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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 |
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] $.
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. |
[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. |
[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. |
[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. |
[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. |
[7] |
P. J. Davis, Circulant Matrices, A Wiley-Interscience Publication. Pure and Applied Mathematics. John Wiley & Sons, New York-Chichester-Brisbane, 1979. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[15] |
T. Hurley,
Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335.
|
[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. |
[17] |
A. Kaya and B. Yildiz,
Constructing formally self-dual codes from block $\lambda$-circulant matrices, Math. Commun., 24 (2019), 91-105.
|
[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. |
show all references
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. |
[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. |
[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. |
[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. |
[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. |
[7] |
P. J. Davis, Circulant Matrices, A Wiley-Interscience Publication. Pure and Applied Mathematics. John Wiley & Sons, New York-Chichester-Brisbane, 1979. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[15] |
T. Hurley,
Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335.
|
[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. |
[17] |
A. Kaya and B. Yildiz,
Constructing formally self-dual codes from block $\lambda$-circulant matrices, Math. Commun., 24 (2019), 91-105.
|
[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. |
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