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Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68
1. | Department of Mathematics, University of Scranton, Scranton, PA 18510, USA |
2. | Department of Mathematical and Physical Sciences, University of Chester, Thornton Science Park, Pool Ln, Chester CH2 4NU, England |
3. | Department of Mathematics Education, Sampoerna University, 12780, Jakarta, Indonesia |
We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over $ \mathbb{F}_4 $. These codes have binary images with parameters $ [32,16,8] $ or $ [32,16,6] $. These are lifted to codes over $ \mathbb{F}_4+u\mathbb{F}_4 $, to obtain codes with Gray images of extremal self-dual binary codes of length 64. Finally, we use a building-up method over $ \mathbb{F}_2+u\mathbb{F}_2 $ to obtain new extremal binary self-dual codes of length 68. We construct 11 new codes via the building-up method and 2 new codes by considering possible neighbors.
References:
[1] |
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. |
[2] |
J. H. Conway and N. J. A. Solane,
A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.
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S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017.
doi: 10.1007/978-3-319-59806-2. |
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S. T. Dougherty, P. Gaborit, M. Harada and P. Sole,
Type Ⅱ codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45.
doi: 10.1109/18.746770. |
[5] |
S. T. Dougherty, J. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings over Frobenius rings, Cryptogr. Commun., (2019), 1–20.
doi: 10.1007/s12095-019-00380-8. |
[6] |
S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak,
Group rings, $G$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.
doi: 10.1007/s10623-017-0440-7. |
[7] |
S. T. Dougherty, T. A. Gulliver and M. Harada,
Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.
doi: 10.1109/18.641574. |
[8] |
S. T. Dougherty, J.-L. Kim, H. Kulosman and H. Liu,
Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 14-26.
doi: 10.1016/j.ffa.2009.11.004. |
[9] |
S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, A. Tylyshchak and B. Yildiz,
Bordered constructions of self-dual codes from group rings and new extremal binary self-dual code, Finite Fields Appl., 57 (2019), 108-127.
doi: 10.1016/j.ffa.2019.02.004. |
[10] |
S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, Binary generator matrices for extremal binary self-dual codes of length $68$, Available from: http://abidinkaya.wixsite.com/math/adrian1. |
[11] |
P. Gaborit, V. Pless, P. Sole and O. Atkin,
"Type Ⅱ codes over $\mathbb{F}_4$", Finite Fields Appl., 8 (2002), 171-183.
doi: 10.1006/ffta.2001.0333. |
[12] |
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. |
[13] |
T. Hurley,
Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335.
|
[14] |
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. |
[15] |
S. Ling and P. Sole,
"Type Ⅱ codes over $\mathbb{F}_4+u \mathbb{F}_4$", European J. Combin., 22 (2001), 983-997.
doi: 10.1006/eujc.2001.0509. |
[16] |
E. M. Rains,
Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.
doi: 10.1109/18.651000. |
show all references
References:
[1] |
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. |
[2] |
J. H. Conway and N. J. A. Solane,
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. |
[3] |
S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017.
doi: 10.1007/978-3-319-59806-2. |
[4] |
S. T. Dougherty, P. Gaborit, M. Harada and P. Sole,
Type Ⅱ codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45.
doi: 10.1109/18.746770. |
[5] |
S. T. Dougherty, J. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings over Frobenius rings, Cryptogr. Commun., (2019), 1–20.
doi: 10.1007/s12095-019-00380-8. |
[6] |
S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak,
Group rings, $G$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.
doi: 10.1007/s10623-017-0440-7. |
[7] |
S. T. Dougherty, T. A. Gulliver and M. Harada,
Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.
doi: 10.1109/18.641574. |
[8] |
S. T. Dougherty, J.-L. Kim, H. Kulosman and H. Liu,
Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 14-26.
doi: 10.1016/j.ffa.2009.11.004. |
[9] |
S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, A. Tylyshchak and B. Yildiz,
Bordered constructions of self-dual codes from group rings and new extremal binary self-dual code, Finite Fields Appl., 57 (2019), 108-127.
doi: 10.1016/j.ffa.2019.02.004. |
[10] |
S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, Binary generator matrices for extremal binary self-dual codes of length $68$, Available from: http://abidinkaya.wixsite.com/math/adrian1. |
[11] |
P. Gaborit, V. Pless, P. Sole and O. Atkin,
"Type Ⅱ codes over $\mathbb{F}_4$", Finite Fields Appl., 8 (2002), 171-183.
doi: 10.1006/ffta.2001.0333. |
[12] |
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. |
[13] |
T. Hurley,
Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335.
|
[14] |
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. |
[15] |
S. Ling and P. Sole,
"Type Ⅱ codes over $\mathbb{F}_4+u \mathbb{F}_4$", European J. Combin., 22 (2001), 983-997.
doi: 10.1006/eujc.2001.0509. |
[16] |
E. M. Rains,
Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.
doi: 10.1109/18.651000. |
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