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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. Anev, M. 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. 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.  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. Dougherty, J.-L. Kim, H. 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. Dougherty, J. Gildea, R. 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. Dougherty, B. 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. Dougherty, S. 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. Dougherty, B. 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. 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.  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

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##### References:
 [1] D. Anev, M. 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. 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.  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. Dougherty, J.-L. Kim, H. 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. Dougherty, J. Gildea, R. 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. Dougherty, B. 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. Dougherty, S. 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. Dougherty, B. 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. 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.  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
$\left[ 64,32,12\right] _{2}$ 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
$\left[ 64,32,12\right] _{2}$ 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
$\left[ 64,32,12\right] _{2}$ 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
$\left[ 64,32,12\right] _{2}$ 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 $\mathcal{C}_{64,38}$ $\left( 3u13u130\right)$ $\left( 0u301u0u\right)$ $2^{7}$ $80$
New extremal binary self-dual codes of length $68$
 $\mathcal{C}_{68,i}$ $\mathcal{C}$ $c$ $X$ $\gamma$ $\beta$ $\mathcal{C}_{68,1}$ $\mathcal{C}_{64,29}$ $1+u$ $\left( uu0u33131u1333130u30uu3130113133\right)$ $3$ $135$ $\mathcal{C}_{68,2}$ $\mathcal{C}_{64,29}$ $1+u$ $\left( 00u013331u1131310u30u0111u331313\right)$ $3$ $139$ $\mathcal{C}_{68,3}$ $\mathcal{C}_{64,29}$ $1+u$ $\left( uu0u33333u131133003u0u113u131331\right)$ $3$ $143$ $\mathcal{C}_{68,4}$ $\mathcal{C}_{64,29}$ $1+u$ $\left( uu0011133u311331uu1u0u3110113111\right)$ $3$ $151$ $\mathcal{C}_{68,5}$ $\mathcal{C}_{64,29}$ $1+u$ $\left( u0uu333310311331uu30u0331u331113\right)$ $3$ $155$ $\mathcal{C}_{68,6}$ $\mathcal{C}_{64,29}$ $1$ $\left( u0u011131u1311110u30u03110113111\right)$ $3$ $161$ $\mathcal{C}_{68,7}$ $\mathcal{C}_{64,38}$ $1+u$ $\left( 33131u0101333uu103310030uu11uu13\right)$ $3$ $186$ $\mathcal{C}_{68,8}$ $\mathcal{C}_{64,38}$ $1$ $\left( 13131uuu0u0u3033u1u130100310u1u0\right)$ $3$ $202$ $\mathcal{C}_{68,9}$ $\mathcal{C}_{64,38}$ $1$ $\left( 133330u301331uu30311003uu0130011\right)$ $3$ $204$
 $\mathcal{C}_{68,i}$ $\mathcal{C}$ $c$ $X$ $\gamma$ $\beta$ $\mathcal{C}_{68,1}$ $\mathcal{C}_{64,29}$ $1+u$ $\left( uu0u33131u1333130u30uu3130113133\right)$ $3$ $135$ $\mathcal{C}_{68,2}$ $\mathcal{C}_{64,29}$ $1+u$ $\left( 00u013331u1131310u30u0111u331313\right)$ $3$ $139$ $\mathcal{C}_{68,3}$ $\mathcal{C}_{64,29}$ $1+u$ $\left( uu0u33333u131133003u0u113u131331\right)$ $3$ $143$ $\mathcal{C}_{68,4}$ $\mathcal{C}_{64,29}$ $1+u$ $\left( uu0011133u311331uu1u0u3110113111\right)$ $3$ $151$ $\mathcal{C}_{68,5}$ $\mathcal{C}_{64,29}$ $1+u$ $\left( u0uu333310311331uu30u0331u331113\right)$ $3$ $155$ $\mathcal{C}_{68,6}$ $\mathcal{C}_{64,29}$ $1$ $\left( u0u011131u1311110u30u03110113111\right)$ $3$ $161$ $\mathcal{C}_{68,7}$ $\mathcal{C}_{64,38}$ $1+u$ $\left( 33131u0101333uu103310030uu11uu13\right)$ $3$ $186$ $\mathcal{C}_{68,8}$ $\mathcal{C}_{64,38}$ $1$ $\left( 13131uuu0u0u3033u1u130100310u1u0\right)$ $3$ $202$ $\mathcal{C}_{68,9}$ $\mathcal{C}_{64,38}$ $1$ $\left( 133330u301331uu30311003uu0130011\right)$ $3$ $204$
FSD $\left[ 72,36,14\right] _{2}^{b-1}$codes by $C_{mn}$ construction over $\mathbb{F}_{2}+u\mathbb{F}_{2}$
 $n$ $m$ $r_{1},\ldots ,r_{n}$ $A_{14}$ $A_{16}$ $A_{18}$ $2$ $9$ $13u10u000,03u100011$ $8820$ $123039$ $1210564$ $2$ $9$ $30u0031uu,10u1u3u3u$ $8856$ $122850$ $1210492$ $2$ $9$ $u00u31uu1,01uu3u013$ $8784$ $123417$ $1207344$ $2$ $9$ $3031uu10u,300333u30$ $8928$ $122436$ $1210776$ $2$ $9$ $uu0003103,300u1303u$ $9288$ $120690$ $1208328$ $2$ $9$ $1333u1313,11u3u31uu$ $9360$ $119583$ $1216936$ $3$ $6$ $11010u,30u1u0,103u11$ $8820$ $123327$ $1207092$ $3$ $6$ $u30u33,333130,0301u3$ $9180$ $121194$ $1209304$ $3$ $6$ $33u101,0u0311,13u1u3$ $9504$ $119151$ $1212760$ $3$ $6$ $3uuu0u,u31u03,10uu13$ $9648$ $118170$ $1215172$
 $n$ $m$ $r_{1},\ldots ,r_{n}$ $A_{14}$ $A_{16}$ $A_{18}$ $2$ $9$ $13u10u000,03u100011$ $8820$ $123039$ $1210564$ $2$ $9$ $30u0031uu,10u1u3u3u$ $8856$ $122850$ $1210492$ $2$ $9$ $u00u31uu1,01uu3u013$ $8784$ $123417$ $1207344$ $2$ $9$ $3031uu10u,300333u30$ $8928$ $122436$ $1210776$ $2$ $9$ $uu0003103,300u1303u$ $9288$ $120690$ $1208328$ $2$ $9$ $1333u1313,11u3u31uu$ $9360$ $119583$ $1216936$ $3$ $6$ $11010u,30u1u0,103u11$ $8820$ $123327$ $1207092$ $3$ $6$ $u30u33,333130,0301u3$ $9180$ $121194$ $1209304$ $3$ $6$ $33u101,0u0311,13u1u3$ $9504$ $119151$ $1212760$ $3$ $6$ $3uuu0u,u31u03,10uu13$ $9648$ $118170$ $1215172$
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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