American Institute of Mathematical Sciences

Advanced
doi: 10.3934/amc.2020096

Additive and linear conjucyclic codes over $ {\mathbb{F}}_4 $

Taher Abualrub 1, and  Steven T. Dougherty 2,,

1. 

Department of Mathematics and Statistics, American University of Sharjah, Sharjah, UAE
2. 

University of Scranton, Scranton, PA 18510, USA

* Corresponding author: Steven T. Dougherty

Received  November 2019 Revised  February 2020 Published  July 2020

Conjucyclic codes were first introduced by Calderbank, Rains, Shor and Sloane in [1] because of their applications in quantum error-correction. In this paper, we study linear and additive conjucyclic codes over the finite field $ {\mathbb{F}}_{4} $ and produce a duality for which the orthogonal, with respect to that duality, of conjucyclic codes is conjucyclic. Moreover, we show that this is not the case for other standard dualities. We show that additive conjucyclic codes are the only non-trivial conjucyclic codes over $ {\mathbb{F}}_{4} $ and we use a linear algebraic approach to classify these codes. We will also show that additive conjucyclic codes of length $ n $ over $ {\mathbb{F}}_{4} $ are isomorphic to binary cyclic codes of length $ 2n. $

Keywords: Conjucyclic code, additive code, codes over $ {\mathbb F}_4 $.
Mathematics Subject Classification: Primary:11T71, 94B15.
Citation: Taher Abualrub, Steven T. Dougherty. Additive and linear conjucyclic codes over $ {\mathbb{F}}_4 $. Advances in Mathematics of Communications, doi: 10.3934/amc.2020096
References:
[1]

A. R. Calderbank, E. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over ${\mathbb{F}}_{4}$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315.  Google Scholar

[2]

S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.  Google Scholar

[3]

S. T. Dougherty, J.-L. Kim and N. Lee, Additive self-dual codes over fields of even order, Bull. Korean Math. Soc., 55 (2018), 341-357.  doi: 10.4134/BKMS.b160842.  Google Scholar

[4]

S. T. Dougherty and S. Meyers, Orthogonality from Group Characters, work in progress. Google Scholar

[5]

T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[6]

J.-L. Kim and N. Lee, Secret sharing schemes based on additive codes over ${\mathbb{F}}_{4}$, Appl. Algebra Engrg. Comm. Comput., 28 (2017), 79-97.  doi: 10.1007/s00200-016-0296-5.  Google Scholar

[7]

D. Radkova and A. J. Van Zanten, Constacyclic codes as invariant subspaces, Linear Algebra Appl., 430 (2009), 855-864.  doi: 10.1016/j.laa.2008.09.036.  Google Scholar

show all references

References:
[1]

A. R. Calderbank, E. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over ${\mathbb{F}}_{4}$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315.  Google Scholar

[2]

S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.  Google Scholar

[3]

S. T. Dougherty, J.-L. Kim and N. Lee, Additive self-dual codes over fields of even order, Bull. Korean Math. Soc., 55 (2018), 341-357.  doi: 10.4134/BKMS.b160842.  Google Scholar

[4]

S. T. Dougherty and S. Meyers, Orthogonality from Group Characters, work in progress. Google Scholar

[5]

T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[6]

J.-L. Kim and N. Lee, Secret sharing schemes based on additive codes over ${\mathbb{F}}_{4}$, Appl. Algebra Engrg. Comm. Comput., 28 (2017), 79-97.  doi: 10.1007/s00200-016-0296-5.  Google Scholar

[7]

D. Radkova and A. J. Van Zanten, Constacyclic codes as invariant subspaces, Linear Algebra Appl., 430 (2009), 855-864.  doi: 10.1016/j.laa.2008.09.036.  Google Scholar

Table 1.  Conjucyclic codes of length 3
The code $ C $ Basis
$ V_{1}=0 $
$ V_{M}=\mathbb{F}_{4}^{3} $ $ \left\{ \left( 1,0,0\right) ,\left( w,0,0\right) ,\left( 0,1,0\right) ,\left( 0,w,0\right) ,\left( 0,0,1\right) ,\left( 0,0,w\right) \right\} $
$ V_{f} $ $ \left\{ \left( 1,1,1\right) \right\} $
$ V_{g} $ $ \left\{ \left( 1,1,0\right) ,\left( 1,0,1\right) \right\} $
$ V_{f^{2}} $ $ \left\{ \left( 1,1,1\right) ,\left( w,1+w,w\right) \right\} $
$ V_{g^{2}} $ $ \left\{ \left( 1,1,0\right) ,\left( 1+\omega ,\omega ,0\right) ,\left( 1,0,1\right) ,\left( \omega ,0,\omega \right) \right\} $
$ V_{fg} $ $ \left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) \right\} $
$ V_{f^{2}g} $ $ \left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) ,\left( w,w,w\right) \right\} $
$ V_{fg^{2}} $ $ \left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) ,\left( w,w,0\right) ,\left( w,0,w\right) \right\} $
Table Options

Download as excel

The code $ C $ Basis
$ V_{1}=0 $
$ V_{M}=\mathbb{F}_{4}^{3} $ $ \left\{ \left( 1,0,0\right) ,\left( w,0,0\right) ,\left( 0,1,0\right) ,\left( 0,w,0\right) ,\left( 0,0,1\right) ,\left( 0,0,w\right) \right\} $
$ V_{f} $ $ \left\{ \left( 1,1,1\right) \right\} $
$ V_{g} $ $ \left\{ \left( 1,1,0\right) ,\left( 1,0,1\right) \right\} $
$ V_{f^{2}} $ $ \left\{ \left( 1,1,1\right) ,\left( w,1+w,w\right) \right\} $
$ V_{g^{2}} $ $ \left\{ \left( 1,1,0\right) ,\left( 1+\omega ,\omega ,0\right) ,\left( 1,0,1\right) ,\left( \omega ,0,\omega \right) \right\} $
$ V_{fg} $ $ \left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) \right\} $
$ V_{f^{2}g} $ $ \left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) ,\left( w,w,w\right) \right\} $
$ V_{fg^{2}} $ $ \left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) ,\left( w,w,0\right) ,\left( w,0,w\right) \right\} $
[1]

Murat Şahİn, Hayrullah Özİmamoğlu. Additive Toeplitz codes over $ \mathbb{F}_{4} $. Advances in Mathematics of Communications, 2020, 14 (2) : 379-395. doi: 10.3934/amc.2020026
[2]

Habibul Islam, Om Prakash, Patrick Solé. $ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020094
[3]

Tingting Wu, Jian Gao, Yun Gao, Fang-Wei Fu. $ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 641-657. doi: 10.3934/amc.2018038
[4]

Raj Kumar, Maheshanand Bhaintwal. Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020135
[5]

Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020123
[6]

Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao, Songsak Sriboonchitta. Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. Advances in Mathematics of Communications, 2018, 12 (2) : 231-262. doi: 10.3934/amc.2018016
[7]

Yuan Cao, Yonglin Cao, Hai Q. Dinh, Ramakrishna Bandi, Fang-Wei Fu. An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021, 15 (2) : 291-309. doi: 10.3934/amc.2020067
[8]

Lingyu Diao, Jian Gao, Jiyong Lu. Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Advances in Mathematics of Communications, 2020, 14 (4) : 555-572. doi: 10.3934/amc.2020029
[9]

Joaquim Borges, Cristina Fernández-Córdoba, Roger Ten-Valls. On ${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 169-179. doi: 10.3934/amc.2018011
[10]

Karim Samei, Arezoo Soufi. Quadratic residue codes over $\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$. Advances in Mathematics of Communications, 2017, 11 (4) : 791-804. doi: 10.3934/amc.2017058
[11]

W. Cary Huffman. Additive cyclic codes over $\mathbb F_4$. Advances in Mathematics of Communications, 2008, 2 (3) : 309-343. doi: 10.3934/amc.2008.2.309
[12]

W. Cary Huffman. Additive cyclic codes over $\mathbb F_4$. Advances in Mathematics of Communications, 2007, 1 (4) : 427-459. doi: 10.3934/amc.2007.1.427
[13]

Junchao Zhou, Yunge Xu, Lisha Wang, Nian Li. Nearly optimal codebooks from generalized Boolean bent functions over $ \mathbb{Z}_{4} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020121
[14]

Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027
[15]

Daniel Heinlein, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. A subspace code of size $ \bf{333} $ in the setting of a binary $ \bf{q} $-analog of the Fano plane. Advances in Mathematics of Communications, 2019, 13 (3) : 457-475. doi: 10.3934/amc.2019029
[16]

Minjia Shi, Yaqi Lu. Cyclic DNA codes over $ \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$. Advances in Mathematics of Communications, 2019, 13 (1) : 157-164. doi: 10.3934/amc.2019009
[17]

Jiao Du, Longjiang Qu, Chao Li, Xin Liao. Constructing 1-resilient rotation symmetric functions over $ {\mathbb F}_{p} $ with $ {q} $ variables through special orthogonal arrays. Advances in Mathematics of Communications, 2020, 14 (2) : 247-263. doi: 10.3934/amc.2020018
[18]

Fanghui Ma, Jian Gao, Fang-Wei Fu. New non-binary quantum codes from constacyclic codes over $ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $. Advances in Mathematics of Communications, 2019, 13 (3) : 421-434. doi: 10.3934/amc.2019027
[19]

W. Cary Huffman. Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order. Advances in Mathematics of Communications, 2007, 1 (3) : 357-398. doi: 10.3934/amc.2007.1.357
[20]

Evangeline P. Bautista, Philippe Gaborit, Jon-Lark Kim, Judy L. Walker. s-extremal additive $\mathbb F_4$ codes. Advances in Mathematics of Communications, 2007, 1 (1) : 111-130. doi: 10.3934/amc.2007.1.111

2019 Impact Factor: 0.734

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences