajm.1999.0024.  Google Scholar

Table 1.  $ \left[ 64,32,12\right] _{2} $ codes via $ C_{mn} $ with $ m = 4,n = 2 $ over $ \mathbb{F}_{4}+u\mathbb{F}_{4} $
$ \mathcal{C}_{64,i} $ $ r_{A_{1}} $ $ r_{A_{2}} $ $ |Aut(\mathcal{C}_{i})| $ $ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,1} $ $ \left( B,4,6,2\right) $ $ \left( E,9,7,0\right) $ $ 2^{6} $ 0
$ \mathcal{C}_{64,2} $ $ \left( B,6,6,0\right) $ $ \left( E,3,5,8\right) $ $ 2^{5} $ 4
$ \mathcal{C}_{64,3} $ $ \left( 9,E,E,0\right) $ $ \left( 6,9,3,A\right) $ $ 2^{5} $ 12
$ \mathcal{C}_{64,4} $ $ \left( B,4,C,A\right) $ $ \left( 6,B,D,0\right) $ $ 2^{5} $ 16
$ \mathcal{C}_{64,5} $ $ \left( 3,6,E,0\right) $ $ \left( E,B,F,A\right) $ $ 2^{5} $ 20
$ \mathcal{C}_{64,6} $ $ \left( 1,E,6,0\right) $ $ \left( C,9,D,8\right) $ $ 2^{5} $ 36
$ \mathcal{C}_{64,7} $ $ \left( 9,C,E,2\right) $ $ \left( C,9,D,8\right) $ $ 2^{5} $ 48
$ \mathcal{C}_{64,8} $ $ \left( 3,6,4,8\right) $ $ \left( E,B,F,A\right) $ $ 2^{5} $ 52
$ \mathcal{C}_{64,i} $ $ r_{A_{1}} $ $ r_{A_{2}} $ $ |Aut(\mathcal{C}_{i})| $ $ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,1} $ $ \left( B,4,6,2\right) $ $ \left( E,9,7,0\right) $ $ 2^{6} $ 0
$ \mathcal{C}_{64,2} $ $ \left( B,6,6,0\right) $ $ \left( E,3,5,8\right) $ $ 2^{5} $ 4
$ \mathcal{C}_{64,3} $ $ \left( 9,E,E,0\right) $ $ \left( 6,9,3,A\right) $ $ 2^{5} $ 12
$ \mathcal{C}_{64,4} $ $ \left( B,4,C,A\right) $ $ \left( 6,B,D,0\right) $ $ 2^{5} $ 16
$ \mathcal{C}_{64,5} $ $ \left( 3,6,E,0\right) $ $ \left( E,B,F,A\right) $ $ 2^{5} $ 20
$ \mathcal{C}_{64,6} $ $ \left( 1,E,6,0\right) $ $ \left( C,9,D,8\right) $ $ 2^{5} $ 36
$ \mathcal{C}_{64,7} $ $ \left( 9,C,E,2\right) $ $ \left( C,9,D,8\right) $ $ 2^{5} $ 48
$ \mathcal{C}_{64,8} $ $ \left( 3,6,4,8\right) $ $ \left( E,B,F,A\right) $ $ 2^{5} $ 52
Table 2.  $ \left[ 64,32,12\right] _{2} $ codes via $ C_{mn} $ $ \ $with $ m = 2,n = 4 $ over $ \mathbb{F}_{4}+u\mathbb{F}_{4} $
$ \mathcal{C}_{64,i} $ $ r_{A_{1}} $ $ r_{A_{2}} $ $ r_{A_{3}} $ $ r_{A_{4}} $ $ |Aut(\mathcal{C}_{64,i})| $ $ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,9} $ $ \left( D,6\right) $ $ \left( 8,F\right) $ $ \left( B,0\right) $ $ \left( 4,A\right) $ $ 2^{6} $ 0
$ \mathcal{C}_{64,10} $ $ \left( D,6\right) $ $ \left( 8,F\right) $ $ \left( B,0\right) $ $ \left( 6,8\right) $ $ 2^{5} $ 4
$ \mathcal{C}_{64,11} $ $ \left( 7,4\right) $ $ \left( E,0\right) $ $ \left( 8,B\right) $ $ \left( 1,6\right) $ $ 2^{5} $ 12
$ \mathcal{C}_{64,12} $ $ \left( 4,A\right) $ $ \left( 0,B\right) $ $ \left( C,B\right) $ $ \left( 6,D\right) $ $ 2^{5} $ 16
$ \mathcal{C}_{64,13} $ $ \left( D,C\right) $ $ \left( 4,2\right) $ $ \left( 8,9\right) $ $ \left( 1,C\right) $ $ 2^{5} $ 20
$ \mathcal{C}_{64,14} $ $ \left( D,4\right) $ $ \left( 6,8\right) $ $ \left( A,3\right) $ $ \left( 3,4\right) $ $ 2^{5} $ 28
$ \mathcal{C}_{64,15} $ $ \left( F,4\right) $ $ \left( 8,5\right) $ $ \left( 9,2\right) $ $ \left( C,A\right) $ $ 2^{6} $ 32
$ \mathcal{C}_{64,16} $ $ \left( 5,6\right) $ $ \left( C,2\right) $ $ \left( A,9\right) $ $ \left( 3,4\right) $ $ 2^{5} $ 36
$ \mathcal{C}_{64,17} $ $ \left( 5,E\right) $ $ \left( C,A\right) $ $ \left( 8,3\right) $ $ \left( 3,E\right) $ $ 2^{5} $ 44
$ \mathcal{C}_{64,18} $ $ \left( 8,9\right) $ $ \left( D,A\right) $ $ \left( C,D\right) $ $ \left( D,3\right) $ $ 2^{5} $ 48
$ \mathcal{C}_{64,19} $ $ \left( D,C\right) $ $ \left( E,0\right) $ $ \left( 8,9\right) $ $ \left( 9,E\right) $ $ 2^{5} $ 52
$ \mathcal{C}_{64,i} $ $ r_{A_{1}} $ $ r_{A_{2}} $ $ r_{A_{3}} $ $ r_{A_{4}} $ $ |Aut(\mathcal{C}_{64,i})| $ $ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,9} $ $ \left( D,6\right) $ $ \left( 8,F\right) $ $ \left( B,0\right) $ $ \left( 4,A\right) $ $ 2^{6} $ 0
$ \mathcal{C}_{64,10} $ $ \left( D,6\right) $ $ \left( 8,F\right) $ $ \left( B,0\right) $ $ \left( 6,8\right) $ $ 2^{5} $ 4
$ \mathcal{C}_{64,11} $ $ \left( 7,4\right) $ $ \left( E,0\right) $ $ \left( 8,B\right) $ $ \left( 1,6\right) $ $ 2^{5} $ 12
$ \mathcal{C}_{64,12} $ $ \left( 4,A\right) $ $ \left( 0,B\right) $ $ \left( C,B\right) $ $ \left( 6,D\right) $ $ 2^{5} $ 16
$ \mathcal{C}_{64,13} $ $ \left( D,C\right) $ $ \left( 4,2\right) $ $ \left( 8,9\right) $ $ \left( 1,C\right) $ $ 2^{5} $ 20
$ \mathcal{C}_{64,14} $ $ \left( D,4\right) $ $ \left( 6,8\right) $ $ \left( A,3\right) $ $ \left( 3,4\right) $ $ 2^{5} $ 28
$ \mathcal{C}_{64,15} $ $ \left( F,4\right) $ $ \left( 8,5\right) $ $ \left( 9,2\right) $ $ \left( C,A\right) $ $ 2^{6} $ 32
$ \mathcal{C}_{64,16} $ $ \left( 5,6\right) $ $ \left( C,2\right) $ $ \left( A,9\right) $ $ \left( 3,4\right) $ $ 2^{5} $ 36
$ \mathcal{C}_{64,17} $ $ \left( 5,E\right) $ $ \left( C,A\right) $ $ \left( 8,3\right) $ $ \left( 3,E\right) $ $ 2^{5} $ 44
$ \mathcal{C}_{64,18} $ $ \left( 8,9\right) $ $ \left( D,A\right) $ $ \left( C,D\right) $ $ \left( D,3\right) $ $ 2^{5} $ 48
$ \mathcal{C}_{64,19} $ $ \left( D,C\right) $ $ \left( E,0\right) $ $ \left( 8,9\right) $ $ \left( 9,E\right) $ $ 2^{5} $ 52
Table 3.  $ \left[ 64,32,12\right] _{2} $ codes via $ D_{8} $ over $ \mathbb{F} _{4}+u\mathbb{F}_{4} $
$ \mathcal{C}_{64,i} $ $ r_{A} $ $ r_{B} $ $ \left\vert Aut\left( \mathcal{C} _{64,i}\right) \right\vert $ $ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,20} $ $ \left( 6,D,7,E\right) $ $ \left( 2,2,4,5\right) $ $ 2^{6} $ 0
$ \mathcal{C}_{64,21} $ $ \left( C,5,F,4\right) $ $ \left( 2,0,C,D\right) $ $ 2^{5} $ 4
$ \mathcal{C}_{64,22} $ $ \left( 0,C,8,8\right) $ $ \left( 9,A,B,D\right) $ $ 2^{5} $ 8
$ \mathcal{C}_{64,23} $ $ \left( 4,B,A,4\right) $ $ \left( E,D,5,C\right) $ $ 2^{4} $ 12
$ \mathcal{C}_{64,24} $ $ \left( F,8,5,E\right) $ $ \left( A,1,D,9\right) $ $ 2^{4} $ 16
$ \mathcal{C}_{64,25} $ $ \left( 6,4,9,2\right) $ $ \left( 7,C,C,F\right) $ $ 2^{4} $ 20
$ \mathcal{C}_{64,26} $ $ \left( E,3,B,1\right) $ $ \left( 0,7,8,1\right) $ $ 2^{5} $ 24
$ \mathcal{C}_{64,27} $ $ \left( 9,9,8,0\right) $ $ \left( 6,6,1,2\right) $ $ 2^{4} $ 28
$ \mathcal{C}_{64,28} $ $ \left( 7,0,A,8\right) $ $ \left( F,8,5,C\right) $ $ 2^{6} $ 32
$ \mathcal{C}_{64,29} $ $ \left( 6,7,7,6\right) $ $ \left( A,2,4,5\right) $ $ 2^{4} $ $ 36 $
$ \mathcal{C}_{64,30} $ $ \left( 0,6,8,2\right) $ $ \left( 6,3,1,1\right) $ $ 2^{5} $ 40
$ \mathcal{C}_{64,31} $ $ \left( 5,F,E,E\right) $ $ \left( 4,1,0,C\right) $ $ 2^{4}\times 3 $ 44
$ \mathcal{C}_{64,32} $ $ \left( F,F,6,6\right) $ $ \left( 4,3,A,4\right) $ $ 2^{5} $ 48
$ \mathcal{C}_{64,33} $ $ \left( D,D,4,4\right) $ $ \left( 6,B,0,6\right) $ $ 2^{5} $ 52
$ \mathcal{C}_{64,i} $ $ r_{A} $ $ r_{B} $ $ \left\vert Aut\left( \mathcal{C} _{64,i}\right) \right\vert $ $ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,20} $ $ \left( 6,D,7,E\right) $ $ \left( 2,2,4,5\right) $ $ 2^{6} $ 0
$ \mathcal{C}_{64,21} $ $ \left( C,5,F,4\right) $ $ \left( 2,0,C,D\right) $ $ 2^{5} $ 4
$ \mathcal{C}_{64,22} $ $ \left( 0,C,8,8\right) $ $ \left( 9,A,B,D\right) $ $ 2^{5} $ 8
$ \mathcal{C}_{64,23} $ $ \left( 4,B,A,4\right) $ $ \left( E,D,5,C\right) $ $ 2^{4} $ 12
$ \mathcal{C}_{64,24} $ $ \left( F,8,5,E\right) $ $ \left( A,1,D,9\right) $ $ 2^{4} $ 16
$ \mathcal{C}_{64,25} $ $ \left( 6,4,9,2\right) $ $ \left( 7,C,C,F\right) $ $ 2^{4} $ 20
$ \mathcal{C}_{64,26} $ $ \left( E,3,B,1\right) $ $ \left( 0,7,8,1\right) $ $ 2^{5} $ 24
$ \mathcal{C}_{64,27} $ $ \left( 9,9,8,0\right) $ $ \left( 6,6,1,2\right) $ $ 2^{4} $ 28
$ \mathcal{C}_{64,28} $ $ \left( 7,0,A,8\right) $ $ \left( F,8,5,C\right) $ $ 2^{6} $ 32
$ \mathcal{C}_{64,29} $ $ \left( 6,7,7,6\right) $ $ \left( A,2,4,5\right) $ $ 2^{4} $ $ 36 $
$ \mathcal{C}_{64,30} $ $ \left( 0,6,8,2\right) $ $ \left( 6,3,1,1\right) $ $ 2^{5} $ 40
$ \mathcal{C}_{64,31} $ $ \left( 5,F,E,E\right) $ $ \left( 4,1,0,C\right) $ $ 2^{4}\times 3 $ 44
$ \mathcal{C}_{64,32} $ $ \left( F,F,6,6\right) $ $ \left( 4,3,A,4\right) $ $ 2^{5} $ 48
$ \mathcal{C}_{64,33} $ $ \left( D,D,4,4\right) $ $ \left( 6,B,0,6\right) $ $ 2^{5} $ 52
Table 4.  $ \left[ 64,32,12\right] _{2} $ codes via $ D_{16} $ over $ \mathbb{F} _{2}+u\mathbb{F}_{2} $
$ \mathcal{C}_{64,i} $ $ r_{A} $ $ r_{B} $ $ \left\vert Aut\left( \mathcal{C} _{64,i}\right) \right\vert $ $ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,34} $ $ \left( 03331uu0\right) $ $ \left( 0003u013\right) $ $ 2^{5} $ 0
$ \mathcal{C}_{64,35} $ $ \left( 3031u110\right) $ $ \left( 0u30100u\right) $ $ 2^{5} $ 16
$ \mathcal{C}_{64,36} $ $ \left( 031u13uu\right) $ $ \left( u01001u1\right) $ $ 2^{5} $ 32
$ \mathcal{C}_{64,37} $ $ \left( 11013uu3\right) $ $ \left( u003111u\right) $ $ 2^{5} $ 48
$ \mathcal{C}_{64,38} $ $ \left( 3u13u130\right) $ $ \left( 0u301u0u\right) $ $ 2^{7} $ $ 80 $
$ \mathcal{C}_{64,i} $ $ r_{A} $ $ r_{B} $ $ \left\vert Aut\left( \mathcal{C} _{64,i}\right) \right\vert $ $ \beta $ in $ W_{64,2} $
$ \mathcal{C}_{64,34} $ $ \left( 03331uu0\right) $ $ \left( 0003u013\right) $ $ 2^{5} $ 0
$ \mathcal{C}_{64,35} $ $ \left( 3031u110\right) $ $ \left( 0u30100u\right) $ $ 2^{5} $ 16
$ \mathcal{C}_{64,36} $ $ \left( 031u13uu\right) $ $ \left( u01001u1\right) $ $ 2^{5} $ 32
$ \mathcal{C}_{64,37} $ $ \left( 11013uu3\right) $ $ \left( u003111u\right) $ $ 2^{5} $ 48
$ \mathcal{C}_{64,38} $ $ \left( 3u13u130\right) $ $ \left( 0u301u0u\right) $ $ 2^{7} $ $ 80 $
Table 5.  New extremal binary self-dual codes of length $ 68 $
$ \mathcal{C}_{68,i} $ $ \mathcal{C} $ $ c $ $ X $ $ \gamma $ $ \beta $
$ \mathcal{C}_{68,1} $ $ \mathcal{C}_{64,29} $ $ 1+u $ $ \left( uu0u33131u1333130u30uu3130113133\right) $ $ 3 $ $ 135 $
$ \mathcal{C}_{68,2} $ $ \mathcal{C}_{64,29} $ $ 1+u $ $ \left( 00u013331u1131310u30u0111u331313\right) $ $ 3 $ $ 139 $
$ \mathcal{C}_{68,3} $ $ \mathcal{C}_{64,29} $ $ 1+u $ $ \left( uu0u33333u131133003u0u113u131331\right) $ $ 3 $ $ 143 $
$ \mathcal{C}_{68,4} $ $ \mathcal{C}_{64,29} $ $ 1+u $ $ \left( uu0011133u311331uu1u0u3110113111\right) $ $ 3 $ $ 151 $
$ \mathcal{C}_{68,5} $ $ \mathcal{C}_{64,29} $ $ 1+u $ $ \left( u0uu333310311331uu30u0331u331113\right) $ $ 3 $ $ 155 $
$ \mathcal{C}_{68,6} $ $ \mathcal{C}_{64,29} $ $ 1 $ $ \left( u0u011131u1311110u30u03110113111\right) $ $ 3 $ $ 161 $
$ \mathcal{C}_{68,7} $ $ \mathcal{C}_{64,38} $ $ 1+u $ $ \left( 33131u0101333uu103310030uu11uu13\right) $ $ 3 $ $ 186 $
$ \mathcal{C}_{68,8} $ $ \mathcal{C}_{64,38} $ $ 1 $ $ \left( 13131uuu0u0u3033u1u130100310u1u0\right) $ $ 3 $ $ 202 $
$ \mathcal{C}_{68,9} $ $ \mathcal{C}_{64,38} $ $ 1 $ $ \left( 133330u301331uu30311003uu0130011\right) $ $ 3 $ $ 204 $
$ \mathcal{C}_{68,i} $ $ \mathcal{C} $ $ c $ $ X $ $ \gamma $ $ \beta $
$ \mathcal{C}_{68,1} $ $ \mathcal{C}_{64,29} $ $ 1+u $ $ \left( uu0u33131u1333130u30uu3130113133\right) $ $ 3 $ $ 135 $
$ \mathcal{C}_{68,2} $ $ \mathcal{C}_{64,29} $ $ 1+u $ $ \left( 00u013331u1131310u30u0111u331313\right) $ $ 3 $ $ 139 $
$ \mathcal{C}_{68,3} $ $ \mathcal{C}_{64,29} $ $ 1+u $ $ \left( uu0u33333u131133003u0u113u131331\right) $ $ 3 $ $ 143 $
$ \mathcal{C}_{68,4} $ $ \mathcal{C}_{64,29} $ $ 1+u $ $ \left( uu0011133u311331uu1u0u3110113111\right) $ $ 3 $ $ 151 $
$ \mathcal{C}_{68,5} $ $ \mathcal{C}_{64,29} $ $ 1+u $ $ \left( u0uu333310311331uu30u0331u331113\right) $ $ 3 $ $ 155 $
$ \mathcal{C}_{68,6} $ $ \mathcal{C}_{64,29} $ $ 1 $ $ \left( u0u011131u1311110u30u03110113111\right) $ $ 3 $ $ 161 $
$ \mathcal{C}_{68,7} $ $ \mathcal{C}_{64,38} $ $ 1+u $ $ \left( 33131u0101333uu103310030uu11uu13\right) $ $ 3 $ $ 186 $
$ \mathcal{C}_{68,8} $ $ \mathcal{C}_{64,38} $ $ 1 $ $ \left( 13131uuu0u0u3033u1u130100310u1u0\right) $ $ 3 $ $ 202 $
$ \mathcal{C}_{68,9} $ $ \mathcal{C}_{64,38} $ $ 1 $ $ \left( 133330u301331uu30311003uu0130011\right) $ $ 3 $ $ 204 $
Table 6.  FSD $ \left[ 72,36,14\right] _{2}^{b-1} $codes by $ C_{mn} $ construction over $ \mathbb{F}_{2}+u\mathbb{F}_{2} $
$ n $ $ m $ $ r_{1},\ldots ,r_{n} $ $ A_{14} $ $ A_{16} $ $ A_{18} $
$ 2 $ $ 9 $ $ 13u10u000,03u100011 $ $ 8820 $ $ 123039 $ $ 1210564 $
$ 2 $ $ 9 $ $ 30u0031uu,10u1u3u3u $ $ 8856 $ $ 122850 $ $ 1210492 $
$ 2 $ $ 9 $ $ u00u31uu1,01uu3u013 $ $ 8784 $ $ 123417 $ $ 1207344 $
$ 2 $ $ 9 $ $ 3031uu10u,300333u30 $ $ 8928 $ $ 122436 $ $ 1210776 $
$ 2 $ $ 9 $ $ uu0003103,300u1303u $ $ 9288 $ $ 120690 $ $ 1208328 $
$ 2 $ $ 9 $ $ 1333u1313,11u3u31uu $ $ 9360 $ $ 119583 $ $ 1216936 $
$ 3 $ $ 6 $ $ 11010u,30u1u0,103u11 $ $ 8820 $ $ 123327 $ $ 1207092 $
$ 3 $ $ 6 $ $ u30u33,333130,0301u3 $ $ 9180 $ $ 121194 $ $ 1209304 $
$ 3 $ $ 6 $ $ 33u101,0u0311,13u1u3 $ $ 9504 $ $ 119151 $ $ 1212760 $
$ 3 $ $ 6 $ $ 3uuu0u,u31u03,10uu13 $ $ 9648 $ $ 118170 $ $ 1215172 $
$ n $ $ m $ $ r_{1},\ldots ,r_{n} $ $ A_{14} $ $ A_{16} $ $ A_{18} $
$ 2 $ $ 9 $ $ 13u10u000,03u100011 $ $ 8820 $ $ 123039 $ $ 1210564 $
$ 2 $ $ 9 $ $ 30u0031uu,10u1u3u3u $ $ 8856 $ $ 122850 $ $ 1210492 $
$ 2 $ $ 9 $ $ u00u31uu1,01uu3u013 $ $ 8784 $ $ 123417 $ $ 1207344 $
$ 2 $ $ 9 $ $ 3031uu10u,300333u30 $ $ 8928 $ $ 122436 $ $ 1210776 $
$ 2 $ $ 9 $ $ uu0003103,300u1303u $ $ 9288 $ $ 120690 $ $ 1208328 $
$ 2 $ $ 9 $ $ 1333u1313,11u3u31uu $ $ 9360 $ $ 119583 $ $ 1216936 $
$ 3 $ $ 6 $ $ 11010u,30u1u0,103u11 $ $ 8820 $ $ 123327 $ $ 1207092 $
$ 3 $ $ 6 $ $ u30u33,333130,0301u3 $ $ 9180 $ $ 121194 $ $ 1209304 $
$ 3 $ $ 6 $ $ 33u101,0u0311,13u1u3 $ $ 9504 $ $ 119151 $ $ 1212760 $
$ 3 $ $ 6 $ $ 3uuu0u,u31u03,10uu13 $ $ 9648 $ $ 118170 $ $ 1215172 $
Table 7.  FSD $ \left[ 72,36,14\right] _{2}^{b-1} $codes by $ C_{mn} $ over $ \mathbb{F}_{2} $
$ n $ $ m $ $ r_{1},\ldots ,r_{n} $ $ A_{14} $ $ A_{16} $ $ A_{18} $
$ 3 $ $ 12 $ $ 000100000010,110001110110,000010011010 $ $ 8496 $ $ 124911 $ $ 1209160 $
$ 3 $ $ 12 $ $ 011110101111,010110100010,101011000100 $ $ 8568 $ $ 124362 $ $ 1211068 $
$ 3 $ $ 12 $ $ 111100000101,100100101100,111000111110 $ $ 9072 $ $ 121653 $ $ 1210816 $
$ 3 $ $ 12 $ $ 100111000011,011000001011,010001011011 $ $ 9144 $ $ 121221 $ $ 1211328 $
$ 3 $ $ 12 $ $ 001110100000,000000111000,110111001000 $ $ 9468 $ $ 119601 $ $ 1209700 $
$ 4 $ $ 9 $ $ 001011011,010011110,001100010,101000011 $ $ 8388 $ $ 125730 $ $ 1206348 $
$ 4 $ $ 9 $ $ 010110011,100110010,101100111,000101011 $ $ 8712 $ $ 123741 $ $ 1209160 $
$ 4 $ $ 9 $ $ 001111011,100100100,010101100,010111000 $ $ 8820 $ $ 123039 $ $ 1210564 $
$ 4 $ $ 9 $ $ 111011010,101110101,111000101,110001001 $ $ 8928 $ $ 122328 $ $ 1212076 $
$ 4 $ $ 9 $ $ 000010001,111001101,101101110,110011100 $ $ 8928 $ $ 122769 $ $ 1206784 $
$ 4 $ $ 9 $ $ 110001100,101110111,001100010,100110110 $ $ 9036 $ $ 121761 $ $ 1211868 $
$ 4 $ $ 9 $ $ 101100101,101110111,010011000,010010110 $ $ 9036 $ $ 121977 $ $ 1208276 $
$ 6 $ $ 6 $ $ 000100,110100,010111,101000,100000,010111 $ $ 8388 $ $ 125973 $ $ 1203436 $
$ 6 $ $ 6 $ $ 001010,011001,110010,111011,010100,110101 $ $ 8784 $ $ 123570 $ $ 1206532 $
$ 6 $ $ 6 $ $ 101001,001110,110110,101000,000110,000100 $ $ 9360 $ $ 120114 $ $ 1210564 $
$ n $ $ m $ $ r_{1},\ldots ,r_{n} $ $ A_{14} $ $ A_{16} $ $ A_{18} $
$ 3 $ $ 12 $ $ 000100000010,110001110110,000010011010 $ $ 8496 $ $ 124911 $ $ 1209160 $
$ 3 $ $ 12 $ $ 011110101111,010110100010,101011000100 $ $ 8568 $ $ 124362 $ $ 1211068 $
$ 3 $ $ 12 $ $ 111100000101,100100101100,111000111110 $ $ 9072 $ $ 121653 $ $ 1210816 $
$ 3 $ $ 12 $ $ 100111000011,011000001011,010001011011 $ $ 9144 $ $ 121221 $ $ 1211328 $
$ 3 $ $ 12 $ $ 001110100000,000000111000,110111001000 $ $ 9468 $ $ 119601 $ $ 1209700 $
$ 4 $ $ 9 $ $ 001011011,010011110,001100010,101000011 $ $ 8388 $ $ 125730 $ $ 1206348 $
$ 4 $ $ 9 $ $ 010110011,100110010,101100111,000101011 $ $ 8712 $ $ 123741 $ $ 1209160 $
$ 4 $ $ 9 $ $ 001111011,100100100,010101100,010111000 $ $ 8820 $ $ 123039 $ $ 1210564 $
$ 4 $ $ 9 $ $ 111011010,101110101,111000101,110001001 $ $ 8928 $ $ 122328 $ $ 1212076 $
$ 4 $ $ 9 $ $ 000010001,111001101,101101110,110011100 $ $ 8928 $ $ 122769 $ $ 1206784 $
$ 4 $ $ 9 $ $ 110001100,101110111,001100010,100110110 $ $ 9036 $ $ 121761 $ $ 1211868 $
$ 4 $ $ 9 $ $ 101100101,101110111,010011000,010010110 $ $ 9036 $ $ 121977 $ $ 1208276 $
$ 6 $ $ 6 $ $ 000100,110100,010111,101000,100000,010111 $ $ 8388 $ $ 125973 $ $ 1203436 $
$ 6 $ $ 6 $ $ 001010,011001,110010,111011,010100,110101 $ $ 8784 $ $ 123570 $ $ 1206532 $
$ 6 $ $ 6 $ $ 101001,001110,110110,101000,000110,000100 $ $ 9360 $ $ 120114 $ $ 1210564 $
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