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Cyclic codes of length $ 2p^n $ over finite chain rings
1. | Departamento de Matemática, Universidade Federal de Viçosa, Viçosa, 36570-000, Brazil |
2. | Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 05311-970, Brazil |
We use group algebra methods to study cyclic codes over finite chain rings and under some restrictive hypotheses, described in section 2, for codes of length $ 2p^n $, $ p $ a prime, we are able to compute the minimum weights of all possible cyclic codes of that length.
References:
[1] |
S. K. Arora and M. Pruthi,
Minimal cyclic codes of length $2p^n$, Finite Fields Appl., 5 (1999), 177-187.
doi: 10.1006/ffta.1998.0238. |
[2] |
Y. L. Cao,
On constacyclic codes over finite chain rings, Finite Fields Appl., 24 (2013), 124-135.
doi: 10.1016/j.ffa.2013.07.001. |
[3] |
H. Q. Dinh and S. R. López-Permouth,
Cyclic and negacyclic codes over finite chain rings, IEEE Transactions on Information Theory, 50 (2004), 1728-1744.
doi: 10.1109/TIT.2004.831789. |
[4] |
S. T. Dougherty, J.-L. Kim and H. W. Liu,
Construction of self-dual codes over finite commutative chain rings, Int. Journal on Information and Coding Theory, 1 (2010), 171-190.
doi: 10.1504/IJICOT.2010.032133. |
[5] |
R. A. Ferraz and C. Polcino Milies,
Idempotents in group algebras and minimal abelian codes, Finite Fields and Their Appl., 13 (2007), 382-393.
doi: 10.1016/j.ffa.2005.09.007. |
[6] |
N. Jacobson, Basic Algebra. II, W. H. Freeman and Company, San Francisco, Calif., 1980. |
[7] |
Z. H. Liu,
Notes on linear codes over finite chain rings, Acta Mathematicae Applicatae Sinica, 27 (2011), 141-148.
doi: 10.1007/s10255-011-0047-0. |
[8] |
E. Martinez-Moro and I. F. Rúa,
On repeated-root multivariable codes over a finite chain ring, Designs, Codes Cryptography, 45 (2007), 219-227.
doi: 10.1007/s10623-007-9114-1. |
[9] |
G. H. Norton and A. Sălăgean-Mandache,
On the structure of linear cyclic codes over finite chain rings, Appl. Algebra Eng. Commun. Comput., 10 (2000), 489-506.
doi: 10.1007/PL00012382. |
[10] |
C. Polcino Milies and S. K. Sehgal, An Introduction to Group Rings, Algebra and Applications, 1. Kluwer Academic Publishers, Dordrecht, 2002.
doi: 10.1007/978-94-010-0405-3. |
[11] |
P. Solé and V. Sison,
Bounds on the minimum homogeneous dis-tance of the $p^r$-ary image of linear block codes over the galois ring $GR(p^r, m)$, IEEE Trans. Information Theory, 53 (2007), 2270-2273.
doi: 10.1109/TIT.2007.896891. |
show all references
References:
[1] |
S. K. Arora and M. Pruthi,
Minimal cyclic codes of length $2p^n$, Finite Fields Appl., 5 (1999), 177-187.
doi: 10.1006/ffta.1998.0238. |
[2] |
Y. L. Cao,
On constacyclic codes over finite chain rings, Finite Fields Appl., 24 (2013), 124-135.
doi: 10.1016/j.ffa.2013.07.001. |
[3] |
H. Q. Dinh and S. R. López-Permouth,
Cyclic and negacyclic codes over finite chain rings, IEEE Transactions on Information Theory, 50 (2004), 1728-1744.
doi: 10.1109/TIT.2004.831789. |
[4] |
S. T. Dougherty, J.-L. Kim and H. W. Liu,
Construction of self-dual codes over finite commutative chain rings, Int. Journal on Information and Coding Theory, 1 (2010), 171-190.
doi: 10.1504/IJICOT.2010.032133. |
[5] |
R. A. Ferraz and C. Polcino Milies,
Idempotents in group algebras and minimal abelian codes, Finite Fields and Their Appl., 13 (2007), 382-393.
doi: 10.1016/j.ffa.2005.09.007. |
[6] |
N. Jacobson, Basic Algebra. II, W. H. Freeman and Company, San Francisco, Calif., 1980. |
[7] |
Z. H. Liu,
Notes on linear codes over finite chain rings, Acta Mathematicae Applicatae Sinica, 27 (2011), 141-148.
doi: 10.1007/s10255-011-0047-0. |
[8] |
E. Martinez-Moro and I. F. Rúa,
On repeated-root multivariable codes over a finite chain ring, Designs, Codes Cryptography, 45 (2007), 219-227.
doi: 10.1007/s10623-007-9114-1. |
[9] |
G. H. Norton and A. Sălăgean-Mandache,
On the structure of linear cyclic codes over finite chain rings, Appl. Algebra Eng. Commun. Comput., 10 (2000), 489-506.
doi: 10.1007/PL00012382. |
[10] |
C. Polcino Milies and S. K. Sehgal, An Introduction to Group Rings, Algebra and Applications, 1. Kluwer Academic Publishers, Dordrecht, 2002.
doi: 10.1007/978-94-010-0405-3. |
[11] |
P. Solé and V. Sison,
Bounds on the minimum homogeneous dis-tance of the $p^r$-ary image of linear block codes over the galois ring $GR(p^r, m)$, IEEE Trans. Information Theory, 53 (2007), 2270-2273.
doi: 10.1109/TIT.2007.896891. |
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