-
Previous Article
Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes
- AMC Home
- This Issue
-
Next Article
Network encoding complexity: Exact values, bounds, and inequalities
Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields
1. | Department of Mathematics and Statistics, Faculty of Science, Thaksin University, Phatthalung Campus, Phatthalung 93110, Thailand |
2. | Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand |
3. | Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand |
Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An alternative algorithm for factorizing $x^n-\lambda $ over ${\mathbb{F}_{{q^2}}}$ is given, where $λ$ is a unit in ${\mathbb{F}_{{q^2}}}$. Based on this factorization, the dimensions of the Hermitian hulls of $\lambda $-constacyclic codes of length $n$ over ${\mathbb{F}_{{q^2}}}$ are determined. The characterization and enumeration of constacyclic Hermitian self-dual (resp., complementary dual) codes of length $n$ over ${\mathbb{F}_{{q^2}}}$ are given through their Hermitian hulls. Subsequently, a new family of MDS constacyclic Hermitian self-dual codes over ${\mathbb{F}_{{q^2}}}$ is introduced.
As a generalization of constacyclic codes, quasi-twisted Hermitian self-dual codes are studied. Using the factorization of $x^n-\lambda $ and the Chinese Remainder Theorem, quasi-twisted codes can be viewed as a product of linear codes of shorter length over some extension fields of ${\mathbb{F}_{{q^2}}}$. Necessary and sufficient conditions for quasi-twisted codes to be Hermitian self-dual are given. The enumeration of such self-dual codes is determined as well.
References:
[1] |
N. Aydin, T. Asamov and T. A. Gulliver, Some open problems on quasi-twisted and related
code constructions and good quaternary codes, in Proc. IEEE ISIT' 2007, Nice, France, 2007,
856–860. |
[2] |
N. Aydin, I. Siap and D. J. Ray-Chaudhuri,
The structure of 1-generator quasi-twisted codes and new linear codes, Des. Codes Crypt., 24 (2001), 313-326.
doi: 10.1023/A:1011283523000. |
[3] |
G. K. Bakshi and M. Raka,
A class of constacyclic codes over a finite field, Finite Fields Appl., 18 (2012), 362-377.
doi: 10.1016/j.ffa.2011.09.005. |
[4] |
E. R. Berlekamp,
Algebraic Coding Theory McGraw-Hill, New York, 1968. |
[5] |
B. Chen, H. Q. Dinh, Y. Fan and S. Ling,
Polyadic constacyclic codes, IEEE Trans. Inf.
Theory, 61 (2015), 1217-1231.
doi: 10.1109/TIT.2015.2451656. |
[6] |
B. Chen, Y. Fan, L. Lin and H. Liu,
Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231.
|
[7] |
E. Z. Chen,
An explicit construction of 2-generator quasi-twisted codes, IEEE Trans. Inf. Theory, 54 (2008), 5770-5773.
doi: 10.1109/TIT.2008.2006430. |
[8] |
V. Chepyzhov, A Gilbert-Vashamov bound for quasitwisted codes of rate $\frac1n$, in Proc. Joint
Swedish-Russian Int. Workshop Inf. Theory, Mölle, Sweden, 1993,214–218. |
[9] |
R. Daskalov and P. Hristov,
New quasi-twisted degenerate ternary linear codes, IEEE Trans. Inf. Theory, 49 (2003), 2259-2263.
doi: 10.1109/TIT.2003.815798. |
[10] |
T. A. Gulliver, M. Harada and H. Miyabayashi,
Double circulant and quasi-twisted self-dual codes over $\mathbb F_5$ and $\mathbb F_7$, Adv. Math. Commun., 1 (2007), 223-238.
doi: 10.3934/amc.2007.1.223. |
[11] |
T. A. Gulliver, N. P. Secord and S. A. Mahmoud,
A link between quasi-cyclic codes and convolution codes, IEEE Trans. Inf. Theory, 44 (1998), 431-435.
doi: 10.1109/18.651076. |
[12] |
Y. Jia,
On quasi-twisted codes over finite fields, Finite Fields Appl., 18 (2012), 237-257.
doi: 10.1016/j.ffa.2011.08.001. |
[13] |
Y. Jia, S. Ling and C. Xing,
On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory, 13 (2011), 2243-2251.
doi: 10.1109/TIT.2010.2092415. |
[14] |
X. Kai, X. Zhu and P. Li,
Constacyclic codes and some new quantum MDS codes, IEEE
Trans. Inf. Theory, 60 (2014), 2080-2086.
doi: 10.1109/TIT.2014.2308180. |
[15] |
T. Kasami,
A Gilbert-Varshamov bound for quasi-cyclic codes of rate $1/2$
IEEE Trans. Inf. Theory 20 (1974), 679. |
[16] |
A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli,
Nonbinary stabilizer codes over finite fields, IEEE Trans. Inf. Theory, 52 (2006), 4892-4914.
doi: 10.1109/TIT.2006.883612. |
[17] |
G. G. La Guardia
On optimal constacyclic codes preprint, arXiv: 1311.2505
doi: 10.1016/j.laa.2016.02.014. |
[18] |
L. Lin, H. Liu and B. Chen,
Existence conditions for self-orthogonal negacyclic codes over finite fields, Adv. Math. Commun., 9 (2015), 1-7.
doi: 10.3934/amc.2015.9.1. |
[19] |
S. Ling, H. Niederreiter and P. Solé,
On the algebraic structure of quasi-cyclic codes Ⅳ: Repeated roots, Des. Codes Crypt., 38 (2006), 337-361.
doi: 10.1007/s10623-005-1431-7. |
[20] |
S. Ling and P. Solé,
On the algebraic structure of quasi-cyclic codes Ⅰ: Finite fields, IEEE
Trans. Inf. Theory, 47 (2001), 2751-2760.
doi: 10.1109/18.959257. |
[21] |
S. Ling and P. Solé,
Good self-dual quasi-cyclic codes exist, IEEE Trans. Inf. Theory, 49 (2003), 1052-1053.
doi: 10.1109/TIT.2003.809501. |
[22] |
S. Ling and P. Solé,
On the algebraic structure of quasi-cyclic codes Ⅲ: Generator theory, IEEE Trans. Inf. Theory, 51 (2005), 2692-2700.
doi: 10.1109/TIT.2005.850142. |
[23] |
G. Nebe, E. M. Rains and N. J. A. Sloane,
Self-Dual Codes and Invariant Theory Springer-Verlag, Berlin, 2006. |
[24] |
H. Özadamb and F. Özbudak,
A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$, Adv. Math. Commun., 3 (2009), 265-271.
doi: 10.3934/amc.2009.3.265. |
[25] |
E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, Elsevier,
North-Holland, 1998,177–294. |
[26] |
E. Sangwisut, S. Jitman, S. Ling and P. Udomkavanich,
Hulls of cyclic and negacyclic codes over finite fields, Finite Fields Appl., 33 (2015), 232-257.
doi: 10.1016/j.ffa.2014.12.008. |
[27] |
J. P. Serre,
A Course in Arithmetic Springer-Verlag, New York, 1973. |
[28] |
G. Solomon and H. C. A. van Tilborg,
A connection between block and convolutional codes, SIAM J. Appl. Math., 37 (1979), 358-369.
doi: 10.1137/0137027. |
[29] |
Y. Yang and W. Cai,
On self-dual constacyclic codes over finite fields, Des. Codes Crypt., 74 (2015), 355-364.
doi: 10.1007/s10623-013-9865-9. |
show all references
References:
[1] |
N. Aydin, T. Asamov and T. A. Gulliver, Some open problems on quasi-twisted and related
code constructions and good quaternary codes, in Proc. IEEE ISIT' 2007, Nice, France, 2007,
856–860. |
[2] |
N. Aydin, I. Siap and D. J. Ray-Chaudhuri,
The structure of 1-generator quasi-twisted codes and new linear codes, Des. Codes Crypt., 24 (2001), 313-326.
doi: 10.1023/A:1011283523000. |
[3] |
G. K. Bakshi and M. Raka,
A class of constacyclic codes over a finite field, Finite Fields Appl., 18 (2012), 362-377.
doi: 10.1016/j.ffa.2011.09.005. |
[4] |
E. R. Berlekamp,
Algebraic Coding Theory McGraw-Hill, New York, 1968. |
[5] |
B. Chen, H. Q. Dinh, Y. Fan and S. Ling,
Polyadic constacyclic codes, IEEE Trans. Inf.
Theory, 61 (2015), 1217-1231.
doi: 10.1109/TIT.2015.2451656. |
[6] |
B. Chen, Y. Fan, L. Lin and H. Liu,
Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231.
|
[7] |
E. Z. Chen,
An explicit construction of 2-generator quasi-twisted codes, IEEE Trans. Inf. Theory, 54 (2008), 5770-5773.
doi: 10.1109/TIT.2008.2006430. |
[8] |
V. Chepyzhov, A Gilbert-Vashamov bound for quasitwisted codes of rate $\frac1n$, in Proc. Joint
Swedish-Russian Int. Workshop Inf. Theory, Mölle, Sweden, 1993,214–218. |
[9] |
R. Daskalov and P. Hristov,
New quasi-twisted degenerate ternary linear codes, IEEE Trans. Inf. Theory, 49 (2003), 2259-2263.
doi: 10.1109/TIT.2003.815798. |
[10] |
T. A. Gulliver, M. Harada and H. Miyabayashi,
Double circulant and quasi-twisted self-dual codes over $\mathbb F_5$ and $\mathbb F_7$, Adv. Math. Commun., 1 (2007), 223-238.
doi: 10.3934/amc.2007.1.223. |
[11] |
T. A. Gulliver, N. P. Secord and S. A. Mahmoud,
A link between quasi-cyclic codes and convolution codes, IEEE Trans. Inf. Theory, 44 (1998), 431-435.
doi: 10.1109/18.651076. |
[12] |
Y. Jia,
On quasi-twisted codes over finite fields, Finite Fields Appl., 18 (2012), 237-257.
doi: 10.1016/j.ffa.2011.08.001. |
[13] |
Y. Jia, S. Ling and C. Xing,
On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory, 13 (2011), 2243-2251.
doi: 10.1109/TIT.2010.2092415. |
[14] |
X. Kai, X. Zhu and P. Li,
Constacyclic codes and some new quantum MDS codes, IEEE
Trans. Inf. Theory, 60 (2014), 2080-2086.
doi: 10.1109/TIT.2014.2308180. |
[15] |
T. Kasami,
A Gilbert-Varshamov bound for quasi-cyclic codes of rate $1/2$
IEEE Trans. Inf. Theory 20 (1974), 679. |
[16] |
A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli,
Nonbinary stabilizer codes over finite fields, IEEE Trans. Inf. Theory, 52 (2006), 4892-4914.
doi: 10.1109/TIT.2006.883612. |
[17] |
G. G. La Guardia
On optimal constacyclic codes preprint, arXiv: 1311.2505
doi: 10.1016/j.laa.2016.02.014. |
[18] |
L. Lin, H. Liu and B. Chen,
Existence conditions for self-orthogonal negacyclic codes over finite fields, Adv. Math. Commun., 9 (2015), 1-7.
doi: 10.3934/amc.2015.9.1. |
[19] |
S. Ling, H. Niederreiter and P. Solé,
On the algebraic structure of quasi-cyclic codes Ⅳ: Repeated roots, Des. Codes Crypt., 38 (2006), 337-361.
doi: 10.1007/s10623-005-1431-7. |
[20] |
S. Ling and P. Solé,
On the algebraic structure of quasi-cyclic codes Ⅰ: Finite fields, IEEE
Trans. Inf. Theory, 47 (2001), 2751-2760.
doi: 10.1109/18.959257. |
[21] |
S. Ling and P. Solé,
Good self-dual quasi-cyclic codes exist, IEEE Trans. Inf. Theory, 49 (2003), 1052-1053.
doi: 10.1109/TIT.2003.809501. |
[22] |
S. Ling and P. Solé,
On the algebraic structure of quasi-cyclic codes Ⅲ: Generator theory, IEEE Trans. Inf. Theory, 51 (2005), 2692-2700.
doi: 10.1109/TIT.2005.850142. |
[23] |
G. Nebe, E. M. Rains and N. J. A. Sloane,
Self-Dual Codes and Invariant Theory Springer-Verlag, Berlin, 2006. |
[24] |
H. Özadamb and F. Özbudak,
A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$, Adv. Math. Commun., 3 (2009), 265-271.
doi: 10.3934/amc.2009.3.265. |
[25] |
E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, Elsevier,
North-Holland, 1998,177–294. |
[26] |
E. Sangwisut, S. Jitman, S. Ling and P. Udomkavanich,
Hulls of cyclic and negacyclic codes over finite fields, Finite Fields Appl., 33 (2015), 232-257.
doi: 10.1016/j.ffa.2014.12.008. |
[27] |
J. P. Serre,
A Course in Arithmetic Springer-Verlag, New York, 1973. |
[28] |
G. Solomon and H. C. A. van Tilborg,
A connection between block and convolutional codes, SIAM J. Appl. Math., 37 (1979), 358-369.
doi: 10.1137/0137027. |
[29] |
Y. Yang and W. Cai,
On self-dual constacyclic codes over finite fields, Des. Codes Crypt., 74 (2015), 355-364.
doi: 10.1007/s10623-013-9865-9. |
Parameters | ||||
Parameters | ||||
[1] |
T. Aaron Gulliver, Masaaki Harada, Hiroki Miyabayashi. Double circulant and quasi-twisted self-dual codes over $\mathbb F_5$ and $\mathbb F_7$. Advances in Mathematics of Communications, 2007, 1 (2) : 223-238. doi: 10.3934/amc.2007.1.223 |
[2] |
Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031 |
[3] |
Nuh Aydin, Nicholas Connolly, Markus Grassl. Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes. Advances in Mathematics of Communications, 2017, 11 (1) : 245-258. doi: 10.3934/amc.2017016 |
[4] |
Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039 |
[5] |
Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229 |
[6] |
Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267 |
[7] |
Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047 |
[8] |
Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2020, 14 (2) : 319-332. doi: 10.3934/amc.2020023 |
[9] |
Keita Ishizuka, Ken Saito. Construction for both self-dual codes and LCD codes. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2021070 |
[10] |
Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028 |
[11] |
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 |
[12] |
Masaaki Harada, Akihiro Munemasa. On the covering radii of extremal doubly even self-dual codes. Advances in Mathematics of Communications, 2007, 1 (2) : 251-256. doi: 10.3934/amc.2007.1.251 |
[13] |
Stefka Bouyuklieva, Iliya Bouyukliev. Classification of the extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2010, 4 (3) : 433-439. doi: 10.3934/amc.2010.4.433 |
[14] |
Hyun Jin Kim, Heisook Lee, June Bok Lee, Yoonjin Lee. Construction of self-dual codes with an automorphism of order $p$. Advances in Mathematics of Communications, 2011, 5 (1) : 23-36. doi: 10.3934/amc.2011.5.23 |
[15] |
Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011 |
[16] |
Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393 |
[17] |
Bram van Asch, Frans Martens. A note on the minimum Lee distance of certain self-dual modular codes. Advances in Mathematics of Communications, 2012, 6 (1) : 65-68. doi: 10.3934/amc.2012.6.65 |
[18] |
Masaaki Harada, Katsushi Waki. New extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2009, 3 (4) : 311-316. doi: 10.3934/amc.2009.3.311 |
[19] |
Katherine Morrison. An enumeration of the equivalence classes of self-dual matrix codes. Advances in Mathematics of Communications, 2015, 9 (4) : 415-436. doi: 10.3934/amc.2015.9.415 |
[20] |
Steven T. Dougherty, Joe Gildea, Adrian Korban, Abidin Kaya. Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68. Advances in Mathematics of Communications, 2020, 14 (4) : 677-702. doi: 10.3934/amc.2020037 |
2021 Impact Factor: 1.015
Tools
Metrics
Other articles
by authors
[Back to Top]