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$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes
On the dual codes of skew constacyclic codes
1. | Universidad de Concepción, Escuela de Educación, Departamento de Ciencias Básicas, Los Ángeles, Chile |
2. | Universidad de Concepción, Facultad de Ciencias Físicas y Matemáticas, Departamento de Matemática, Concepción, Chile |
Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $\theta : \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with skew constacyclic codes, that is, linear codes of $\mathbb{F}_q^n$ which are invariant under the action of a semi-linear map $ \phi _{\alpha,\theta }:\mathbb{F}_q^n\to\mathbb{F}_q^n$, defined by $ \phi _{\alpha,\theta }(a_0,...,a_{n-2}, a_{n-1}): = (\alpha \theta (a_{n-1}),\theta (a_0),...,\theta (a_{n-2}))$ for some $\alpha \in \mathbb{F}_q\setminus\{0\}$ and $n≥2$. In particular, we study some algebraic and geometric properties of their dual codes and we give some consequences and research results on $1$-generator skew quasi-twisted codes and on MDS skew constacyclic codes.
References:
[1] |
T. Abualrub, A. Ghrayeb, N. Aydin and I. Siap,
On the construction of skew quasi-cyclic codes, IEEE Trans. Inform. Theory, 56 (2010), 2081-2090.
doi: 10.1109/TIT.2010.2044062. |
[2] |
M. Bhaintwal,
Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.
doi: 10.1007/s10623-011-9494-0. |
[3] |
A. Blokhuis, A. A. Bruen and J. A. Thas,
Arcs in $ PG(n,q)$, MDS-codes and three fundamental problems of B. Segre - some extensions, Geom. Dedic., 35 (1990), 1-11.
doi: 10.1007/BF00147336. |
[4] |
A. Blokhuis, A. A. Bruen and J. A. Thas,
On MDS-codes, arcs in $ PG(n,q)$ with $ q$ even, and a solution of three fundamental problems of B. Segre, Invent. Math., 92 (1988), 441-459.
doi: 10.1007/BF01393742. |
[5] |
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. |
[6] |
D. Boucher, W. Geiselmann and F. Ulmer,
Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 379-389.
doi: 10.1007/s00200-007-0043-z. |
[7] |
D. Boucher, P. Solé and F. Ulmer,
Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.
doi: 10.3934/amc.2008.2.273. |
[8] |
D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, Cryptography and coding, Lecture Notes in Comput. Sci., 5921, Springer, Berlin, 2009, 38–55.
doi: 10.1007/978-3-642-10868-6_3. |
[9] |
D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, Cryptography and Coding, Lecture Notes in Comput. Sci., 7089, Springer, Heidelberg, 2011,230–243.
doi: 10.1007/978-3-642-25516-8_14. |
[10] |
D. Boucher and F. Ulmer,
Self-dual skew codes and factorization of skew polynomials, J. Symbolic Comput., 60 (2014), 47-61.
doi: 10.1016/j.jsc.2013.10.003. |
[11] |
M'Hammed Boulagouaz and A. Leroy,
(σ, δ)-codes, Adv. Math. Commun., 7 (2013), 463-474.
doi: 10.3934/amc.2013.7.463. |
[12] |
A. Cherchem and A. Leroy,
Exponents of skew polynomials, Finite Fields and Their Appl., 37 (2016), 1-13.
doi: 10.1016/j.ffa.2015.08.004. |
[13] |
N. Fogarty and H. Gluesing-Luerssen,
A circulant approach to skew-constacyclic codes, Finite Fields Appl., 35 (2015), 92-114.
doi: 10.1016/j.ffa.2015.03.008. |
[14] |
J. W. P. Hirschfeld,
Projective Geometries Over Finite Fields, Claredon Press - Oxford, 1979. |
[15] |
S. Jitman, S. Ling and P. Udomkavanich,
Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 39-63.
doi: 10.3934/amc.2012.6.39. |
[16] |
H. Kaneta and T. Maruta,
An elementary proof and extension of Thas' theorem on k-arcs, Mat. Proc. Camb. Philos. Soc., 105 (1989), 459-462.
doi: 10.1017/S0305004100077823. |
[17] |
A. Leroy, Noncommutative polynomial maps,
J. Algebra Appl., 11 (2012), 1250076, 16 pp.
doi: 10.1142/S0219498812500764. |
[18] |
T. Maruta, A geometric approach to semi-cyclic codes, Advances in Finite Geometries and Designs (Chelwood Gate, 1990), Oxford Sci. Publ., Oxford Univ. Press, New York, (1991), 311–318. |
[19] |
T. Maruta,
On the existence of cyclic and pseudo-cyclic MDS codes, Europ. J. Combinatorics, 19 (1998), 159-174.
doi: 10.1006/S0195-6698(97)90000-7. |
[20] |
T. Maruta, M. Shinohara and M. Takenaka,
Constructing linear codes from some orbits of projectivities, Discrete Math., 308 (2008), 832-841.
doi: 10.1016/j.disc.2007.07.045. |
[21] |
L. Storme and J. A. Thas,
M.D.S. codes and arcs in $ PG(n,q)$ with $ q$ even: An improvement of the bounds of Bruen, Thas, and Blokhuis, J. Comb. Theory, Ser. A, 62 (1993), 139-154.
doi: 10.1016/0097-3165(93)90076-K. |
[22] |
L. F. Tapia Cuitiño and A. L. Tironi,
Dual codes of product semi-linear codes, Linear Algebra Appl., 457 (2014), 114-153.
doi: 10.1016/j.laa.2014.05.011. |
[23] |
L. F. Tapia Cuitiño and A. L. Tironi,
Some properties of skew codes over finite fields, Des. Codes Cryptogr., 85 (2017), 359-380.
doi: 10.1007/s10623-016-0311-7. |
[24] |
J. A. Thas,
Normal rational curves and $ (q+2)$-arcs in a Galois space $ S_{q-2,q}(q=2^h)$, Atti Accad. Naz. Lincei Rend., 47 (1969), 249-252.
|
show all references
References:
[1] |
T. Abualrub, A. Ghrayeb, N. Aydin and I. Siap,
On the construction of skew quasi-cyclic codes, IEEE Trans. Inform. Theory, 56 (2010), 2081-2090.
doi: 10.1109/TIT.2010.2044062. |
[2] |
M. Bhaintwal,
Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.
doi: 10.1007/s10623-011-9494-0. |
[3] |
A. Blokhuis, A. A. Bruen and J. A. Thas,
Arcs in $ PG(n,q)$, MDS-codes and three fundamental problems of B. Segre - some extensions, Geom. Dedic., 35 (1990), 1-11.
doi: 10.1007/BF00147336. |
[4] |
A. Blokhuis, A. A. Bruen and J. A. Thas,
On MDS-codes, arcs in $ PG(n,q)$ with $ q$ even, and a solution of three fundamental problems of B. Segre, Invent. Math., 92 (1988), 441-459.
doi: 10.1007/BF01393742. |
[5] |
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. |
[6] |
D. Boucher, W. Geiselmann and F. Ulmer,
Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 379-389.
doi: 10.1007/s00200-007-0043-z. |
[7] |
D. Boucher, P. Solé and F. Ulmer,
Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.
doi: 10.3934/amc.2008.2.273. |
[8] |
D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, Cryptography and coding, Lecture Notes in Comput. Sci., 5921, Springer, Berlin, 2009, 38–55.
doi: 10.1007/978-3-642-10868-6_3. |
[9] |
D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, Cryptography and Coding, Lecture Notes in Comput. Sci., 7089, Springer, Heidelberg, 2011,230–243.
doi: 10.1007/978-3-642-25516-8_14. |
[10] |
D. Boucher and F. Ulmer,
Self-dual skew codes and factorization of skew polynomials, J. Symbolic Comput., 60 (2014), 47-61.
doi: 10.1016/j.jsc.2013.10.003. |
[11] |
M'Hammed Boulagouaz and A. Leroy,
(σ, δ)-codes, Adv. Math. Commun., 7 (2013), 463-474.
doi: 10.3934/amc.2013.7.463. |
[12] |
A. Cherchem and A. Leroy,
Exponents of skew polynomials, Finite Fields and Their Appl., 37 (2016), 1-13.
doi: 10.1016/j.ffa.2015.08.004. |
[13] |
N. Fogarty and H. Gluesing-Luerssen,
A circulant approach to skew-constacyclic codes, Finite Fields Appl., 35 (2015), 92-114.
doi: 10.1016/j.ffa.2015.03.008. |
[14] |
J. W. P. Hirschfeld,
Projective Geometries Over Finite Fields, Claredon Press - Oxford, 1979. |
[15] |
S. Jitman, S. Ling and P. Udomkavanich,
Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 39-63.
doi: 10.3934/amc.2012.6.39. |
[16] |
H. Kaneta and T. Maruta,
An elementary proof and extension of Thas' theorem on k-arcs, Mat. Proc. Camb. Philos. Soc., 105 (1989), 459-462.
doi: 10.1017/S0305004100077823. |
[17] |
A. Leroy, Noncommutative polynomial maps,
J. Algebra Appl., 11 (2012), 1250076, 16 pp.
doi: 10.1142/S0219498812500764. |
[18] |
T. Maruta, A geometric approach to semi-cyclic codes, Advances in Finite Geometries and Designs (Chelwood Gate, 1990), Oxford Sci. Publ., Oxford Univ. Press, New York, (1991), 311–318. |
[19] |
T. Maruta,
On the existence of cyclic and pseudo-cyclic MDS codes, Europ. J. Combinatorics, 19 (1998), 159-174.
doi: 10.1006/S0195-6698(97)90000-7. |
[20] |
T. Maruta, M. Shinohara and M. Takenaka,
Constructing linear codes from some orbits of projectivities, Discrete Math., 308 (2008), 832-841.
doi: 10.1016/j.disc.2007.07.045. |
[21] |
L. Storme and J. A. Thas,
M.D.S. codes and arcs in $ PG(n,q)$ with $ q$ even: An improvement of the bounds of Bruen, Thas, and Blokhuis, J. Comb. Theory, Ser. A, 62 (1993), 139-154.
doi: 10.1016/0097-3165(93)90076-K. |
[22] |
L. F. Tapia Cuitiño and A. L. Tironi,
Dual codes of product semi-linear codes, Linear Algebra Appl., 457 (2014), 114-153.
doi: 10.1016/j.laa.2014.05.011. |
[23] |
L. F. Tapia Cuitiño and A. L. Tironi,
Some properties of skew codes over finite fields, Des. Codes Cryptogr., 85 (2017), 359-380.
doi: 10.1007/s10623-016-0311-7. |
[24] |
J. A. Thas,
Normal rational curves and $ (q+2)$-arcs in a Galois space $ S_{q-2,q}(q=2^h)$, Atti Accad. Naz. Lincei Rend., 47 (1969), 249-252.
|
Generator Matrix | ||||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
6 | 1 | 2 | ||
6 | 1 | 2 | ||
10 | 1 | 2 | ||
10 | 1 | 2 | ||
8 | 1 | 2 | ||
12 | 1 | 2 | ||
12 | 1 | 2 | ||
30 | 1 | 2 | ||
62 | 1 | 2 | ||
2 | ||||
7 | 2 | |||
4 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 3 | |||
6 | 3 |
Generator Matrix | ||||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 2 | |||
6 | 1 | 2 | ||
6 | 1 | 2 | ||
10 | 1 | 2 | ||
10 | 1 | 2 | ||
8 | 1 | 2 | ||
12 | 1 | 2 | ||
12 | 1 | 2 | ||
30 | 1 | 2 | ||
62 | 1 | 2 | ||
2 | ||||
7 | 2 | |||
4 | 2 | |||
5 | 2 | |||
5 | 2 | |||
5 | 3 | |||
6 | 3 |
Generator Matrix | ||||
6 | 2 | |||
10 | 2 | |||
10 | 2 | |||
12 | 2 | |||
12 | 2 | |||
30 | 2 | |||
30 | 2 | |||
62 | 2 |
Generator Matrix | ||||
6 | 2 | |||
10 | 2 | |||
10 | 2 | |||
12 | 2 | |||
12 | 2 | |||
30 | 2 | |||
30 | 2 | |||
62 | 2 |
Polynomial |
|||||
Polynomial |
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Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273 |
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