
-
Previous Article
Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields
- AMC Home
- This Issue
-
Next Article
Self-duality of generalized twisted Gabidulin codes
A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India |
In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over $R = \mathbb{Z}_4+u\mathbb{Z}_4;u^2 = 1$, with an automorphism $θ$ and a derivation $δ_θ$. We generalize the notion of cyclic codes to skew-cyclic codes with derivation, and call such codes as $δ_θ$-cyclic codes. Some properties of skew polynomial ring $R[x, θ, {δ_θ}]$ are presented. A $δ_θ$-cyclic code is proved to be a left $R[x, θ, {δ_θ}]$-submodule of $\frac{R[x, θ, {δ_θ}]}{\langle x^n-1 \rangle}$. The form of a parity-check matrix of a free $δ_θ$-cyclic codes of even length $n$ is presented. These codes are further generalized to double $δ_θ$-cyclic codes over $R$. We have obtained some new good codes over $\mathbb{Z}_4$ via Gray images and residue codes of these codes. The new codes obtained have been reported and added to the database of $\mathbb{Z}_4$-codes [
References:
[1] |
M. Araya, M. Harada, H. Ito and K. Saito,
On the classification of Z4-codes, Adv. Math. Commun., 11 (2017), 747-756.
doi: 10.3934/amc.2017054. |
[2] |
N. Aydin and T. Asamov, http://www.z4codes.info The database of Z4 codes (Accessed March, 2018). |
[3] |
N. Aydin and T. Asamov,
A database of Z4 codes, J. Comb. Inf. Syst. Sci., 34 (2009), 1-12.
|
[4] |
M. Bhaintwal,
Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.
doi: 10.1007/s10623-011-9494-0. |
[5] |
I. F. Blake,
Codes over certain rings, Information and Control., 20 (1972), 396-404.
doi: 10.1016/S0019-9958(72)90223-9. |
[6] |
I. F. Blake,
Codes over integer residue rings, Information and Control., 29 (1975), 295-300.
doi: 10.1016/S0019-9958(75)80001-5. |
[7] |
W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of magma functions, Edition, 2 (2010), 5017 pages. |
[8] |
D. Boucher and F. Ulmer,
Coding with skew polynomial rings, J. of Symbolic Comput., 44 (2009), 1644-1656.
doi: 10.1016/j.jsc.2007.11.008. |
[9] |
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. |
[10] |
D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, In Proc. of 12th IMA International Conference, Cryptography and Coding, Cirencester, UK, LNCS, 5921 (2009), 38-55.
doi: 10.1007/978-3-642-10868-6_3. |
[11] |
D. Boucher, P. Sol$\acute{e}$ and F. Ulmer,
Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.
doi: 10.3934/amc.2008.2.273. |
[12] |
D. Boucher and F. Ulmer,
Linear codes using skew polynomials with automorphisms and derivations, Des. Codes Cryptogr., 70 (2014), 405-431.
doi: 10.1007/s10623-012-9704-4. |
[13] |
S. T. Dougherty and K. Shiromoto,
Maximum distance codes over rings of order 4, IEEE Trans. Info Theory, 47 (2001), 400-404.
doi: 10.1109/18.904544. |
[14] |
F. Gursoy, I. Siap and B. Yildiz,
Construction of skew cyclic codes over $\mathbb{F}_q+v\mathbb{F}_q$, Adv. Math. Commum., 8 (2014), 313-322.
doi: 10.3934/amc.2014.8.313. |
[15] |
Jr. A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. Sloane and P. Sol$\acute{e}$,
The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[16] |
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. |
[17] |
B. R. McDonald,
Finite Rings with Identity, Marcel Dekker Inc, New York, 1974. |
[18] |
M. Ozen, F. Z. Uzekmek, N. Aydin and N. T. Ozzaim,
Cyclic and some constacyclic codes over the ring $\frac{Z_4[u]}{\langle u^2-1\rangle}$, Finite Fields Appl., 38 (2016), 27-39.
doi: 10.1016/j.ffa.2015.12.003. |
[19] |
E. Prange,
Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN, (1957), 57-103.
|
[20] |
M. Shi, L. Qian, L. Sok, N. Aydin and P. Sole,
On constacyclic codes over $\frac{Z_4[u]}{\langle u^2-1 \rangle}$ and their Gray images, Finite Fields Appl., 45 (2017), 86-95.
doi: 10.1016/j.ffa.2016.11.016. |
[21] |
I. Siap, T. Abualrub, N. Aydin and P. Seneviratne,
Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory, 2 (2011), 10-20.
doi: 10.1504/IJICOT.2011.044674. |
[22] |
E. Spiegel,
Codes over $\mathbb{Z}_m$, Information and Control., 35 (1977), 48-51.
doi: 10.1016/S0019-9958(77)90526-5. |
[23] |
E. Spiegel,
Codes over $\mathbb{Z}_m$ (revisited), Information and Control., 37 (1978), 100-104.
doi: 10.1016/S0019-9958(78)90461-8. |
[24] |
B. Yildiz and N. Aydin,
On codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$ and their $\mathbb{Z}_4$-images, Int. J. Inf. Coding Theory, 2 (2014), 226-237.
doi: 10.1504/IJICOT.2014.066107. |
[25] |
B. Yildiz and S. Karadeniz,
Linear codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$: MacWilliams identities, projections, and formally self dual codes, Finite Fields Appl., 27 (2014), 24-40.
doi: 10.1016/j.ffa.2013.12.007. |
show all references
References:
[1] |
M. Araya, M. Harada, H. Ito and K. Saito,
On the classification of Z4-codes, Adv. Math. Commun., 11 (2017), 747-756.
doi: 10.3934/amc.2017054. |
[2] |
N. Aydin and T. Asamov, http://www.z4codes.info The database of Z4 codes (Accessed March, 2018). |
[3] |
N. Aydin and T. Asamov,
A database of Z4 codes, J. Comb. Inf. Syst. Sci., 34 (2009), 1-12.
|
[4] |
M. Bhaintwal,
Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.
doi: 10.1007/s10623-011-9494-0. |
[5] |
I. F. Blake,
Codes over certain rings, Information and Control., 20 (1972), 396-404.
doi: 10.1016/S0019-9958(72)90223-9. |
[6] |
I. F. Blake,
Codes over integer residue rings, Information and Control., 29 (1975), 295-300.
doi: 10.1016/S0019-9958(75)80001-5. |
[7] |
W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of magma functions, Edition, 2 (2010), 5017 pages. |
[8] |
D. Boucher and F. Ulmer,
Coding with skew polynomial rings, J. of Symbolic Comput., 44 (2009), 1644-1656.
doi: 10.1016/j.jsc.2007.11.008. |
[9] |
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. |
[10] |
D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, In Proc. of 12th IMA International Conference, Cryptography and Coding, Cirencester, UK, LNCS, 5921 (2009), 38-55.
doi: 10.1007/978-3-642-10868-6_3. |
[11] |
D. Boucher, P. Sol$\acute{e}$ and F. Ulmer,
Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.
doi: 10.3934/amc.2008.2.273. |
[12] |
D. Boucher and F. Ulmer,
Linear codes using skew polynomials with automorphisms and derivations, Des. Codes Cryptogr., 70 (2014), 405-431.
doi: 10.1007/s10623-012-9704-4. |
[13] |
S. T. Dougherty and K. Shiromoto,
Maximum distance codes over rings of order 4, IEEE Trans. Info Theory, 47 (2001), 400-404.
doi: 10.1109/18.904544. |
[14] |
F. Gursoy, I. Siap and B. Yildiz,
Construction of skew cyclic codes over $\mathbb{F}_q+v\mathbb{F}_q$, Adv. Math. Commum., 8 (2014), 313-322.
doi: 10.3934/amc.2014.8.313. |
[15] |
Jr. A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. Sloane and P. Sol$\acute{e}$,
The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[16] |
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. |
[17] |
B. R. McDonald,
Finite Rings with Identity, Marcel Dekker Inc, New York, 1974. |
[18] |
M. Ozen, F. Z. Uzekmek, N. Aydin and N. T. Ozzaim,
Cyclic and some constacyclic codes over the ring $\frac{Z_4[u]}{\langle u^2-1\rangle}$, Finite Fields Appl., 38 (2016), 27-39.
doi: 10.1016/j.ffa.2015.12.003. |
[19] |
E. Prange,
Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN, (1957), 57-103.
|
[20] |
M. Shi, L. Qian, L. Sok, N. Aydin and P. Sole,
On constacyclic codes over $\frac{Z_4[u]}{\langle u^2-1 \rangle}$ and their Gray images, Finite Fields Appl., 45 (2017), 86-95.
doi: 10.1016/j.ffa.2016.11.016. |
[21] |
I. Siap, T. Abualrub, N. Aydin and P. Seneviratne,
Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory, 2 (2011), 10-20.
doi: 10.1504/IJICOT.2011.044674. |
[22] |
E. Spiegel,
Codes over $\mathbb{Z}_m$, Information and Control., 35 (1977), 48-51.
doi: 10.1016/S0019-9958(77)90526-5. |
[23] |
E. Spiegel,
Codes over $\mathbb{Z}_m$ (revisited), Information and Control., 37 (1978), 100-104.
doi: 10.1016/S0019-9958(78)90461-8. |
[24] |
B. Yildiz and N. Aydin,
On codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$ and their $\mathbb{Z}_4$-images, Int. J. Inf. Coding Theory, 2 (2014), 226-237.
doi: 10.1504/IJICOT.2014.066107. |
[25] |
B. Yildiz and S. Karadeniz,
Linear codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$: MacWilliams identities, projections, and formally self dual codes, Finite Fields Appl., 27 (2014), 24-40.
doi: 10.1016/j.ffa.2013.12.007. |
Set of generators | Code | |||
|
||||
|
Set of generators | Code | |||
|
||||
|
Set of generators | Name | |||
Set of generators | Name | |||
[1] |
Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41 |
[2] |
Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018 |
[3] |
Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79 |
[4] |
Yanyan Gao, Qin Yue, Xinmei Huang, Yun Yang. Two classes of cyclic extended double-error-correcting Goppa codes. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022003 |
[5] |
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 |
[6] |
Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004 |
[7] |
Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177 |
[8] |
Fatmanur Gursoy, Irfan Siap, Bahattin Yildiz. Construction of skew cyclic codes over $\mathbb F_q+v\mathbb F_q$. Advances in Mathematics of Communications, 2014, 8 (3) : 313-322. doi: 10.3934/amc.2014.8.313 |
[9] |
Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55 |
[10] |
Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025 |
[11] |
Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017 |
[12] |
Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83 |
[13] |
San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038 |
[14] |
Yan Liu, Xiwang Cao, Wei Lu. Two classes of new optimal ternary cyclic codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021033 |
[15] |
Heide Gluesing-Luerssen, Hunter Lehmann. Automorphism groups and isometries for cyclic orbit codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021040 |
[16] |
Enhui Lim, Frédérique Oggier. On the generalised rank weights of quasi-cyclic codes. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022010 |
[17] |
Gustavo Terra Bastos, Reginaldo Palazzo Júnior, Marinês Guerreiro. Abelian non-cyclic orbit codes and multishot subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 631-650. doi: 10.3934/amc.2020035 |
[18] |
Yunwen Liu, Longjiang Qu, Chao Li. New constructions of systematic authentication codes from three classes of cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 1-16. doi: 10.3934/amc.2018001 |
[19] |
Gerardo Vega, Jesús E. Cuén-Ramos. The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes. Advances in Mathematics of Communications, 2020, 14 (3) : 525-533. doi: 10.3934/amc.2020059 |
[20] |
Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571 |
2021 Impact Factor: 1.015
Tools
Metrics
Other articles
by authors
[Back to Top]